diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzopvh" "b/data_all_eng_slimpj/shuffled/split2/finalzzopvh" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzopvh" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec1}\n\nThe discovery of Giant Magnetoteresistance in 1998 by the groups\nof Fert and Gr\\\"unberg led to new reading heads for hard disks\n\\cite{Fert,Grunberg}. Moreover for the first time, a device based\non magnetic phenomena replaced a conventional electronics device\nbased on the movement of the electrons charge and thus opened the\nway to the field of spintronics or magnetoelectronics. The aim is\nto replace conventional electronics with new devices where\nmagnetism plays a central role leading to smaller energy\nconsumption. Several architectures have been proposed\n\\cite{Zutic,Zabel} but only in 2009 Dash and collaborators managed\nto inject spin-polarized current from a metallic electrode into\nSi, which is a key issue in current research in this field.\nshowing that spintronic devices can be incorporated into\nconventional electronics \\cite{Dash}.\n\nIn order to maximize the efficiency of spintronic devices, the\ninjected current should have as high spin-polarization as possible\n\\cite{Zutic,Zabel}. To this respect half-metallic compounds have\nattracted a lot of interest (for a review see reference\n\\cite{Katsnelson}). These alloys are ferromagnets where the\nmajority spin channel is metallic while the minority-spin band\nstructure is that of a semiconductor leading to 100\\%\\\nspin-polarization of the electrons at the Fermi level and thus to\npossibly 100\\%\\ spin-polarized current into a semiconductor when\nhalf metals are employed as the metallic electrode. The term\nhalf-metal was initially used by de Groot et al in the case of the\nNiMnSb Heusler alloy \\cite{Groot}.\n\nAb-initio (also known as first-principles) calculations have been\nwidely used to explain the properties of these alloys and to\npredict new half-metallic compounds. An interesting case is the\ntransition-metal pnictides like CrAs and MnAs. Akinaga and\ncollaborators found in 2000 that when a CrAs thin film is grown on\ntop of a zinc-blende semiconductor like GaAs, the metallic film\nadopts the lattice of the substrate and it crystallizes in a\nmeta-stable half-metallic zinc-blende phase \\cite{Akinaga}\nstructure. Later CrAs was successfully synthesized in the\nzinc-blence structure in the form of multilayers with GaAs\n\\cite{Mizuguchi} and other successful experiments include the\ngrowth of zinc-blende MnAs in the form of dots \\cite{Ono} and CrSb\nin the form of films \\cite{Zhao,Li}.\n\nExperiments agree with predictions of ab-initio calculations\nperformed by several groups\n\\cite{MavropoulosZB,GalaZB,calculations,Shirai}. In the case of\nthe half-metallic ferromagnets like CrAs or CrSe, the gap in the\nminority-spin band arises from the hybridization between the\n$p$-states of the $sp$ atom and the triple-degenerated $t_{2g}$\nstates of the transition-metal and as a result the total\nspin-moment, $M_t$, follows the Slater-Pauling (SP) behavior being\nequal in $\\mu_B$ to $Z_t-8$ where $Z_t$ the total number of\nvalence electrons in the unit cell \\cite{MavropoulosZB}. Recently\ntheoretical works have appeared attacking also some crucial\naspects of these alloys like the exchange bias in\nferro-\/antiferromagnetic interfaces \\cite{Nakamura2006}, the\nstability of the zinc-blende structure \\cite{Xie2003}, the\ndynamical correlations \\cite{Chioncel2006}, the interfaces with\nsemiconductors \\cite{InterGala,Interfaces}, the exchange\ninteraction \\cite{Sasioglu-Gala}, the emergence of half-metallic\nferrimagnetism \\cite{Rapid} and the temperature effects\n\\cite{MavropoulosTemp}. An extended overview on the properties of\nthese alloys can be found in reference \\cite{Review}.\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{lekk-fig1.eps}\n\\end{center} \\caption{(Color online) Schematic representation of the structure of the\nmultilayers under study. We have assumed that the growth direction\nof the zinc-blende structure is the (001) and the period consists\nof four CrAs and four InAs(CdSe) layers. Each plane contains atoms\nof a single chemical type plus a void site. We denote as Void1 the\nvacant site in the same layer with the Cr atoms at the interfaces,\nVoid2 in the interfacial As(Se) layer and Void3 in the interfacial\nIn(Cd) layer. The distance between two consecutive plane is\n$\\frac{1}{4}$ of the lattice constant. Note that in the case of\nthe CrAs\/CdSe we have two non-equivalent interfaces: (i) when the\nsequence of the atoms is ...-Cr-As-Cd-... denoted as CrAs\/CdSe-1\nand (ii) when the sequence is ...-Cr-Se-Cd-... denoted as\nCrAs\/CdSe-2. Finally we should note that we have assumed the\nlattice constant of the two semiconductors (0.606 nm).\n\\label{fig1}}\n\\end{figure}\n\n\\begin{table*}\n\\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the perfect CrAs and CrSe alloys and for the three perfect\nmultilayers under study. For the position of each atom in the\nmultilayer please refer to figure \\ref{fig1}.}\n \\begin{tabular}{l|ccccc} \\hline \\hline\nPerfect Systems& CrAs(Bulk) & CrSe(Bulk)& CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\\nCr & 3.267 & 3.825 & 3.248& 3.117 & 3.462 \\\\\nVoid1 & -0.029& 0.049& -0.008& -0.025& -0.002 \\\\\nAs(Se) [CrAs layer] & -0.382& -0.103(Se)& -0.383& -0.388 & \\\\\nAs(Se) [Interface] &&& -0.209& -0.365 & -0.093(Se)\\\\\n As(Se) [SemiConducting layer]& && -0.066& & -0.048(Se) \\\\\nVoid2 & 0.080& 0.162 & 0.043& 0.027& 0.056\\\\\n In(Cd)&& &-0.008& -0.045& 0.003 \\\\\n Void3&&& -0.022 &-0.033 & -0.001\n \\\\\\hline \\hline\n\\end{tabular}\n\\label{table1}\n\\end{table*}\n\nRecently we have published a work on the role of defects in the\ncase of half-metallic CrAs, CrSb and CrSe alloys crystallizing in\nthe zinc-blende structure and adopting the lattice constant of\nInAs binary semiconductor of 0.606 nm \\cite{Pouliasis}. All\ndefects under study in this reference, with the exception of void\nimpurities at Cr and $sp$ sites and Cr impurities at $sp$ sites,\nwere found to induce new states within the gap and the Fermi level\ncan be pinned within these new minority states destroying the\nhalf-metallic character of the perfect bulk compounds. These\nimpurity states are localized in space around the impurity atoms\nand very fast the bulk behavior is regained. But in realistic\ndevices interfaces with semiconductors will occur and the\ninteraction of these bulk-impurity states with interface states\ncan destroy the spin-polarization of the injected current.\nAb-initio calculations show that interfaces are in principle\nhalf-metallic for pnictides containing Cr or V atoms even when the\nsemiconductor is a II-VI like CdSe and not a III-V one like InAs,\nand no interface states occur \\cite{InterGala}. But impurities at\nthe interfaces can induce new impurity-interfaces states within\nthe minority spin-gap which can couple to the bulk impurity states\nand lead to loss of the half-metallic character.\n\nIn this communication we expand our previous study to cover also\nthe case of impurities at interfaces. We have decided to consider\nCrAs as the half-metallic spacer since it is the most-widely\nstudied transition-metal zinc-blende pnictide and both InAs and\nCdSe as the zinc-blende semiconducting spacer to cover both the\ncase of III-V and II-VI semiconductors. In and Cd atoms, as well\nas As and Se atoms, are in the same row of the periodic table and\nthey have one valence electron difference and thus both alloys\nhave the same lattice constant of 0.606 nm and for this value of\nthe lattice constant bulk CrAs shows a Fermi level exactly in the\nmiddle of the minority-spin gap making clear the appearance of\nimpurity states. For our calculation we employ the\n the Korringa-Kohn-Rostoker method (KKR) method \\cite{Pap02} as in the case of\n the perfect bulk \\cite{MavropoulosZB} and interfaces\n \\cite{InterGala} and we have treated the impurities as in\n references \\cite{Pouliasis,imp}.\n\n\nTo model the interface we have considered a multilayer with a\nperiod of 4 CrAs and 4 InAs(CdSe) monolayers (MLs) which is in\naccordance with the experimental data in reference\n\\cite{Mizuguchi} and a schematic illustration is shown in figure\n\\ref{fig1}. As growth direction we have chosen the (001) and thus\nits layer is made up of atoms of a single chemical type. Moreover\nat each layer we have considered a void to describe better the\nspace in our calculation. In the case of CrAs\/InAs multilayers all\ninterfaces are equivalent while in CrAs\/CdSe case we have two\nnon-equivalent interfaces; the interface where As is between the\nCr and Cd layers which we denote as Interface-1 and the interface\nwhere Se separates the Cr and Cd layers (Interface-2). Moreover we\nshould note that the distance between two successive layers is 1\/4\nof the lattice constant and thus atoms in successive layers are\nnearest atoms while within the same atomic layer the nearest atoms\nare second neighbors. We have also denoted as Void1 the vacant\nsite at the Cr interface layer, Void2 in the As(Se) interface\nlayer and Void3 in the In(Cd) interface layer. We have considered\nall possible impurities and in section \\ref{sec2} we present the\ncase of perfect bulk and mulitilayers while in section \\ref{sec3}\nwe present the properties of the Cr impurities at various sites at\nthe interface, in section \\ref{sec4} the case of As(Se)\nimpurities, in section \\ref{sec5} the case of voids and in section\n\\ref{sec6} the case of In(Cd) impurities. Finally in section\n\\ref{sec7} we summarize and conclude. Throughout the discussion we\ncompare our results with the case of impurities in perfect bulk\nCrAs presented in reference \\cite{Pouliasis}. In the case of the\nCrAs\/CdSe-2 interface the Cr atoms have As atoms from one side and\nSe atoms from the other side but as it was shown in\n\\cite{Pouliasis} both CrAs and CrSe show similar behavior with\nrespect to the defects-induced states.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig2.eps}\n\\caption{(Color online) Atom-resolved density of states (DOS) for\nthe bulk CrAs and the perfect multilayers under study presented in\nfigure \\ref{fig1}. Panels (a), (b), (c) and (d) refer to the\nCrAs\/InAs multilayer and (e),(f), (g) and (h) to the CrAs\/CdSe\nmultilayer. In panel (a) and (b) we have also included the DOS of\nCr and As atoms in perfect bulk CrAs for comparison. Note that in\nthe case of the CrAs\/CdSe we have two interfaces one containing As\none containing Se atoms (see figure \\ref{fig1} for explanation).\nThe Fermi energy has been set as the zero of the energy axis.\nPositive DOS values correspond to\n the spin-up (majority-spin) electrons and negative values to the spin-down\n (minority-spin) electrons.\\label{fig2}}\n\\end{figure}\n\n\n\n\\section{Perfect systems}\\label{sec2}\n\n\nWe will start our discussion from the perfect systems. In table\n\\ref{table1} we have gathered the atom-resolved spin magnetic\nmoment in $\\mu_B$ for the three interfaces and for the bulk CrAs\nand CrSe alloys. Notice that for the bulk systems, where we have\ntwo vacant sites, we denote as Void1 the vacant site with the\nsame symmetry as the Void1 site at the interface and the same\nstands for the Void2 site. Bulk CrAs has 11 valence electrons per\nunit cell while CrSe has 12 valence electrons and thus\nhalf-metallic CrAs has a total spin moment of 3 $\\mu_B$ and CrSe\nof 4 $\\mu_B$. As it was shown in \\cite{MavropoulosZB} Cr atoms in\nCrSe accommodate half of the extra electron in the majority-spin\nband leading to a larger Cr-spin moment which now reaches about\n3.8 $\\mu_B$ instead of $\\sim$3.3 $\\mu_B$ in CrAs. The other half\nelectron is accommodated in the majority-spin band of the Se and\nVoid2 atoms as it easily deduced from the spin moments presented\nin table \\ref{table1} (Void2 is not a real atom but orbitals from\nthe nearest Cr atoms penetrate in the Void2 site).\n\nIn the case of the CrAs\/InAs multilayer, both Cr atoms have\nexactly the same nearest environment as in bulk CrAs and almost an\nidentical spin magnetic moment. The same is true also for the As\natom within the CrAs layer (As layer between the two Cr layer as\nshown in figure \\ref{fig1}) which has four Cr and four Void1 sites\nas nearest neighbors as in the bulk CrAs and exactly the same spin\nmagnetic moment of -0.38 $\\mu_B$. As at the interface layer has\none Cr layer from one side an In layer from the other side. In\natoms have two valence electrons which mainly are of $s$ character\nand have no contribution to the In spin magnetic moment. The spin\nmoment at the In sites is induced by the As atoms since\n$p$-orbitals of As penetrate in the In sites. Thus As atoms at the\ninterface show a smaller absolute value of the spin magnetic\nmoment since now they have only two instead of four Cr nearest\nneighbors (the spin-moment at As is induced by the Cr atoms\nthrough the hybridization of the As-$p$ and Cr-$t_{2g}$ states)\nand In atoms at the interface show a negligible spin moment\ninduced by the As interface atoms which in reality is the\nreflection of the Cr spin moment. As atoms within the\nsemiconducting layer have four In atoms as nearest neighbors and a\nvery small spin magnetic moment.\n\nIn the case of the CrAs\/CdSe multilayer we have two inequivalent\ninterfaces and the two Cr layer are no-more equivalent. The Cr\natoms at the As interface behave similarly to bulk CrAs although\nthe spin moment is somewhat smaller while in the case of the Se\ninterface Cr atoms have two As and two Se atoms as nearest\nneighbors exhibiting a spin magnetic moment of about 3.5$\\mu_B$ a\nmean value between the bulk CrAs and CrSe atoms. Within the CrAs\nlayer As shows a spin magnetic moment similar to the CrAs\/InAs\nmultilayer. At the 1$^\\mathrm{st}$ interface As atoms show similar\nbehavior to the CrAs\/InAs case although their spin moment is\nlarger in absolute value since Cd atoms have one valence electron\nless that In ones and the leakage of charge from the As interface\natoms to the Cd ones is smaller than to the In ones. In the\n2$^\\mathrm{nd}$ interface Se atoms have one electron more than As\none and thus the Se-$p$ orbitals are less polarized from the\nCr-$t_{2g}$ ones and show a smaller spin magnetic moment by a\nfactor of four. Within the CdSe semiconducting layer the Se atoms\nbehave like the As atoms in the InAs layer in the first interface\nand show a small value of the spin moment. Finally Cd atoms show a\nnegligible spin-magnetic moment as the In ones.\n\nIn figure \\ref{fig2} we have gathered the density of states (DOS)\nin states\/eV for both multilayers under study and the bulk CrAs.\nIn the left column we present the case of the CrAs\/InAs\nmultilayer. Cr atoms at the interface show a similar DOS to the\nbulk case with a very small weight of occupied minority-spin\nstates while in the majority band both $e_g$ and a fraction of the\nthree $t_{2g}$ states are occupied. The gap is smaller in the case\nof the multilayer since the occupied minority-spin states are\nbroader in energy and this is reflected to all other atoms. As\natoms within the CrAs layer show an almost identical DOS to the\nbulk CrAs case as was also the case for the spin magnetic moments.\nAs we mote to the interface and then to the InAs layer the shape\nof the As states changes due to the different local environment\nand the bands move towards the Fermi level. The states shown are\nthe $p$ states since the $s$-states of all atoms are located at\nabout -9 to -12 eV below the Fermi level and are not shown in the\nfigures. Note that for the In atoms we have used a different scale\nin the vertical axis. In the case of the CrAs\/CdSe multilayer the\nhalf-metallicity is again present as for the CrAs\/InAs layer but\ndue to the character of the II-VI semiconductor the gap in the\nminority-spin band is smaller and the Fermi level is near the\nleft-edge of the gap as it is clearly seen in the case of the Cr\nDOS. The reduction of the gap-width is larger for the As than the\nSe interface and this agrees with the fact that the bulk CrSe\npresents a larger gap than the bulk CrAs due to the larger\nelectronegativity of the Se atom as it was shown in reference\n\\cite{MavropoulosZB}. As a consequence the states of the As atoms\nat the first interface have moved closer to the Fermi level in a\nrigid way with respect to the As atoms in the CrAs layer. Cd atoms\nat both interfaces show similar DOS to the In atoms in the\nCrAs\/InAs interface. Se atoms within the CdSe layer show a similar\nshape to the As atoms in the InAs layer while Se atoms at the\nsecond interface show states deeper in energy than the As atom in\nthe first interface in agreement with the larger gap-width shown\nby the Cr atoms in the second interface with respect to the first\nCrAs\/CdSe interface.\n\\begin{table*}\n\\centering \\caption{Atom-resolved spin magnetic moments for the\ncase of Cr impurity atoms at interfacial As(Se) and Void2 sites in\nthe case of bulk CrAs and the three interfaces under study. \"imp\"\nstands for impurity and \"1st\" stands for nearest neighbor atoms\n(similar for 2nd and 3rd). We present results for both cases of\ncoupling of the Cr impurity spin moment with respect to the spin\nmoments of the Cr nearest-neighbors (ferromagnetic-FM and\nantiferromagnetic-AFM cases). Note that As(Se) atoms can be found\nin the CrAs layer [CrAs], the interfaces [Inter] and the\nsemiconductiong layer [SC]. }\n \\begin{tabular}{l|c|c|c|c|c|c|c|c|c} \\hline \\hline\n\\underline{\\bf Cr at As(Se) site} & \\multicolumn{2}{c|}{BULK CrAs}\n& \\multicolumn{2}{c|}{CrAs\/InAs} &\n\\multicolumn{2}{c|}{CrAs\/CdSe-1}&\n\\multicolumn{2}{c|}{CrAs\/CdSe-2}\\\\\n& \\underline{FM} & \\underline{AFM} & \\underline{FM} &\n\\underline{AFM} & \\underline{FM} &\n\\underline{AFM} & \\underline{FM} & \\underline{AFM} \\\\\n Cr-imp & 4.388 & -3.690 & 4.334 & -3.802 & 4.248& -4.058 & 4.335 & -3.988 \\\\\n Cr-1st & 3.602 & 3.364 & 3.704 & 3.425 & 3.543 & 3.343 & 3.735 & 3.485 \\\\\n In(Cd)-1st & & & 0.043 & -0.067 & 0.044 & -0.108 & 0.063 & -0.104 \\\\\n As [CrAs]-3rd & -0.405 & -0.405 & -0.412 & -0.413 & -0.418 & -0.418 &\n -0.025 & -0.074\\\\\nAs(Se) [Inter]-3rd & & & -0202 &-0.237 & -0.305 & -0.352 &-0.105(Se) & -0.110(Se) \\\\\nAs(Se) [SC]-3rd & & & -0.025 & -0.067 & -0.032(Se) & -0.059(Se) &-0.417(Se) & -0.418(Se) \\\\\n\\hline \\hline \\underline{\\bf Cr at Void2 site} &\n\\multicolumn{2}{c|}{BULK CrAs} & \\multicolumn{2}{c|}{CrAs\/InAs} &\n\\multicolumn{2}{c|}{CrAs\/CdSe-1}&\n\\multicolumn{2}{c|}{CrAs\/CdSe-2}\\\\\n& \\underline{FM} & \\underline{AFM} &\\underline{FM} &\n\\underline{AFM} & \\underline{FM} &\n \\underline{AFM} & \\underline{FM} & \\underline{AFM} \\\\\nCr-imp & 3.901 & -2.908 & 3.821 & -3.262& 3.729& -3.263& 4.171 & -3.551 \\\\\n Cr-1st & 3.419 & 3.267& 3.412 &3.225 & 3.300 &3.103 & 3.467 &3.335 \\\\\n In(Cd)-1st & & & 0.014 & -0.176 & -0.118 &-0.030 & 0.023 & -0.038\\\\\n As [CrAs]-2nd & -0.341 &-0.285 & -0.364 &-0.323 & -0.371 & -0.330 & -0.470 &-0.371 \\\\\n As(Se) [Inter]-2nd & & & -0.167 &-0.141 & -0.308 &-0.214 & -0.102(Se) & -0.104(Se) \\\\\n As(Se) [SC]-2nd & & & -0.009 &-0.039& -0.013(Se) &-0.033(Se) & -0.033(Se) & -0.026(Se)\\\\\n\\hline \\hline\n\\end{tabular}\n\\label{table2}\n\\end{table*}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig3.eps}\n\\caption{(Color online) Cr-resolved DOS for the case of Cr\nimpurity atoms at interfacial As(Se) sites at in the case of the\nferromagnetic (FM) [left panel] and antiferromagnetic coupling\n(AFM) [right panel) between the Cr impurity atom and its Cr\nnearest-neighbors (see text for details). Details as in figure\n\\ref{fig2}. \\label{fig3}}\n\\end{figure}\n\n\n\\section{Cr impurities}\\label{sec3}\n\nWe will start our discussion from the case of Cr impurities. The\nfirst case under study is when Cr impurity atoms are located at\nAs(Se) or Void2 sites at the interface. With respect to the\nperfect Cr sites where the first neighbors were four sp atoms and\nfour voids, the Cr impurity atoms have now as nearest neighbors\ntwo Cr atoms and two In(Cd) atoms together with two Void1 and two\nVoid3 sites. Thus the local environment of the Cr impurities is\nfundamentally different. In the case of Cr impurities at As and\nVoid2 sites in the bulk, we have shown that the spin magnetic\nmoment could be either ferromagnetically (FM) coupled to the spin\nmoment of the Cr atoms at the perfect sites or\nantiferromagnetically (AFM) coupled with the latter one being the\nenergetically favorable case \\cite{Pouliasis}. The AFM was\noccurring due to the short distance between the Cr impurity atoms\nand its four nearest Cr neighbors (Cr as well as Mn atoms are know\nto have antiparallel spin moments under a critical distance). In\nthe case of the interfaces under study the Cr impurity atoms have\nnow two instead of four nearest Cr neighbors as in the bulk. But\nit seems that this is enough to have again two stable solutions, a\nFM and an AFM one, depending on the starting potential, for both\nCr at interfacial As(Se) and Void2 sites. The AFM solution seems\nenergetically more favorable although we have not converged the\nenergetics of the defects (for the reason see discussion in\nreference \\cite{Pouliasis}).\n\nIn figure \\ref{fig3} we have gathered the atom-resolved DOS for\nthe Cr impurity atoms and its nearest Cr neighbors for all three\ninterfaces under study and bulk CrAs for the case of Cr impurity\nat As(Se) site and for both the FM and AFM cases. Results are\nsimilar when the Cr impurity is located at the Void2 site and thus\nare not shown here. Also results are in all cases similar to the\nbulk one with minor changes at the position of the picks due to\nthe variation in the environment surrounding the impurities. In\nthe FM case the impurities show almost zero occupied minority-spin\nstates while the Cr nearest neighbors show a very small weight of\nminority-spin states below the Fermi level. In all cases the\nhalf-metallic character of the interface is not affected. In the\nAFM case the Cr impurities exhibit a large spin-splitting of the\nbands and almost all the spin-down states are occupied while\nalmost all spin-up states are empty. This leads to a large gap in\nthe spin-down band between the spin-down occupied bands of the\nimpurity atom and the unoccupied spin-down bands of the Cr nearest\nneighbors and the half-metallic character of the interface is not\naffected. In table \\ref{table2} we have gathered the atomic spin\nmagnetic moments for all cases mentioned above. We can see that\nspin moments behave similar to the case of the Cr impurities at As\nor Void2 sites in the perfect bulk. In the FM case the Cr impurity\natoms have a larger spin moment with respect to the Cr atoms in\nthe perfect systems since the weight of the minority occupied\nstates has vanished as a result of the hybridization with its\nfirst neighbors. The Cr atoms which are first neighbors of the\nimpurity atoms present spin moments almost identical to the\nperfect compounds in table \\ref{table1}. In the AFM case although\nit seems from the DOS that all five spin-down $d$-states of the\nimpurity atom are occupied the spin moment is less than 5 $\\mu_B$\nsince some spin-up states also appear below the Fermi level\nreflecting the band-structure of the Cr nearest neighbors. We\nshould note that spin moments are slightly smaller when the\nimpurity is located at a Void2 instead of a As(Se) site because\nalthough the nearest-neighbors are the same, the next-nearest and\nfurther neighbors are different. We should finally note here that\nthe AFM case is the most interesting case for applications since\nthe occurrence of half-metallic ferrimagnetism leads to smaller\nexternal fields and exhibiting smaller energy losses with respect\nto ferromagnets.\n\n\\begin{table}\n\\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the case of Cr impurity atoms at interfacial Void1, In(Cd) and\nVoid3 sites in the case of the three interfaces under study.\nNotation as in table \\ref{table2}. }\n \\begin{tabular}{l|c|c|c} \\hline \\hline\n\\underline{\\bf Cr at Void1 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nCr-imp & 3.522& 3.363 & 3.686 \\\\\n As(Se) [CrAs]-1st & -0.300 & -0.299 & -0.296 \\\\\n As(Se) [Inter]-1st & -0.151& -0.275& -0.042 \\\\\n Cr [Inter]-2nd & 3.448& 3.352 & 3.622\\\\ \\hline\n\\underline{\\bf Cr at In(Cd) site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\ Cr-imp & 3.185 & 3.311& 4.015 \\\\\n As(Se) [SC]-1st & -0.157 & -0.095& -0.147 \\\\\n As(Se) ([Inter]-1st &-0.324 & -0.406& -0.254 \\\\\n Cr [Inter]-3rd & 3.328& 3.252 & 4.219 \\\\ \\hline\n\\underline{\\bf Cr at Void3 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nCr-imp & 3.508& 3.501& 3.833\\\\\n As(Se) [SC]-1st &-0.024 &-0.004 & 0.009 \\\\\n As(Se) [Inter]-1st & -0.135 & -0.275 & -0.025 \\\\\n Cr [Inter]-2nd & 3.478 & 3.384 & 3.635 \\\\\n \\hline \\hline\n\\end{tabular}\n\\label{table3}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig4.eps}\n\\caption{(Color online) Density of states of the Cr impurity atom\nat Void3(upper panel) and In(Cd) (lower panel) interfacial sites\nfor all the interfaces under study. Details as in figure\n\\ref{fig2}. \\label{fig4}}\n\\end{figure}\n\nThe next case under study is when the Cr impurity is located at a\nVoid1 site. Although the Void1 site is located in the same layer\nas the Cr atoms, they are second neighbors having the same nearest\nneighbors (four sp atoms and four voids). Thus we do not expect Cr\nimpurities at Void1 sites to alter the electronic and magnetic\nproperties of the interface similarly to the situation of Cr\nimpurities at Void1 sites in the bulk CrAs \\cite{Pouliasis}. The\nDOS is similar to the Cr atoms in the perfect systems and thus we\ndo not present them since the gap is almost unaltered and the\nhalf-metallicity is not affected. The spin magnetic moment of the\nCr impurity atoms which are shown in table \\ref{table3} are\nslightly larger than the Cr atoms in the perfect system by about\n0.2 $\\mu_B$. The Cr nearest-neighbors have spin moment almost\nidentical to the Cr impurities and thus part of the charge of the\nCr impurity atoms is used to increase the spin moment of the Cr\nsecond neighbors with respect to the perfect interfaces.\n\nThe final case under study in this section is when the Cr impurity\noccurs at the In(Cd) sites at the interface or the Void3 sites\nwhich are located in the same layer with the In(Cd) ones. Although\none could think that these impurities should destroy the\nhalf-metallic character of the interface, this is not true. Cr\nimpurities at these sites have as nearest neighbors two As(Se)\natoms in the semiconducting space and two As(Se) site at the\ninterface. Thus their local environment is quite close to the one\nof the Cr atoms at the perfect interface sites. As a result the\nhalf-metallic character is not destroyed as shown by the DOS in\nfigure \\ref{fig4}. When the impurity is located at a Void3 site,\nthere is almost no-occupied spin-down states while when the\nimpurity is located at the In(Cd) site the weight of the occupied\nspin-down states is very small and the width of the gap is\ncomparable to the perfect compounds in figure \\ref{fig2}. In the\nlatter case we see that for the CrAs\/CdSe-2 interface the bands\nare slightly shifted to higher energy as in a rigid-band model\nwith respect to the other interfaces but the gap has even larger\nwidth. This is reflected also to the spin-magnetic moments\npresented in table \\ref{table3} where for this case the Cr\nimpurity spin moment exceeds the 4 $\\mu_B$ while for the other\ninterfaces it is slightly larger than the perfect systems.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig5.eps}\n\\caption{(Color online) In the case of As(Se) impurities at Void2\nsites we present the atom-resolved DOS of the nearest Cr and\nIn(Cd) neighbors for all three interfaces. Details as in figure\n\\ref{fig2}. \\label{fig5}}\n\\end{figure}\n\n\\section{As(Se) impuritites}\\label{sec4}\n\nNext we will present our results on the As impurities in the case\nof the CrAs\/InAs and CrAs\/CdSe-1 interfaces and on the Se\nimpurities in the case of the CrAs\/CdSe-2 interface. We have\nstudied all possible cases and in table \\ref{table4} we have\nconcentrated the atom-resolved spin magnetic moments for the\nimpurities and their nearest neighbors. The DOS of the impurities\nin general present no large variation with respect to the As(Se)\natom at the interface or the case of impurities at the perfect\nbulk and thus we present only the case of As(Se) atom at Void2\nsites, where the As(Se) impurity atom has the same\nnearest-neighbors as at the perfect interfacial As(Se) sites. In\nall other cases the impurity atom has four As(Se) atoms as nearest\nneighbors (two at the interface and two at the CrAs, when the\nimpurity is located at the Cr or Void1 sites, and two at the\nInAs(CdSe) layer, when it is located at the In(Cd) or Void3\nsites).\n\nWe will first discuss the case of As(Se) impurities at the Cr and\nVoid1 sites. At the same energy window with the $d$ electrons of\nthe Cr atoms are located the $p$ electrons of the As(Se) atoms.\nThe DOS of the impurity atoms is similar to the case of As(Se)\nimpurities at Cr and Void1 sites in bulk CrAs(CrSe) presented in\n\\cite{Pouliasis} and thus we do not present them. For the\ninterfacial As(Se) atoms in the perfect interfaces the minority\n$p$ states of As(Se) are completely occupied leading to small\nnegative spin moments (see table \\ref{table1}). The impurity\nAs(Se) atoms at Cr or Void1 sites, on the other hand, have four\nother As(Se) atoms as nearest neighbors and thus the $p$ states of\nthe As(Se) impurity atoms have to hybridize with the $p$ states of\nthe neighboring As(Se) atoms which are almost completely occupied\ninstead of the Cr $t_2g$ states for which only the majority states\nare occupied leading to occupancy of the antibonding $p$ spin-up\nstates and to positive spin magnetic spin moments as shown in\ntable \\ref{table4}. Due to the reorganization of the charge\ninduced by the impurity As(Se) states, the number of the occupied\nspin-up $p$-states of the neighboring As(Se) atoms increases and,\nalthough the spin magnetic moments remain negative, their absolute\nvalue decreases. In all cases discussed just above, the impurity\nstates are not located in the gap similar to the what happened for\nthe same impurities in the bulk systems (see \\cite{Pouliasis}) and\nthe half-metallicity is preserved.\n\n\n\\begin{table}\n\\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the case of As(Se) impurity atoms at various interfacial sites\nin the case of the three interfaces under study. Notation as in\ntable \\ref{table2}. }\n \\begin{tabular}{l|c|c|c} \\hline \\hline\n\\underline{\\bf As(Se) at Cr site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nAs(Se)-imp & 0.108 & 0.118& 0.127 \\\\\n As(Se) [CrAs]-1st & -0.243 & -0.255 &-0.271 \\\\\n As(Se) [Inter)]-1st & -0.061 & -0.177 & -0.093 \\\\ \\hline\n\n\\underline{\\bf As(Se) at Void1 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nAs(Se)-imp & 0.143 & 0.084 & 0.070 \\\\\nAs(Se) [CrAs]-1st & -0.222 &-0.267 & -0.123 \\\\\nAs(Se) [Inter]-1st & -0.114 & -0.208& -0.087 \\\\ \\hline\n\n\\underline{\\bf As(Se) at Void2 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\\nAs(Se)-imp & 0.269& 0.207& -0.046\\\\\n Cr-1st& 3.361&3.212& 3.367\\\\\n In(Cd)-1st & 0.012& -0.007& -0.010\\\\\nAs(Se) [CrAs]-2nd& -0.318& -0.314& -0.494\\\\\n As(Se) [Inter]-2nd& -0.104& -0.229& -0.114\\\\\n As(Se) [SC]-2nd & 0.056&0.050& -0.074\\\\ \\hline\n\n\\underline{\\bf As(Se) at In(Cd) site} & CrAs\/InAs & CrAs\/CdSe-1\n& CrAs\/CdSe-2\n\\\\ As(Se)-imp & 0.038& 0.269& 0.048\\\\\n As(Se) [Inter]-1st&-0.128& -0.013& -0.584\\\\\n As(Se) [SC]-1st& -0.008&0.287& -0.249\\\\ \\hline\n\n\\underline{\\bf As(Se) at Void3 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2\\\\ As(Se)-imp & 0.093&\n0.068& 0.197\\\\\n As(Se) [Inter]-1st& -0.069& -0.142&-0.003\\\\\n As(Se) [SC]-1st& -0.009 & -0.013& -0.008\\\\\n \\hline \\hline\n\\end{tabular}\n\\label{table4}\n\\end{table}\n\nWhen the As(Se) atoms migrate to Void2 sites, their nearest\nneighbors remain two Cr and two In(Cd) atoms as for the perfect\nAs(Se), but the further neighbors change and the interfacial As\natoms are now next-nearest neighbors and the hybridization between\ntheir $p$ states is intense leading to occupation of the\nantibonding spin-up $p$ states and positive values of the spin\nmagnetic moments of the impurity atoms as shown in table\n\\ref{table4} for the case of the As impurities in CrAs\/InAs and\nCrAs\/CdSe-1 interfaces or to a large variation of the negative Se\nspin moment in the case the CrAs\/CdSe-2 interface (the spin moment\nof the impurity is half with respect to the Se atom at the perfect\nsites). The Cr atoms at the interface have now 5 instead of 4\nAs(Se) atoms as nearest neighbors but the hybridization of the\n$t_{2g}$ $d$-states is little affected and the nearest-neighboring\nCr spin moments are almost identical to the perfect interfaces. A\nsimilar behavior is exhibited by the spin moments of the As(Se)\nsecond neighbors. The behavior of the spin moments is reflected\nalso in the DOS presented in figure \\ref{fig5}. The DOS of the Cr\nnearest neighbors presents for all interfaces under study a very\nwide minority-spin gap similar to the perfect compounds with a\nlarge spin-splitting between the occupied majority- and the\nunoccupied minority-spin bands. The shape of the Cr DOS is similar\nto the Cr atoms at the perfect interfaces shown in figure\n\\ref{fig2} revealing the small influence of the hybridization with\nthe impurity As(Se) $p$-states. The same is also valid for the DOS\nof the nearest In(Cd) atoms shown also in the same figure although\nthe gap is smaller for these atoms similarly to what occurred in\nthe perfect interfaces.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig6.eps}\n\\caption{((Color online) In the case of Void impurity at an\ninterfacial Cr site we present the atom-resolved DOS of the\nnearest As(Se) neighbors in the CrAs layer and at the interface\nfor all three interfaces and for the bulk CrAs. Details as in\nfigure \\ref{fig2}. \\label{fig6}}\n\\end{figure}\n\nThe case of As(Se) impurities at In(Cd) and Void3 sites is more\ndifficult to be understood and although the half-metallicity is\npreserved the spin moments of the As(Se) impurity atoms and their\nnearest As(Se) atoms show large variations. In the case of the\nCrAs\/InAs interface the As impurity atom has 4 As atoms as nearest\nneighbors, in the case of the CrAs\/CdSe-1 atom the As impurity\natom has two As at the interface and two Se atoms within the CdSe\nlayer as nearest neighbors while in the case of the CrAs\/CdSe-2\ninterface the Se impurity atom has four Se atoms as nearest\nneighbors. Se atoms have one electron more than the As ones and\nthus in general a larger number of spin-up states is occupied as\nrevealed also from the spin magnetic moments in table\n\\ref{table1}. Thus especially when the impurity is located at the\nIn(Cd) site no trend can be found in table \\ref{table4} regarding\nthe spin magnetic moments and the situation is\ninterface-dependent.\n\n\n\n\n\\section{Void impurities}\\label{sec5}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig7.eps}\n\\caption{(Color online) In the case of Void impurity at an\ninterfacial In(Cd) site we present the atom-resolved DOS of the\nnearest As(Se) neighbors in the semiconducting layer and at the\ninterface for two of the studied interfaces. Details as in figure\n\\ref{fig2}. \\label{fig7}}\n\\end{figure}\n\nUp to now we have studied the case of the Cr and As(Se) impurities\nand in all cases the half-metallicity was preserved. In this\nsection we will study the effect of Voids. Voids are Schottky-type\ndefects with large formation energies. A Void impurity in reality\nmeans that an atom is missing leading to a reorganization of the\ncharge of the neighboring atoms. We will start our discussion from\nthe case of a Void at a Cr site and in figure \\ref{fig6} we\npresent the DOS of the nearest As(Se) neighbors for all three\ninterfaces and for the bulk CrAs. In the case of the bulk the\nmissing Cr atom means that As atoms have now three instead of four\nCr nearest neighbors and the $p$-bands of As move closer to the\nFermi level without crossing it. In the case of interfaces we\nshould distinguish between the As atoms within the CrAs layer\nwhich show a behavior identical to the bulk case and the As(Se)\natoms at the interface layer. The latter ones in the perfect\nmultilayers have two Cr atoms and two In(Cd) as nearest neighbors\nand when a Void occurs at a Cr site they lose one of the two Cr\nneighbors resulting to a large shift of the $p$-bands which almost\ncross the Fermi level. Although one could argue that\nhalf-metallicity is present, a small change of the lattice\nconstant would lead to loss of the half-metallic character of the\nperfect interfaces. As the bands move closer to the Fermi level\nthe weight of the spin-up states decreases while the bonding\nspin-down states are completely occupied leading to larger\nabsolute values of the negative spin moments of the nearest As(Se)\nneighbors with respect to the perfect case as shown in table\n\\ref{table5}.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig8.eps}\n\\caption{(Color online) In the case of Void impurity at an\ninterfacial As(Se) site we present the atom-resolved DOS of the\nnearest Cr and In(Cd) neighbors in the CrAs layer for all three\ninterfaces and for the bulk CrAs. Note that the In(Cd) DOS has\nbeen multiplied by a factor of 10. Details as in figure\n\\ref{fig2}.\\label{fig8}}\n\\end{figure}\n\nThe same phenomenon occurs also when the Void is located at an\nIn(Cd) site as shown in figure \\ref{fig7}. The missing In(Cd)\nneighbors lead to a shift of the $p$-bands of the\nnearest-neighboring As(Se) atoms towards higher energies and now\nclearly the spin-down band crosses the Fermi level destroying the\nhalf-metallicity. Especially in the case of the CrAs\/InAs\ninterface a spin-down pick is pinned exactly at the Fermi level.\nIn the case of the CrAs\/CdSe-1 interface the bands are more\nnarrow; the spin-down pick is located below the Fermi level but a\nsmall contraction of the lattice constant could lead to the\npinning of the Fermi level at this pick. As discussed for the case\nof Void impurities at Cr sites, also here the absolute values of\nthe negative spin moments of the neighboring As(Se) atoms increase\n(see table \\ref{table5}).\n\n\\begin{table} \\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the case of Void impurities at interfacial Cr, As(Se) and\nIn(Cd) sites in the case of the three interfaces under study.\nNotation as in table \\ref{table2}.}\n \\begin{tabular}{l|c|c|c} \\hline \\hline\n\\underline{\\bf Void at Cr site} & CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\\n Void-imp &\n-0.097& -0.129& -0.061\\\\ As [CrAs]-1st& -0.592&\n-0.606& -0.613\\\\ As(Se) [Inter]-1st& -0.350& -0.644& -0.159\\\\\n\\hline\n \\underline{\\bf Void at Ae(Se) site} & CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\\nVoid-imp & 0.201& 0.154& 0.190\\\\\n Cr-1st & 3.748& 3.691& 3.812\\\\\n In(Cd)-1st &0.033& -0.008& 0.017 \\\\ \\hline\n\\underline{\\bf Void at In(Cd) site} & CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\ Void-imp & -0.094& -0.073& 0.066\\\\\n As(Se) [Inter]-1st &\n-0.468& -0.535& -0.210\\\\\n As(Se) [SC]-1st & -0.141&\n-0.115& -0.127 \\\\\n \\hline \\hline\n\\end{tabular}\n\\label{table5}\n\\end{table}\n\n\nThe last possible case which can occur is when the Void is located\nat the interface As(Se) site. The loss of the As(Se) atom only\nmarginally affects the DOS of the Cr atoms at the interface which\nnow have three instead of four neighboring atoms. As shown in\nfigure \\ref{fig8} Cr nearest-neighbors at the interface show\nsimilar behavior as in the bulk and the weight of the occupied\nminority-spin states is vanishing and the half-metallicity remains\nrobust. The In(Cd) nearest-neighbors exhibit also a large gap as\nshown in the same figure where the DOS of the In(Cd) has been\nmultiplied by 10 to be visible with respect to the Cr nearest\nneighbors. The loss of its As(Se) neighbor leads to an increase of\nthe local charge of the Cr nearest-neighbors (in a way the regain\nthe charge which was participating at the bonds with the missing\nAs(Se) atom). This extra local charge occupies spin-up states\nleading to and increase of the Cr spin moment with respect to the\nperfect interfaces by about 0.4-0.5 $\\mu_B$.\n\n\n\n\\begin{table} \\centering \\caption{Atom-resolved spin magnetic moments in $\\mu_B$\nfor the case of In(Cd) impurities at various interfacial sites in\nthe case of the three interfaces under study. Notation as in table\n\\ref{table2}.}\n \\begin{tabular}{l|c|c|c} \\hline \\hline\n\\underline{\\bf In(Cd) at Cr site} & CrAs\/InAs &\nCrAs\/CdSe-1 & CrAs\/CdSe-2 \\\\\nIn(Cd)-imp & <0.001& -0.070& -0.025\\\\\n As(Se) [CrAs]-1st&\n-0.357& -0.459& -0.434\\\\ As(Se) [Inter]-1st& -0.146& -0.376&\n-0.074\\\\ Cr [CrAs]-3rd& 3.299& 3.456& 3.173\\\\\nCr [Inter]-3rd& 3.321& 3.176& 3.464\\\\ In(Cd) [Inter]-3rd\n& -0.014& -0.048& -0.006 \\\\ \\hline\n\n\\underline{\\bf In(Cd) at Void1 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\ In(Cd)-imp & 0.068 & 0.004 & 0.033\\\\ As(Se)\n[CrAs]-1st & -0.238 & -0.318 & -0.283\\\\ As(Se) [Inter]-1st &\n-0.130 & -0.269 & -0.059\\\\ Cr [CrAs]-2nd & 3.429 & 3.571 &\n3.590\\\\ Cr [Inter]-2nd & 3.427 & 3.272 & 3.262 \\\\In(Cd)\n[Inter]-2nd & -0.002 & -0.032& 0.004\\\\ \\hline\n\n\\underline{\\bf In(Cd) at As(Se) site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2\n\\\\ In(Cd)-imp & 0.100& 0.148& 0.188\\\\ Cr-1st& 3.590&\n3.590& 3.746\\\\ In(Cd)-1st& -0.024& -0.001& 0.019\\\\ As(Se)\n[CrAs]-3rd -0.414& -0.421& -0.416\\\\ As(Se) [Inter]-3rd\n&-0.209& -0.384& -0.096\\\\ As(Se) [SC]-3rd& -0.040& -0.032&\n-0.023\\\\ \\hline\n\n\\underline{\\bf In(Cd) at Void2 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2 \\\\ In(Cd)-imp & -0.041& 0.088& 0.063\\\\ Cr-1st &\n3.185& 3.256& 3.446\\\\ In(Cd)-1st& -0.006& -0.020&\n-0.007\\\\ As(Se) [CrAs]-2nd& -0.398& -0.362& -0.475\\\\ As(Se)\n[Inter]-2nd & -0.176& -0.288& -0.102\\\\ As(Se) [SC]-2nd&\n-0.044& -0.039& -0.067\\\\ \\hline\n\n\\underline{\\bf In(Cd) at Void3 site} & CrAs\/InAs & CrAs\/CdSe-1 &\nCrAs\/CdSe-2\n\\\\ In(Cd)-imp & 0.058& -0.008& 0.045\\\\ As(Se) [Inter]-1st&\n-0.073& -0.230& -0.016\\\\ As(Se) [SC]-1st &-0.011&\n-0.028& -0.011 \\\\Cr [Inter]-2nd& 3.473& 3.286& 3.638\\\\\nIn(Cd) [Inter]-2nd& 0.001& -0.031& 0.004\\\\ In(Cd) [SC]-2nd &\n-0.003& 0.002& -0.038\\\\\n\n \\hline \\hline\n\\end{tabular}\n\\label{table6}\n\\end{table}\n\n\\section{In(Cd) impurities}\\label{sec6}\n\nFinally, in the last section we will present our results\nconcerning the case of In, for the CrAs\/InAs interface, and Cd,\nfor both CrAs\/CdSe interfaces, impurities at various sites. All\nthree interfaces show similar behavior and thus in figure\n\\ref{fig9} we present the DOS for all possible In impurities for\nthe CrAs\/InAs multilayer. We should note that with respect to the\nconservation of the half-metallicity this is the most interesting\ncase since for the other two CrAs\/CdSe interfaces the\nhalf-metallic character is conserved for all cases under study. In\ntable \\ref{table6} we have gathered the atom-resolved spin moments\nfor all cases under study and as it can be easily deduced from the\ntable the variation of the spin moments for the same position of\nthe In(Cd) impurity is similar for all three interfaces and thus\nwe will restrict our discussion to the CrAs\/InAs case.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{lekk-fig9.eps}\n\\caption{(Color online) In the case of In(Cd) impurities at\nvarious interfacial sites, we present the atom-resolved DOS of the\nimpurity atom and its neighbors. Note that the In(Cd) DOS has been\nmultiplied by a factor of 5 or 10 in all cases. Details as in\nfigure \\ref{fig2}. \\label{fig9}}\n\\end{figure}\n\nWe expect that the most frequent case to occur would be the In\nimpurity at the Cr site since such an impurity does not disrupt\nthe zinc-blende structure. In atoms have only two valence\nelectrons occupying the deep-energy-lying $s$-states and thus for\nthe energy window which we examine the $p$-states, which we\nobserve, have their origin at the nearest As neighbors whose\n$p$-states penetrate in the In sites (Cd has only one valence\n$s$-electron). Thus the In impurity acts similarly to a Void,\nalthough it does not lead to such large reorganization of the\ncharge of the neighboring atoms, leading to slightly larger spin\nmoment of the neighboring atoms with respect to the perfect\ninterfaces as shown in table \\ref{table6}. Due to the small weight\nof the In $p$-states we have multiplied the corresponding DOS with\na factor 5 or 10 in figure \\ref{fig9} to make it visible. With\nrespect to the case of Void impurity at the Cr site, here the\nshift of the bands of the nearest-neighboring As atoms is smaller\nkeeping the half-metallic character of the interface although the\ngap is considerably shrinking.\n\nWhen the In impurity is located at the Void1 site, the disturbance\nof the lattice is smaller with respect to the case just presented,\nalthough both Cr and Void1 sites have the same nearest-neighbors\nand as shown in figure \\ref{fig9} the width of the gap remains\nunchanged. Due to the negligible weight of the In $p$ states also\nthe occurrence of In impurities at Void2 and Void3 sites leads to\na slight variation of the spin moments but the half-metallicity is\npreserved and the gap retains a large width. To conclude we should\ndiscuss also the case of In impurities at As sites. As atoms at\nthe interface have two Cr atoms as nearest neighbors and the\nhybridization between the As $p$- and Cr $t_{2g}$-orbitals is\nstrong. The substitution of an As atom by an In one leads to\nreduced hybridization for the Cr orbitals and Cr atoms at the\ninterface regain the charge participating at the bonds with the\nmissing As atom. This extra charge is accommodated at the Cr\nspin-up states leading to larger spin magnetic moments of the Cr\natoms at the interface which now are about 3.59 $\\mu_B$ instead of\n3.25 $\\mu_B$ in the case of the perfect CrAs\/InAs interface\npresented in table \\ref{table1}. The In impurity atom and its\nnearest-neighboring In atoms have states within the gap, as shown\nin figure \\ref{fig9} but if we take into account that we have\nmultiplied the In DOS by 10 their real weight at the Fermi level\nis negligible with respect to the Cr majority-spin DOS. Thus we\ncan safely consider that the half-metallicity is conserved\nalthough as shown by the Cr DOS, the gap in the minority-spin band\nseriously shrinks and the Fermi level is near the right edge of\nthe gap.\n\n\n\\section{Conclusion}\\label{sec7}\n\nWe have studied using the Korringa-Kohn-Rostoker method the\nappearance of single impurities at interfaces between the\nhalf-metallic ferromagnet CrAs and the binary InAs and CdSe\nsemiconductors. In the case of bulk CrAs studied in reference\n\\cite{Pouliasis} we had shown that most impurities affect the\nhalf-metallic character of CrAs inducing states either at the\nedges of the gap or in the middle of the gap. But multilayers are\nvery thin as experiments show \\cite{Mizuguchi}\n and thus we cannot use the impurity calculations for the bulk to\n discuss the case of interfaces. We have studied al possible\n single impurities at the interfaces. Contrary to the bulk systems\n almost all defects were not affecting the half-metallic character\n of the perfect interfaces. The only exception were Void\n impurities at Cr or In(Cd) sites. The missing Cr or In(Cd) atom\n leads to a reorganization of the charge of the surrounding atoms\n and as a result the $p$ bands of the nearest neighboring As(Se)\n atom shift to higher energies crossing the Fermi level and\n leading to loss of the half-metallicity. This is the opposite\n behavior that the one exhibited by the Void impurities at Cr site in the\n bulk CrAs alloy where the DOS remained almost unchanged\n \\cite{Pouliasis}. The different behavior of the Void impurities\n should be attributed to the lower dimensionality of the\n interfaces with respect to the bulk. But Void impurities are\n Schottky-type and we expect them to exhibit high formation\n energies and thus their number should be small in the\n experimental systems. Thus contrary to the bulk CrAs and\n eventually thick films showing a bulk-like behavior,\n impurities are expected to affect only marginally the\n half-metallic character of the interfaces in the case of thin multilayers\nand the latter ones are promising for spintronic devices.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and known results}\n\n\nThe study of random field $\\xi_x$ from a lattice graph $\\mathbb L$ (usually $\\mathbb Z^d$\nor a Cayley tree $\\Gamma^k$) to a measure space $(E, \\mathcal E)$ is a\ncentral component of ergodic theory and statistical physics.\n\nIn many classical models\nfrom physics (e.g., the Ising model, the Potts model), $E$ is a finite set (i.e., with a finite underlying measure $\\lambda$), and $\\xi_x$\nhas a physical interpretation as the spin of a particle\nat location $x$ in a crystal lattice.\n\nFollowing \\cite{BiKo}, \\cite{BEvE}, \\cite{FS97}, \\cite{Ge}, \\cite{HKLR}, \\cite{HKa}, \\cite{HKb}, \\cite{KS}, \\cite{Sh}, \\cite{Z1}, let us\ngive basic definitions and some known facts related to (gradient) Gibbs measures.\n\n{\\bf $\\sigma$-algebra, Hamiltonian.}\nIn general, $(E, \\mathcal E)$ is a space with an infinite underlying measure $\\lambda$ (i.e. $\\mathbb L$ with counting measure),\nwhere $\\mathcal E$ is the Borel $\\sigma$-algebra of $E$ and $\\xi_x$ usually\nhas a physical interpretation as the spatial position of a particle at location $x$\nin a lattice. In \\cite{FS97} first such models were considered.\n\nThe prime examples of unbounded spin systems are harmonic\noscillators. Another example is the Ginzburg-Landau\ninterface model; which is obtained from the harmonic oscillators \\cite{Ge}, \\cite{Sh}.\n\nDenote by $\\Omega$ the set of functions from $\\mathbb L$ to $E$, such a function also is called a configuration.\n\nAssume random field $(\\xi_x)_{x\\in \\mathbb L}$ on $\\Omega$ given as the projection onto the coordinate $x\\in \\mathbb L$:\n$$\\xi_x(\\omega)=\\omega(x)=\\omega_x, \\ \\ \\omega \\in \\Omega.$$\n\nFor $\\Lambda\\subset \\mathbb L$, denote by $\\mathcal F_\\Lambda$\nthe smallest $\\sigma$-algebra with respect to which $\\xi_x$ is measurable for all $x\\in \\Lambda$. Write $\\mathcal T_\\Lambda=\\mathcal F_{\\mathbb L\\setminus\\Lambda}$.\n\nA subset of $\\Omega$,\nis called a cylinder set if it belongs to $\\mathcal F_\\Lambda$ for some finite set $\\Lambda \\subset \\mathbb L$.\n\nLet $\\mathcal F$ be the smallest\n$\\sigma$-algebra on $\\Omega$ containing the cylinder sets.\n\nWrite $\\mathcal T$ for the tail-$\\sigma$-algebra, i.e., intersection of $\\mathcal T_\\Lambda$\nover all finite subsets $\\Lambda$ of $\\mathbb L$ the sets in $\\mathcal T$ are called tail-measurable sets.\n\nAssume that we are given a family of measurable potential\nfunctions $\\Phi_\\Lambda:\\Omega\\to \\mathbb R\\cup \\{\\infty\\}$ (one for each finite subset $\\Lambda$ of $\\mathbb L$)\n each $\\Phi_\\Lambda$ is $\\mathcal F_\\Lambda$\nmeasurable.\n\nFor each finite subset $\\Lambda$ of $\\mathbb L$ define a Hamiltonian:\n$$ H_\\Lambda(\\sigma)=\\sum_{S\\subset \\mathbb L:\\atop S\\cap \\Lambda\\ne \\emptyset}\\Phi_{S}(\\sigma),$$\n where the sum is taken over finite subsets $S$.\n\n{\\bf Gibbs Measures.}\nTo define Gibbs measures and gradient Gibbs measures, we will need some additional\nnotation \\cite{Ge}, \\cite{Sh}.\n\nLet $(X, \\mathcal X)$ and $(Y, \\mathcal Y)$ be general measure spaces.\n\nA function $\\pi : \\mathcal X\\times Y\\to [0, \\infty]$ is called a probability kernel from $(Y, \\mathcal Y)$ to $(X, \\mathcal X)$ if\n\\begin{itemize}\n\\item[1.] $\\pi(\\cdot | y)$ is a probability measure on $(X, \\mathcal X)$ for each fixed $y\\in Y$, and\n\\item[2.] $\\pi(A | \\cdot)$ is $\\mathcal Y$-measurable for each fixed $A\\in \\mathcal X$.\n\\end{itemize}\n\nSuch a kernel maps each measure $\\mu$, on $(Y, \\mathcal Y)$ to a measure $\\mu\\pi$ on $(X, \\mathcal X)$ by\n$$\\mu\\pi(A)=\\int \\pi(A | \\cdot)d\\mu$$\n\nThe following is a probability kernel from $(\\Omega, \\mathcal T_\\Lambda)$ to $(\\Omega, \\mathcal F)$:\n$$\\gamma_\\Lambda(A,\\omega)=Z_\\Lambda(\\omega)^{-1}\\int\\exp(-H_\\Lambda(\\sigma_\\Lambda\\omega_{\\Lambda^c}))\n\\mathbf{1}_A(\\sigma_\\Lambda\\omega_{\\Lambda^c})\\nu^{\\otimes\\Lambda}(d\\sigma_\\Lambda),$$\nwhere $\\nu=\\{\\nu(i)>0, i\\in E\\}$ is a counting measure.\n\nA configuration $\\sigma$ has finite energy if $ \\Phi_\\Lambda(\\sigma) < \\infty$ for all finite $\\Lambda$. Moreover, $\\sigma$ is $\\Phi$-admissible\nif each $Z_\\Lambda(\\sigma)$ is finite and non-zero.\n\nGiven a measure $\\mu$ on $(\\Omega, \\mathcal F)$, define a new\nmeasure $\\mu\\gamma_\\Lambda$ by\n$$\\mu\\gamma_\\Lambda(A)=\\int\\gamma_\\Lambda(A, \\cdot)d\\mu$$\n\n\\begin{defn} A probability measure $\\mu$ on $(\\Omega, \\mathcal F)$ is called a Gibbs measure if $\\mu$ is supported on the set of $\\Phi$-admissible configurations in $\\Omega$ and for all finite subset $\\Lambda$ we have\n$$\\mu \\gamma_\\Lambda=\\mu.$$\n\\end{defn}\n{\\bf Gradient Gibbs measure.} For any configuration $\\omega = (\\omega(x))_{x \\in \\mathbb L} \\in E^\\mathbb L$ and edge $b = \\langle x,y \\rangle$ of $\\mathbb L$\nthe \\textit{difference} along the edge $b$ is given by $\\nabla \\omega_b = \\omega_y - \\omega_x$ and $\\nabla \\omega$ is called the \\textit{gradient field} of $\\omega$.\n\nThe gradient spin variables are now defined by $\\eta_{\\langle x,y \\rangle} = \\omega_y - \\omega_x$ for each $\\langle x,y \\rangle$.\n\nThe space of \\textit{gradient configurations} denoted by $\\O^\\nabla$. The measurable structure on the space $\\Omega^{\\nabla}$ is given by $\\sigma$-algebra $$\\mathcal{F}^\\nabla:=\\sigma(\\{ \\eta_b \\, \\vert \\, b \\in \\mathbb L \\}).$$\nNote that $\\mathcal F^\\nabla$ is the subset of\n$\\mathcal F$ containing those sets that are invariant under translations $\\omega\\to \\omega +c$ for $c\\in E$.\n\nSimilarly, we define\n$$\\mathcal T^\\nabla_\\Lambda=\\mathcal T_\\Lambda \\cap \\mathcal{F}^\\nabla, \\ \\ \\mathcal F^\\nabla_\\Lambda=\\mathcal F_\\Lambda \\cap \\mathcal{F}^\\nabla.$$\n\nLet $\\Phi$ be a translation invariant\ngradient potential. Since, given any $A\\in \\mathcal F^\\nabla$, the kernels $\\gamma^\\Phi_\\Lambda(A, \\omega)$ are $\\mathcal F^\\nabla$-measurable\nfunctions of $\\omega$, it follows that the kernel sends a given measure $\\mu$ on $(\\Omega, \\mathcal F^\\nabla)$ to another\nmeasure $\\mu\\gamma_\\Lambda^\\Phi$ on $(\\Omega, \\mathcal F^\\nabla)$.\n\n\\begin{defn} A measure $\\mu$ on $(\\Omega, \\mathcal F^\\nabla)$ is called a gradient Gibbs measure\nif for all finite subset $\\Lambda$ we have\n$$\\mu \\gamma^\\Phi_\\Lambda=\\mu.$$\n\\end{defn}\nNote that, if $\\mu$ is a Gibbs measure on $(\\Omega, \\mathcal F)$, then its restriction to $\\mathcal F^\\nabla$ is a gradient\nGibbs measure.\n\nA gradient Gibbs measure is said to be localized or smooth if it arises\nas the restriction of a Gibbs measure in this way. Otherwise, it is non-localized or\nrough.\n\nIt is known\\cite{Ge}, \\cite[Theorem 8.19.]{FV} that\nmany natural Gibbs measures on $\\mathbb Z^d$ are rough when $d\\in \\{1, 2\\}$.\n\n{\\bf Construction of gradient Gibbs measure on Cayley trees}. Following \\cite{KS}\nwe consider models where spin-configuration $\\omega$ is a function from the\nvertices of the Cayley tree $\\Gamma^k=(V, \\vec L)$ to the set $E= \\Z$, where\n$V$ is the set of vertices and $\\vec L$ is the set of oriented edges (bonds) of the tree\n(see Chapter 1 of \\cite{Ro} for properties of the Cayley tree).\n\nFor nearest-neighboring (n.n.) interaction potential $\\Phi=(\\Phi_b)_b$, where\n$b=\\langle x,y \\rangle$ is an edge, define symmetric transfer matrices $Q_b$ by\n\\begin{equation}\\label{Qd}\nQ_b(\\omega_b) = e^{- \\big(\\Phi_b(\\omega_b) + | \\partial x|^{-1} \\Phi_{\\{x\\}}(\\omega_x) + |\\partial y |^{-1} \\Phi_{\\{y\\}} (\\omega_y) \\big)},\n\\end{equation}\nwhere $\\partial x$ is the set of all nearest-neighbors of $x$ and $|S|$ denotes the number of elements of the set $S$.\n\nDefine the Markov (Gibbsian) specification as\n$$\n\\gamma_\\Lambda^\\Phi(\\sigma_\\Lambda = \\omega_\\Lambda | \\omega) = (Z_\\Lambda^\\Phi)(\\omega)^{-1} \\prod_{b \\cap \\Lambda \\neq \\emptyset} Q_b(\\omega_b).\n$$\n\nIf for any bond $b=\\langle x,y \\rangle$ the transfer operator $Q_b(\\omega_b)$ is\na function of gradient spin variable $\\zeta_b=\\omega_y-\\omega_x$ then the underlying potential $\\Phi$ is called\na \\textit{gradient interaction potential}.\n\n \\emph{Boundary laws} (see \\cite{Z1}) which allow to describe the set $\\mathcal{G}(\\gamma)$ of all Gibbs measures (that are Markov chains on trees).\n\n\\begin{defn}\n\tA family of vectors $\\{ l_{xy} \\}_{\\langle x,y \\rangle \\in \\vec L}$ with $l_{xy}=\\left(l_{xy}(i): i\\in \\Z\\right) \\in (0, \\infty)^\\Z$ is called a {\\em boundary law for the transfer operators $\\{ Q_b\\}_{b \\in \\vec L}$} if for each $\\langle x,y \\rangle \\in \\vec L$ there exists a constant $c_{xy}>0$ such that the consistency equation\n\t\\begin{equation}\\label{eq:bl}\n\tl_{xy}(i) = c_{xy} \\prod_{z \\in \\partial x \\setminus \\{y \\}} \\sum_{j \\in \\Z} Q_{zx}(i,j) l_{zx}(j)\n\t\\end{equation}\n\tholds for every $i \\in \\Z$.\n\nA boundary law is called {\\em $q$-periodic} if $l_{xy} (i + q) = l_{xy}(i)$ for every oriented edge $\\langle x,y \\rangle \\in \\vec L$ and each $i \\in \\Z$.\n\t\n\\end{defn}\n\nIt is known that there is a one-to-one correspondence between boundary laws\nand tree-indexed Markov chains if the boundary laws are {\\em normalisable} in the sense of Zachary \\cite{Z1}:\n\n\\begin{defn} A boundary law $l$ is said to be {\\em normalisable} if and only if\n\\begin{equation}\\label{Norm}\n\\sum_{i \\in \\Z} \\Big( \\prod_{z \\in \\partial x} \\sum_{j \\in \\Z} Q_{zx}(i,j) l_{zx}(j) \\Big) < \\infty\n\\end{equation} at any $x \\in V$.\n\\end{defn}\nFor any $\\Lambda \\subset V$ we define its outer boundary as\n\\begin{equation*}\n\\partial \\Lambda := \\{ x \\notin \\Lambda : \\langle x,y\\rangle \\, \\mbox{ for some } \\, y \\in \\Lambda\\}.\n\\end{equation*}\nThe correspondence now reads the following:\n\n\\begin{thm} \\cite{Z1}\nFor any Markov specification $\\gamma$ with associated family of transfer matrices $(Q_b)_{b \\in L}$ we have\n\\begin{enumerate}\n\\item Each {\\it normalisable} boundary law $(l_{xy})_{x,y}$ for $(Q_b)_{b \\in L}$ defines a unique tree-indexed Markov chain $\\mu \\in \\mathcal{G}(\\gamma)$ via the equation given for any connected set $\\Lambda \\subset V$\n\\begin{equation}\\label{BoundMC}\n\\mu(\\sigma_{\\Lambda \\cup \\partial \\Lambda}=\\omega_{\\Lambda \\cup \\partial \\Lambda}) = (Z_\\Lambda)^{-1} \\prod_{y \\in \\partial \\Lambda} l_{y y_\\Lambda}(\\omega_y) \\prod_{b \\cap \\Lambda \\neq \\emptyset} Q_b(\\omega_b),\n\\end{equation}\nwhere for any $y \\in \\partial \\Lambda$, $y_\\Lambda$ denotes the unique $n.n.$ of $y$ in $\\Lambda$.\n\\item Conversely, every tree-indexed Markov chain $\\mu \\in \\mathcal{G}(\\gamma)$ admits a representation of the form (\\ref{BoundMC}) in terms of a {\\it normalisable} boundary law (unique up to a constant positive factor).\n\\end{enumerate}\n\\end{thm}\n\n\nThe Markov chain $\\mu$ defined in \\eqref{BoundMC} has the transition probabilities\n\\begin{equation}\\label{last}\nP_{xy}(i,j)=\\mu(\\s_y = j \\mid \\s_x =i)\n= \\frac{l_{yx}(j) Q_{yx}(j, i)}{\\sum_s l_{yx}(s) Q_{yx}(s, i)}.\n\\end{equation}\nThe expressions \\eqref{last} may exist even in situations where the underlying boundary\nlaw $(l_{xy})_{x,y}$ is not normalisable. However, the Markov chain given by (\\ref{last}), in general, does not have an invariant probability measure.\nTherefore in \\cite{HKLR}, \\cite{HKa}, \\cite{HKb}, \\cite{KS}\nsome non-normalisable boundary laws are used to give gradient Gibbs measures.\n\n Now we give some results of above mentioned papers. Consider a model on Cayley tree $\\Gamma^k=(V, \\vec L)$, where the spin takes values in\nthe set of all integer numbers $\\mathbb Z$. The set of all configurations is $\\Omega:=\\mathbb Z^V$.\n\nFor $\\Lambda\\subset V$, fix a site $w \\in \\Lambda$. If the boundary law $l$ is assumed to be $q$-periodic, then take $s \\in \\mathbb{Z}_q=\\{0,1,\\dots,q-1\\}$ and define probability measure $\\nu_{w,s}$ on $\\mathbb{Z}^{\\{b \\in \\vec L \\mid b \\subset \\Lambda\\}}$ by\n$$\n\\nu_{w,s}(\\eta_{\\Lambda \\cup \\partial \\Lambda}=\\zeta_{\\Lambda \\cup \\partial \\Lambda})=\n$$\n$$Z^\\Lambda_{w,s}\\prod_{y \\in \\partial \\Lambda} l_{yy_\\L}\\Bigl (T_q(\ns+\\sum_{b\\in \\Gamma(w,y)}\\zeta_b)\n\\Bigr) \\prod_{b \\cap \\Lambda \\neq \\emptyset}Q_b(\\zeta_b),\n$$\nwhere $Z^\\Lambda_{w,s}$ is a normalization constant, $\\Gamma(w,y)$ is the unique path from $w$ to $y$\nand $T_q: \\mathbb{Z} \\rightarrow \\mathbb{Z}_q$ denotes the coset projection.\n\n\\begin{thm} \\cite{KS}\n\tLet $(l_{})_{ \\in \\vec L}$ be any $q$\n-periodic boundary law to some gradient interaction potential.\nFix any site $w \\in V$ and any class label $s \\in \\mathbb{Z}_q$. Then\n$$\t\\nu_{w,s}(\\eta_{\\Lambda \\cup \\partial \\Lambda}=\\zeta_{\\L\\cup\\partial\\L})\n\t=$$\n\\begin{equation}\nZ^\\Lambda_{w,s} \\prod_{y \\in \\partial \\Lambda} l_{yy_\\L}\\Bigl (T_q(\n\ts+\\sum_{b\\in \\Gamma(w,y)}\\zeta_b)\n\t\\Bigr) \\prod_{b \\cap \\Lambda \\neq \\emptyset}\n\tQ_b(\\zeta_b),\n\t\\end{equation}\ngives a consistent family of probability measures on the gradient space $\\Omega^\\nabla$.\nHere $\\Lambda$ with $w \\in \\L \\subset V$ is any finite connected set,\n$\\zeta_{\\L\\cup\\partial\\L} \\in \\Z^{\\{b \\in \\vec L \\mid b \\subset (\\L \\cup \\partial\\L)\\}}$ and $Z^\\Lambda_{w,s}$ is a normalization constant.\\\\\n\\end{thm}\n\tThe measures $\\nu_{w,s}$ will be called pinned gradient measures.\n\n\nIf $q$-periodic boundary law and the underlying potential are translation invariant then it is possible to obtain\nprobability measure $\\nu$ on the gradient space by mixing the pinned gradient measures:\n\n\\begin{thm}\\cite{KS}\t\nLet a $q$-periodic boundary law $l$ and its gradient interaction potential are translation invariant.\nLet $\\Lambda \\subset V$ be any finite connected set and let $w\\in \\Lambda$ be any vertex. Then the measure $\\nu $ with marginals given by\n\\begin{equation}\n\\nu (\\eta_{\\L\\cup\\partial \\L} = \\zeta_{\\L\\cup\\partial\\L}) = Z_\\L \\ \\left(\\sum_{s\\in\\Z_q} \\prod_{y \\in \\partial\\L} l \\big(s + \\sum_{b \\in \\Gamma(w,y)} \\zeta_{b}\\big) \\right)\\prod_{b \\cap \\L \\neq \\emptyset} Q(\\zeta_b),\n\\end{equation}\nwhere $Z_\\L$ is a normalisation constant, defines a translation invariant gradient Gibbs measure on $\\Omega^\\nabla$.\n\\end{thm}\n\n{\\bf SOS model.} The (formal) Hamiltonian of the SOS model is\n\\begin{equation}\\label{nu1}\n H(\\omega)=-J\\sum_{\\langle x,y\\rangle \\in L}\n|\\omega_x-\\omega_y|, \\ \\ \\omega \\in\\Omega,\n\\end{equation}\nwhere $J \\in \\mathbb R_+$ is a constant.\n\nIn \\cite{HKLR}, using Theorem 3 some gradient Gibbs measures are found.\n\n Let $\\beta>0$ be inverse temperature and $\\theta:= \\exp(-J\\beta)<1$.\n The transfer operator $Q$ then reads $Q(i-j)=\\theta^{\\vert i-j \\vert }$ for any $i,j \\in \\mathbb{Z}$,\n and a translation invariant boundary law, denoted by $\\mathbf z$, is any positive function on $\\mathbb{Z}$\n solving the consistency equation, whose values we will denote by $z_i$ instead of $z(i)$.\n By definition of the boundary law it is only unique up to multiplication with any positive prefactor.\n Hence we may choose this constant in a way such that we have $z_0=1$.\n\n Set $\\mathbb{Z}_0:= \\mathbb{Z} \\setminus \\{0\\}$. Then the boundary law equation (for translation-invariant case, i.e.\n $l_b\\equiv l$, for all $b\\in L$) reads\n\\begin{equation}\\label{nu11}\nz_i=\\left({\\theta^{|i|}+\n\\sum_{j\\in \\mathbb Z_0}\\theta^{|i-j|}z_j\n\\over\n1+\\sum_{j\\in \\mathbb Z_0}\\theta^{|j|}z_j}\\right)^k, \\ \\ i\\in\\mathbb Z_0.\n \\end{equation}\n\nLet $\\mathbf z(\\theta)=(z_i=z_i(\\theta), i\\in \\mathbb Z_0)$ be a solution to (\\ref{nu11}).\n\nDenote $u_i=\\sqrt[k]{z_i}$ and assume $u_0=1$.\n\n\\begin{pro} \\cite{HKLR} If $z_0=1$ (i.e. $u_0=1$) then the equation (\\ref{nu11}) is equivalent to the following\n\\begin{equation}\\label{V}\nu_i^k={u_{i-1}+u_{i+1}-\\tau u_i\\over u_{-1}+u_{1}-\\tau}, \\ \\ i\\in \\mathbb Z,\n\\end{equation}\nwhere $\\tau=\\theta^{-1}+\\theta$.\n\\end{pro}\n\nIn general, solutions of (\\ref{V}) are not known. But in class of periodic solutions, some results are obtained.\nThe following theorem is proved for $k=2$ and $4$-periodic boundary laws:\n \t\n\\begin{thm} \\cite{HKLR} For the SOS model (\\ref{nu1}) on the binary tree (i.e. $k=2$) with parameter $\\tau=\\theta+\\theta^{-1}$ the following assertions hold\n\t\\begin{itemize}\n\t\t\\item[1.] If $\\tau \\leq 4$ then there is precisely one GGM associated to a 4-periodic boundary law.\n\t\t\\item[2.] If $4< \\tau \\leq 6$ then there are precisely two GGMs.\n\t\t\\item[3.] If $6<\\tau < 2+2\\sqrt{5}$ then there are precisely three GGMs.\n\t\t\\item[4.] If $\\tau \\geq 2+2\\sqrt{5}$ then there are precisely four such measures.\n\t\\end{itemize}\n\\end{thm}\n\nThe following theorem is proved for any $k\\geq 2$ and $3$-periodic boundary laws.\n\nDenote\n$$\\tau_0:={2k+1\\over k-1}.$$\n\n\\begin{thm} \\cite{HKLR} For the SOS-model on the $k$-regular tree, $k \\geq 2$, with parameter $\\tau$ there is $\\tau_c$ such that\n$0<\\tau_c<\\tau_0$ and the following holds:\n\\begin{itemize}\n\t\\item[1.] If $\\tau<\\tau_c$ then there is no any GGM corresponding to\n\ta nontrivial 3-periodic boundary.\n\t\\item[2.] At $\\tau=\\tau_c$ there is a unique GGM corresponding to a\n\tnontrivial $3$-periodic boundary law.\n\t\\item[3.] For $\\tau>\\tau_c$, $\\tau\\ne \\tau_0$ (resp. $\\tau=\\tau_0$) there are exactly\n\ttwo such (resp. one) GGMs.\n\\end{itemize}\t\n\n The GGMs described above are all different from the GGMs mentioned in Theorem 4.\n\\end{thm}\n\n{\\bf General case.}\n Assume that the transfer operator $\\{Q_b\\}_{b\\in \\mathbb L}$, defined in (\\ref{Qd}), is summable, i.e.\n$$\\sum_{i\\in \\mathbb Z}Q_b(i) <\\infty \\ \\ \\mbox{for all} \\ \\ b\\in \\mathbb L.$$\n\nThe following is the main result of \\cite{HKa}:\n\n\\begin{thm} For any summable $Q$ and any degree $k\\geq 2$ there is a finite period\n$q_0(k)$ such that for all $q\\geq q_0(k)$ there are GGMs of\nperiod $q$ which are not translation invariant.\n\\end{thm}\n\nMoreover, in \\cite{HKb} the authors provided general conditions in terms of the\nrelevant $p$-norms of the associated transfer operator $Q$ which ensure the existence of\na countable family of proper Gibbs measures. The existence of delocalized GGMs is proved, under natural conditions on $Q$.\nThis implies coexistence of both types of measures for large classes of models including the SOS-model,\nand heavy-tailed models arising for instance for potentials of logarithmic growth.\n\n\\section{4-periodic boundary laws for $k\\geq 2$}\n\nIn this section our goal is to find solutions of (\\ref{V}) which have the form\n\\begin{equation}\\label{up}\nu_n=\\left\\{ \\begin{array}{lll}\n1, \\ \\ \\mbox{if} \\ \\ n=2m,\\\\[2mm]\na, \\ \\ \\mbox{if} \\ \\ n=4m-1, \\ \\ m\\in \\mathbb Z\\\\[2mm]\nb, \\ \\ \\mbox{if} \\ \\ n=4m+1,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $a$ and $b$ some positive numbers.\n\nThen from (\\ref{V}) for $a$ and $b$ we get the following system of equations\n\\begin{equation}\\label{ab}\n\\begin{array}{ll}\n(a+b-\\tau)b^k+\\tau b-2=0\\\\[2mm]\n(a+b-\\tau)a^k+\\tau a-2=0.\n\\end{array}\n\\end{equation}\n\nThe case $k=2$ is fully analyzed in \\cite{HKLR} and the following is proved\n\\begin{pro}\\label{pps} For $k=2$ the periodic solutions of the form (\\ref{up}) (i.e. solutions of the system (\\ref{ab}))\ndepend on the parameter $\\tau=2\\cosh(\\beta)$ in the following way.\n\t\\begin{enumerate}\n\t\t\\item If $\\tau \\leq 4$ then there is a unique solution.\n\t\t\\item If $4<\\tau \\leq 6$ then there are exactly two solutions.\n\t\t\\item If $6<\\tau < 2+2\\sqrt{5}$ then there are exactly four solutions.\n\t\t\\item If $\\tau \\geq 2+2\\sqrt{5}$ then there are exactly five solutions.\n\t\\end{enumerate}\n\twhere explicit formula of each solution is found.\n\\end{pro}\n\nNow we reduce the system (\\ref{ab}) to a polynomial equation with one unknown $a$. To do this\nfrom the first (resp. second) equation of (\\ref{ab}) find $a$ (resp. $b$):\n\n\\begin{equation}\\label{ab1}\n\\begin{array}{ll}\na=f(b):=\\tau-b+(2-\\tau b)b^{-k}\\\\[2mm]\nb=f(a).\n\\end{array}\n\\end{equation}\nThus the system (\\ref{ab}) is reduced to\n\\begin{equation}\\label{ab2}\na=f(f(a)).\n\\end{equation}\nNote that solutions of $a=f(a)$ are solutions to (\\ref{ab2}) too. It is easy to see that $a=f(a)$ is equivalent to\n\\begin{equation}\\label{ab3}\nQ(a):=2a^{k+1}-\\tau a^k+\\tau a-2=0\n\\end{equation}\n\n\nThe equation (\\ref{ab3}) has the solution $a=1$ independently of the parameters $(\\tau, k)$. Dividing both sides of (\\ref{ab3}) by $a-1$\nwe get\n\\begin{equation}\\label{uy22}\n2a^k+(2-\\tau)(a^{k-1}+a^{k-2}+\\dots+a)+2=0.\n\\end{equation}\nThe following lemma gives the number of solutions to equation (\\ref{uy22}) (compare with Lemma 4.7 in \\cite{HKLR}):\n\\begin{lemma}\\label{l6} For each $k\\geq 2$,\n\tthere is exactly one critical value of $\\tau$, i.e., $\\tau_c=\\tau_c(k):=2\\cdot {k+1\\over k-1}$, such that\n\t\\begin{enumerate}\n\t\t\\item if $\\tau<\\tau_c$ then (\\ref{uy22}) has no positive solution;\n\t\t\\item if $\\tau=\\tau_c$ then the equation has a unique solution $a=1$;\n\t\t\\item if $\\tau>\\tau_c$,\tthen it has exactly two solutions (both different from 1);\n\t\t\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFrom (\\ref{uy22}) we get\n\t\\[\\tau=\\psi_k(a):=2+\\frac{2(a^k+1)}{a^{k-1}+a^{k-2}+\\dots+a}. \\]\n\tWe have $\\psi_k(a)>2$, $a>0$ and $\\psi_k'(a)=0$ is equivalent to\n\t\\begin{equation}\\label{ho}\n\t\\sum_{j=1}^{k-1}(k-j)a^{k+j-1}-\\sum_{j=1}^{k-1}ja^{j-1}=0.\n\t\\end{equation}\n\tThe last polynomial equation has exactly one positive solution,\n\tbecause signs of its coefficients changed only one time,\n\tand at $a=0$ it is negative, i.e. -1 and at $a=+\\infty$ it is positive.\n\tMoreover, this unique solution is $a=1$, because putting $a=1$ in (\\ref{ho}) we get\n $$\\sum_{j=1}^{k-1}(k-j)-\\sum_{j=1}^{k-1}j=\\sum_{j=1}^{k-1}k-2\\sum_{j=1}^{k-1}j = k(k-1)-2\\cdot {k(k-1)\\over 2}=0.$$\n\n Thus $\\psi_k(a)$ has unique minimum\n\tat $a=1$, and $\\lim_{a\\to 0}\\psi_k(a)=\\lim_{a\\to +\\infty}\\psi_k(a)=+\\infty$.\nConsequently,\n\t\\[\\tau_c=\\tau_c(k)=\\min_{a>0}\\psi_k(a)=\\psi_k(1)=2\\cdot {k+1\\over k-1}.\\]\n\tThese properties of $\\psi_k(a)$ completes the proof.\n\\end{proof}\n\\begin{rk} The equation (\\ref{uy22}) was also considered in \\cite{BH}.\nLemma \\ref{l6} improves their result (see Theorem 5.2 of \\cite{BH}), because we found explicit\nformula for the critical value $\\tau_c$.\n\n\\end{rk}\nNow we want to find solutions of (\\ref{ab2}) which are different from solutions of $a=f(a)$ (i.e., $Q(a)=0$).\n\nBy simple calculations the equation (\\ref{ab2}) can be rewritten as\n\\begin{equation}\\label{ab4}\nP(a):=(2-\\tau a)[2-\\tau a+\\tau a^k-a^{k+1}]^k-a^{k^2}[\\tau a^{k+1}+(2-\\tau^2)a^k+\\tau^2a-2\\tau]=0.\n\\end{equation}\nRecall that $Q(a)$ divides $P(a)$. Now we shall find ${P(a)\\over Q(a)} $.\nIt is easy to see that $P(a)$ can be written as\n$$P(a)=(2-\\tau a)[a^{k+1}-Q(a)]^k-a^{k^2}[2a^k-\\tau a^{k+1}+\\tau Q(a)]$$\n$$=(2-\\tau a)\\sum_{j=0}^{k}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j}(a)-a^{k^2+k}(2-\\tau a)-\\tau a^{k^2}Q(a)$$\n$$=(2-\\tau a)\\sum_{j=0}^{k-1}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j}(a)+(2-\\tau a)a^{k^2+k}-a^{k^2+k}(2-\\tau a)-\\tau a^{k^2}Q(a)$$\n$$=(2-\\tau a)\\sum_{j=0}^{k-1}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j}(a)-\\tau a^{k^2}Q(a)$$\n$$=Q(a)\\left((2-\\tau a)\\sum_{j=0}^{k-1}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j-1}(a)-\\tau a^{k^2}\\right).$$\n\nConsequently, the equation (\\ref{ab2}) in case $Q(a)\\ne 0$ is reduced to\n\\begin{equation}\\label{kt}\nU(a):=\\tau a^{k^2}-(2-\\tau a)\\sum_{j=0}^{k-1}(-1)^{k-j}{k\\choose j}a^{(k+1)j}Q^{k-j-1}(a)=0.\n\\end{equation}\nBut $U(a)$ may have some roots coinciding with roots of $Q(a)$. This is result of the following lemma.\n\n\\begin{lemma} Let $a=\\hat a$ be a root of $Q(a)$. Then $\\hat a$ is a root of $U(a)$ iff\n$\\hat a={2k\\over \\tau (k-1)}$ and $\\tau$ satisfies\n\\begin{equation}\\label{ha}\n(k-1)^k\\tau^{k+1}-(k-1)2^{k-1}k^k\\tau^2+(2k)^{k+1}=0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} For the root $\\hat a$ we have $Q(\\hat a)=0$. Therefore from (\\ref{kt}), i.e. $U(\\hat a)=0$, we get\n$$\\tau \\hat a^{k^2}+(2-\\tau \\hat a)k \\hat a^{k^2-1}=0 \\ \\ \\Leftrightarrow \\ \\ \\tau \\hat a+(2-\\tau \\hat a)k=0 \\ \\ \\Leftrightarrow \\ \\ \\hat a={2k\\over \\tau (k-1)}.$$\n\nThen $Q(\\hat a)=Q\\left({2k\\over \\tau (k-1)}\\right)=0$ gives (\\ref{ha}).\n\n\\end{proof}\n\n\\begin{lemma}\\label{s3} For each fixed $k\\geq 2$ the equation (\\ref{ha}) has exactly two\nsolutions $\\tau_1(k)={2k\\over k-1}$ and $\\tau_2(k)>{2(k+1)\\over k-1}$.\n\\end{lemma}\n\\begin{proof} It is easy to see that $\\tau={2k\\over k-1}$ is a solution to (\\ref{ha}),\nfor any $k\\geq 2$. Moreover, the corresponding\n$\\hat a$ is $1$.\n\nDenote $L=(k-1)\\tau-2k$\n then from (\\ref{ha}) we get\n$$0=(L+2k)^{k+1}-2^{k-1}k^k(L+2k)^2+(k-1)(2k)^{k+1}$$\n$$=\\sum_{j=0}^{k+1}{k+1\\choose j}L^{k+1-j}(2k)^j-2^{k-1}k^kL^2-(2k)^{k+1}L-2^{k+1}k^{k+2}+(k-1)(2k)^{k+1}\n$$ $$=\\sum_{j=0}^{k}{k+1\\choose j}L^{k+1-j}(2k)^j+(2k)^{k+1}-2^{k-1}k^kL^2-(2k)^{k+1}L-2^{k+1}k^{k+2}+(k-1)(2k)^{k+1}$$\n$$=\\sum_{j=0}^{k}{k+1\\choose j}L^{k+1-j}(2k)^j-2^{k-1}k^kL^2-(2k)^{k+1}L$$\n$$=L\\left(\\sum_{j=0}^{k}{k+1\\choose j}L^{k-j}(2k)^j-2^{k-1}k^kL-(2k)^{k+1}\\right)$$\n$$=L\\left(\\sum_{j=0}^{k-2}{k+1\\choose j}L^{k-j}(2k)^j+{k(k-1)\\over 2}L-(k-1)(2k)^{k}\\right).$$\nThus $L=0$ (i.e. $\\tau=\\tau_1(k)$) or\n\\begin{equation}\\label{L}\\sum_{j=0}^{k-2}{k+1\\choose j}L^{k-j}(2k)^j+{k(k-1)\\over 2}L-(k-1)(2k)^{k}=0.\n\\end{equation}\nBy Lemma \\ref{l6} we know that $Q(a)=0$ has a solution different\nfrom 1 iff $\\tau>\\tau_c={2(k+1)\\over k-1}$. Therefore $L=(k-1)\\tau-2k>(k-1)\\cdot {2(k+1)\\over k-1}-2k=2>0$.\nThe polynomial equation (\\ref{L}) has exactly one positive solution, denoted by $L^*$,\n\tbecause signs of its coefficients changed only one time. Moreover, we have $L^*>2$, because at $L=2$ the LHS of (\\ref{L}) is negative:\n$$2^k[(k+1)^{k+1}-(3k+1)k^k]<0$$ and at $L=+\\infty$ it is positive. Thus $\\tau_2(k)={L^*+2k\\over k-1}$.\n\\end{proof}\n\\begin{ex} For $k=2,3,4$ we have\n$$\\tau_1(2)=4, \\ \\ \\tau_2(2)=2+2\\sqrt{5}.$$\n$$\\tau_1(3)=3, \\ \\ \\tau_2(3)=3\\sqrt{2}.$$\n$$\\tau_1(4)=8\/3, \\ \\ \\tau_2(4)\\approx 3.497.$$\n\n\\end{ex}\n\n\\begin{rk} It seems impossible to solve equation (\\ref{kt}) for each $k\\geq 2$. But one can use numerical\nmethods to give some its solutions for concrete values of parameters.\nTherefore, one can try to solve it for small values of $k$. In \\cite{HKLR} the case $k=2$ is fully analyzed.\nBelow we shall consider the case $k=3$. Cases $k\\geq 4$ remains open.\n\\end{rk}\n\n{\\bf Case $k=3$}. In this case the equation (\\ref{kt}) has the form\n\n$$g(a,\\tau):=(\\tau^2+2)a^8-(\\tau^2+2)\\tau a^7+\\tau^2 a^6+2(\\tau^2+2)\\tau a^5$$\n\\begin{equation}\\label{g}-4(2\\tau^2+1)a^4-(\\tau^2-8)\\tau a^3+6\\tau^2a^2-12\\tau a+8=0.\n\\end{equation}\nIt is well known (see \\cite{Pra}, p.28) that the number of positive\nroots of the polynomial (\\ref{g}) does not exceed the number of sign\nchanges of its coefficients.\nSince $\\tau>2$, the number of positive roots of\nthe polynomial (\\ref{g}) is at most $6$. Numerical analysis shows that\nfor some values of $\\tau$ there are exactly 6 solutions (see Fig.\\ref{k3}).\nIndeed, rewrite (\\ref{g}) as\n$$a=Y(a,\\tau):={(\\tau^2+2)\\tau a^7+4(2\\tau^2+1)a^4+(\\tau^2-8)\\tau a^3\n+13\\tau a-8\\over(\\tau^2+2)a^7+\\tau^2 a^5+2(\\tau^2+2)\\tau a^4+6\\tau^2 a+\\tau}.$$\nNote that, for fixed $\\tau>2$, the function $Y(a, \\tau)$ is continuous with respect\nto both arguments $a>0$, $\\tau>2$ and is a bounded function.\nMoreover, $Y(0,\\tau)=-{8\\over \\tau}$ and $Y(+\\infty, \\tau)=\\tau$.\n\n\\begin{figure}[h!]\n\\vspace{-.5pc} \\centering\n\\includegraphics[width=7.2cm]{k3tau6.eps}\n\\includegraphics[width=7.2cm]{k3tau6-davomi.eps}\n\\caption{Graph of function $Y(x,6)-x$, for $x\\in (0.3,0.9)$ and $x\\in (0.9,10)$. Hence, $x=Y(x,6)$ has 6 (the maximal number) positive solutions.}\\label{k3}\n\\end{figure}\n\n\n\\begin{rk} \\label{rk} For $k=3$ one can explicitly find all positive solutions of (\\ref{ab3}), i.e.\n$$a_1=1, \\ \\ 01.$$\nMoreover, $a_2a_3=1$.\n\\end{rk}\n\nIn case $a\\ne b$, we do not know explicit solutions of (\\ref{g}).\nSince $b=f(a)$ may be negative for some positive solutions $a$,\nwe have to check positivity of $b$ for each positive $a$. To avoid this difficulty, in case $k=3$ and $a\\ne b$, we solve (\\ref{ab}) as\nfollows.\n\nWe rewrite the system of equations (\\ref{ab}) for the case $k=3$.\n\\begin{equation}\\label{eq1.1}\\left\\{\n \\begin{array}{ll}\n (a+b-\\tau)a^3+\\tau a-2=0; \\\\\n (a+b-\\tau)b^3+\\tau b-2=0.\n \\end{array}\n\\right.\n\\end{equation}\n\\begin{lemma}\\label{new} For the system (\\ref{eq1.1}) there are critical\nvalues $\\tau^{(1)}_{cr}\\approx 3.13039$ and $\\tau^{(2)}_{cr}\\approx 4,01009$ of $\\tau$ such that the following assertions hold\n\\begin{enumerate}\n\\item If $\\tau\\in [2, \\tau^{(1)}_{cr}]$ then there is no any solution.\n\\item If $\\tau\\in (\\tau^{(1)}_{cr}, \\tau^{(2)}_{cr}]$ then there is precisely one solution.\n\\item If $\\tau\\in(\\tau^{(2)}_{cr},\\infty)$ then there are precisely two solution to (\\ref{eq1.1}).\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nWe add the first and second equations of (\\ref{eq1.1}), i.e.,\n$$(a+b-\\tau)(a^3+b^3)+\\tau(a+b)-4=0.$$\nIf $a+b-\\tau=0$ then there is one solution to (\\ref{eq1.1}), i.e., $a=b=\\frac{2}{\\tau}$. We have to find $(a,b), a\\neq b$ solutions, so we can suppose $a+b-\\tau\\neq 0$. Consequently,\n$$a^3+b^3=\\frac{4-\\tau(a+b)}{a+b-\\tau}.$$\nFrom the last equality, one gets\n\\begin{equation}\\label{eq1.2} 3ab=(a+b)^2+\\frac{\\tau(a+b)-4}{(a+b)(a+b-\\tau)}.\n\\end{equation}\nNow, we subtract the second equation of (\\ref{eq1.1}) from the first one. Then\n$$(a+b-\\tau)(a^3-b^3)+\\tau(a-b)=0.$$\nSince $a\\neq b$, both sides can be divided by $a-b$ and we obtain the following\n\\begin{equation}\\label{eq1.3} ab=(a+b)^2+\\frac{\\tau}{a+b-\\tau}.\n\\end{equation}\nLet $a+b=x$. Then by (\\ref{eq1.2}) and (\\ref{eq1.3}), we have\n$$2x^2+\\frac{3\\tau}{x-\\tau}=\\frac{\\tau x-4}{x(x-\\tau)}.$$\nThe last equation can be written as\n\\begin{equation}\\label{eq1.4} x^4-\\tau x^3+\\tau x+2=0.\\end{equation}\nBy Ferrari's method for solving a quartic equation, the equation (\\ref{eq1.4}) can be written as\n\\begin{equation}\\label{eq1.5} \\left(x^2-\\frac{\\tau}{2}x+\\frac{c(\\tau)}{2}\\right)^2-\n\\left[\\left(\\frac{\\tau^2}{4}+c(\\tau)\\right)x^2-\\left(\\frac{\\tau c(\\tau)}{2}+\\tau\\right)x+\\left(\\frac{c^2(\\tau)}{4}-2\\right)\\right]=0,\\end{equation}\nwhere $c(\\tau)$ is a real root of the following polynomial\n$$P(z):=z^3-(\\tau^2+8)z-3\\tau^2=0.$$\nDenote\n$$F(\\tau) = \\dfrac {-(\\tau^2+8)} {3}, \\quad R(\\tau)= \\dfrac {3\\tau^2} {2}.$$\nTo find a certain view of $c(\\tau)$, we shall find a real root of $P(z)$. By Cardano's formula, roots of $P(z)$ are:\n$$z_1(\\tau) =S(\\tau)+T(\\tau),$$\n$$z_2(\\tau)= - \\dfrac {S(\\tau) + T(\\tau)} 2 + \\dfrac {i \\sqrt 3} 2 (S(\\tau)- T(\\tau)),$$\n$$z_3(\\tau)= - \\dfrac {S(\\tau) + T(\\tau)} 2 - \\dfrac {i \\sqrt 3} 2 (S(\\tau)- T(\\tau)),$$\nwhere\n$$S(\\tau)= \\sqrt [3] {R(\\tau)+ \\sqrt {F^3(\\tau)+R^2(\\tau)} }, \\quad T(\\tau)= \\sqrt [3] {R(\\tau)- \\sqrt {F^3(\\tau)+ R^2(\\tau)} }.$$\nIt's known that the expression $D(\\tau)=F^3(\\tau)+R^2(\\tau)$ is called the discriminant of the equation.\n\n \\begin{itemize}\n\\item If $D(\\tau)>0$, then one root is real and two are complex conjugates.\n\\item If $D(\\tau)=0$, then all roots are real, and at least two are equal.\n\\item If $D(\\tau)<0$, then all roots are real and unequal.\n\\end{itemize}\nWe have\n$$D(\\tau)=F^3(\\tau)+R^2(\\tau)=-\\frac{1}{27}\\tau^6+\\frac{49}{36}\\tau^4-\\frac{64}{9}\\tau^2-\\frac{512}{27}.$$\nNote that there are two positive solutions of $D(\\tau)$ on $[2,+\\infty)$ i.e., $\\tau_1\\approx 2.994$ and $\\tau_2\\approx 5.45$ and $D(\\tau)<0$ for $\\tau\\in[2, \\tau_1)\\cup (\\tau_2, +\\infty)$ and $D(\\tau)>0$ for $\\tau\\in(\\tau_1, \\tau_2).$\nFor all cases, $S(\\tau)+T(\\tau)$ is a real solution and we can choose $c(\\tau)$ as $S(\\tau)+T(\\tau)$. In addition, $c(\\tau)=S(\\tau)+T(\\tau)>0$.\nThe equation (\\ref{eq1.5}) can be written as\n$$\\left(x^2+\\left(\\sqrt{\\frac{\\tau^2}{4}+c(\\tau)}-\\frac{\\tau}{2}\\right)x+\\frac{c(\\tau)}{2}-\\sqrt{\\frac{c^2(\\tau)}{4}-2}\\right)\\times$$\n$$\\times\\left(x^2-\\left(\\sqrt{\\frac{\\tau^2}{4}+c(\\tau)}+\\frac{\\tau}{2}\\right)x+\\frac{c(\\tau)}{2}+\\sqrt{\\frac{c^2(\\tau)}{4}-2}\\right)=0.$$\nIt's easy to check that\n$$x^2+\\left(\\sqrt{\\frac{\\tau^2}{4}+c(\\tau)}-\\frac{\\tau}{2}\\right)x+\\frac{c(\\tau)}{2}-\\sqrt{\\frac{c^2(\\tau)}{4}-2}>0, \\ x\\in \\mathbb{R}.$$\nHence, from the last equality we obtain\n$$x_{1,2}(\\tau)=\\frac{1}{2}\\left[\\sqrt{\\frac{\\tau^2}{2}+c(\\tau)}+\\frac{\\tau}{2}\\pm \\sqrt{\\left(\\sqrt{\\frac{\\tau^2}{2}+c(\\tau)}+\\frac{\\tau}{2}\\right)^2-2\\left(c(\\tau)+\\sqrt{c^2(\\tau)-8}\\right)}\\right].$$\n$$\\left(\\sqrt{\\frac{\\tau^2}{2}+c(\\tau)}+\\frac{\\tau}{2}\\right)^2-2\\left(c(\\tau)+\\sqrt{c^2(\\tau)-8}\\right)\\geq 0\\ \\Leftrightarrow \\ \\tau\\geq\\tau_1\\approx 2.994.$$\nFrom the equation (\\ref{eq1.3})\n$$ab=x^2_{i}(\\tau)+\\frac{\\tau}{x^2_{i}(\\tau)-\\tau}, \\ i\\in \\{1,2\\}.$$\nNamely,\n$$a(x_{i}(\\tau)-a)=x^2_{i}(\\tau)+\\frac{\\tau}{x^2_{i}(\\tau)-\\tau}.$$\nConsequently,\n$$a_{1,2}^{(i)}(\\tau)=\\frac{x_{i}(\\tau)\\pm\\sqrt{x^2_{i}(\\tau)-4\\left(x^2_i(\\tau)+\\frac{\\tau}{x_{i}(\\tau)-\\tau}\\right)}}{2}.$$\nFrom numerical analysis, $$x^2_{1}(\\tau)-4\\left(x^2_1(\\tau)+\\frac{\\tau}{x_{1}(\\tau)-\\tau}\\right)< 0, \\ \\tau\\in[\\tau_1,+\\infty).$$\nAlso,\n$$x^2_{2}(\\tau)-4\\left(x^2_2(\\tau)+\\frac{\\tau}{x_{2}(\\tau)-\\tau}\\right)\\geq 0, \\ \\tau\\in[\\tau^{(1)}_{cr},+\\infty), \\ \\tau^{(1)}_{cr}\\approx 3.13039.$$\nHence, we have only two cases $a\\in \\{a_1^{(2)}(\\tau), a_2^{(2)}(\\tau)\\}$. When $a_1^{(2)}(\\tau)$ (resp $a_2^{(2)}(\\tau)$) belongs to the interval $(0, x_{2}(\\tau))$ then $b_1^{(2)}(\\tau)$ (resp. $b_2^{(2)}(\\tau)$) is also positive. Again we use numerical analysis and obtain the following results: if $\\tau\\in[2,\\tau^{(1)}_{cr}]$ then $a_2^{(2)}(\\tau)\\geq x_2(\\tau)$, i.e., the equation $(\\ref{eq1.1})$ has no any positive solution such that $a\\neq b$. Also, for all $\\tau\\in(\\tau^{(1)}_{cr}, \\tau^{(2)}_{cr}]$ we have $a_2^{(2)}(\\tau)0, i\\in \\mathbb Z_0)$ be a solution to (\\ref{di1}). Denote\n\\begin{equation}\\label{lr}\nl_i\\equiv l_i(\\theta)=\\sum_{j=-\\infty}^{-1}\\theta^{|i-j|}z_j, \\ \\\nr_i\\equiv r_i(\\theta)=\\sum_{j=1}^{\\infty}\\theta^{|i-j|}z_j, \\ \\ i\\in\\mathbb Z_0.\n\\end{equation}\nIt is clear that each $l_i$ and $r_i$ can be a finite positive number or $+\\infty$.\n\n\\begin{lemma}\\label{l1} \\cite{HKLR} For each $i\\in \\mathbb Z_0$ we have\n\\begin{itemize}\n\\item $l_i<+\\infty$ if and only if $l_0<+\\infty$;\n\n\\item $r_i<+\\infty$ if and only if $r_0<+\\infty$.\n\\end{itemize}\n\\end{lemma}\n\n\n\\begin{pro} \\label{pps: 1} Assume $h(0)=1$.\n\tA vector $\\mathbf z=(z_i,i\\in \\mathbb Z)$, with $z_0=1$, is a solution to (\\ref{di1})\nif and only if for $s_i=\\sqrt[k]{{z_i\\over h(i)}}$ the following holds\n\t\\begin{equation}\\label{Va}\n\th(i)s_i^k=\\frac{{s_{i-1}+s_{i+1}-\\tau s_i}}{s_{-1}+s_{1}-\\tau}, \\ \\ i\\in \\mathbb Z,\n\t\\end{equation}\n\twhere $\\tau=\\theta^{-1}+\\theta=2\\cosh(\\beta)$.\n\\end{pro}\n\\begin{proof} (cf. with the proof of Proposition 4.3 of \\cite{HKLR}).\nTake some $C>0$ and denote\n$$v_i=C\\cdot h^{1\/k}(i)\\left(\\theta^{|i|}+\n\\sum_{j\\in \\mathbb Z_0}\\theta^{|i-j|}z_j\\right), \\ \\ i\\in \\mathbb Z.$$\n\nThen from (\\ref{di1}) we get $z_i=\\left({v_i\\over v_0}\\right)^k$ and consequently,\n\\begin{equation}\\label{di2}\n\\left({v_i\\over v_0}\\right)^k=\\frac{h(i)}{h(0)}\\left({\\theta^{|i|}+\n\\sum_{j\\in \\mathbb Z_0}\\theta^{|i-j|}\\left({v_j\\over v_0}\\right)^k\n\\over\n1+\\sum_{j\\in \\mathbb Z_0}\\theta^{|j|}\\left({v_j\\over v_0}\\right)^k}\\right)^k, \\ \\ i\\in\\mathbb Z_0.\n \\end{equation}\nFrom (\\ref{di2}) we obtain\n$$v_i=C\\cdot \\sqrt[k]{h(i)}\\left(\\sum_{j=1}^{+\\infty}\\theta^jv_{i-j}^k+v_i^k+ \\sum_{j=1}^{+\\infty}\\theta^jv_{i+j}^k\\right), \\ \\ i \\in \\mathbb{Z}.\n$$\nBy the last equality we get\n$${v_{i-1}\\over \\sqrt[k]{h(i-1)}}+{v_{i+1}\\over \\sqrt[k]{h(i+1)}}-\\tau\\cdot {v_{i}\\over \\sqrt[k]{h(i)}}=C\\cdot (\\theta-{1\\over \\theta}) v^k_i.$$\nThis equality by the notation $s_i=\\sqrt[k]{{z_i\\over h(i)}}$ gives (\\ref{Va}). Conversely, from (\\ref{Va}) one gets (\\ref{di1}).\n\\end{proof}\n\\subsection{A case of 4-periodic external field for $k=2$.}\nHere we shall find solutions of (\\ref{Va}) for external field\n\\begin{equation}\\label{uh}\nh(i)=\\left\\{ \\begin{array}{lll}\n1, \\ \\ \\mbox{if} \\ \\ i=2m,\\\\[2mm]\nh_1, \\ \\ \\mbox{if} \\ \\ i=4m-1, \\ \\ m\\in \\mathbb Z\\\\[2mm]\nh_2, \\ \\ \\mbox{if} \\ \\ i=4m+1,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $h_1$ and $h_2$ are some positive numbers and\na solution which has the form\n\n\\begin{equation}\\label{upa}\nu_n=\\left\\{ \\begin{array}{lll}\n1, \\ \\ \\mbox{if} \\ \\ n=2m,\\\\[2mm]\na, \\ \\ \\mbox{if} \\ \\ n=4m-1, \\ \\ m\\in \\mathbb Z\\\\[2mm]\nb, \\ \\ \\mbox{if} \\ \\ n=4m+1,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $a$ and $b$ some positive numbers.\n\nThen from (\\ref{Va}) for $a$ and $b$ we get the following system of equations\n\\begin{equation}\\label{abd}\n\\begin{array}{ll}\n(a+b-\\tau)h_2b^k+\\tau b-2=0\\\\[2mm]\n(a+b-\\tau)h_1a^k+\\tau a-2=0.\n\\end{array}\n\\end{equation}\n\nFor simplicity we consider the case $k=2$ and $h_1=h_2=h$.\nIn this case, subtracting from the first equation of the system (\\ref{abd}) the second one we get\n\\[(b-a)[h(a+b)^2-h\\tau(a+b)+\\tau]=0.\\]\nThis gives three possible cases:\n\\begin{equation}\n\\label{b}\na=b, \\ \\ \\mbox{and} \\ \\ a+b=\\frac{1}{2h}(h\\tau\\pm \\sqrt{h\\tau(h\\tau-4)}) \\ \\ \\mbox{for} \\ \\ h\\tau\\geq 4.\n\\end{equation}\n\n \\textbf{Case $a=b$}. In this case from the first equation of (\\ref{abd}) we get\n\t\\begin{equation}\n\t\\label{a3}\n\t2ha^3-h\\tau a^2+\\tau a-2=0.\n\t\\end{equation}\n\\begin{lemma}\\label{lp} Any real solution of (\\ref{a3}) is positive.\n\\end{lemma}\n\\begin{proof} Since $h>0$, $\\tau>2$, if $a\\leq 0$ then LHS of (\\ref{a3}) is strictly negative.\n\\end{proof}\nRewrite the cubic equation (\\ref{a3}) as\n$$\na^{3}+\\a a^{2}+\\b a+\\c=0\n$$\nwhere $\\a=-\\tau\/2$, $\\b=\\tau\/2h$, $\\c=-1\/h$.\n\nLet\n$$\np=\\b-\\frac{\\a^{2}}{3} \\quad \\text { and } \\quad q=\\frac{2 \\a^{3}}{27}-\\frac{\\a \\b}{3}+\\c\n$$\nThen the discriminant $\\Delta$ of the cubic equation is\n$$\n\\Delta=\\frac{q^{2}}{4}+\\frac{p^{3}}{27}.\n$$\n\n\n\\begin{lemma} The following assertions hold\n\\begin{itemize}\n\\item[Case: $\\Delta>0 .$] In this case there is only one \\textbf{positive} real solution. It is\n$$\na_1=\\left(-\\frac{q}{2}+\\sqrt{\\Delta}\\right)^{\\frac{1}{3}}+\\left(-\\frac{q}{2}-\\sqrt{\\Delta}\\right)^{\\frac{1}{3}}+\\frac{\\tau}{6}\n$$\n\n\\item[Case: $\\Delta=0 .$] In this case there are two \\textbf{positive} real solutions. These roots are\n$$\na_{1}=-2\\left(\\frac{q}{2}\\right)^{\\frac{1}{3}}+\\frac{\\tau}{6} \\quad \\text { and } \\quad a_{2}=a_{3}=\\left(\\frac{q}{2}\\right)^{\\frac{1}{3}}+\\frac{\\tau}{6}\n$$\n\n\\item[Case: $\\Delta<0 .$] In this case $-p>0$ and there are three \\textbf{positive} real solutions:\n$$\n\\begin{array}{l}\na_{1}=\\frac{2}{\\sqrt{3}} \\sqrt{-p} \\sin \\left(\\frac{1}{3} \\sin ^{-1}\\left(\\frac{3 \\sqrt{3} q}{2(\\sqrt{-p})^{3}}\\right)\\right)+\\frac{\\tau}{6} \\\\\na_{2}=-\\frac{2}{\\sqrt{3}} \\sqrt{-p} \\sin \\left(\\frac{1}{3} \\sin ^{-1}\\left(\\frac{3 \\sqrt{3} q}{2(\\sqrt{-p})^{3}}\\right)+\\frac{\\pi}{3}\\right)+\\frac{\\tau}{6} \\\\\na_{3}=\\frac{2}{\\sqrt{3}} \\sqrt{-p} \\cos \\left(\\frac{1}{3} \\sin ^{-1}\\left(\\frac{3 \\sqrt{3} q}{2(\\sqrt{-p})^{3}}\\right)+\\frac{\\pi}{6}\\right)+\\frac{\\tau}{6}\n\\end{array}\n$$\n\\end{itemize}\n\\end{lemma}\n\\begin{proof} The conditions of existence of real solutions are well-known\\footnote{https:\/\/en.wikipedia.org\/wiki\/Cubic$_-$equation}. The positivity of each solution follows from Lemma \\ref{lp}.\n\\end{proof}\n\n\n\\textbf{Case $a+b=\\frac{1}{2h}(h\\tau+ \\sqrt{h\\tau(h\\tau-4)})$}.\n\tIn this case from the second equation of (\\ref{abd}) we get\n\t\\[(h\\tau-\\sqrt{h\\tau(h\\tau-4)})a^2-2\\tau a +4=0.\\]\n\tNote that this equation has only positive real solutions.\nMoreover, one can see that if\n$$(\\tau, h)\\in A:=\\left\\{(\\tau, h)\\in \\mathbb R^2_+ \\,: \\,h\\geq \\left[\\begin{array}{ll}\n{\\tau^3\\over 8(\\tau^2-8)}, \\ \\ \\mbox{if} \\ \\ 2\\sqrt{2}<\\tau<4\\\\[2mm]\n{4\\over \\tau}, \\ \\ \\mbox{if} \\ \\ \\tau\\geq 4\n\\end{array}\\right.\\right\\}$$\nthen the quadratic equation has the following positive solutions\n\t\\[a_4=\\frac{\\tau-\\sqrt{\\tau^2-4h\\tau+4\\sqrt{h\\tau(h\\tau-4)}}}{h\\tau-\\sqrt{h\\tau(h\\tau-4)}}, \\ \\ a_5=\\frac{\\tau+\\sqrt{\\tau^2-4h\\tau+4\\sqrt{h\\tau(h\\tau-4)}}}{h\\tau-\\sqrt{h\\tau(h\\tau-4)}}. \\]\n\n\tUsing (\\ref{b}) we get $b_4=a_5$ and $b_5=a_4$.\n\n\\textbf{Case $a+b=\\frac{1}{2h}(h\\tau- \\sqrt{h\\tau(h\\tau-4)})$}.\n\tIn this case we obtain\n\t\\begin{equation*}\n\t(h\\tau+ \\sqrt{h\\tau(h\\tau-4)})a^2-2\\tau a+4=0\n\t\\end{equation*}\n\twhich\n\tfor\n$$(\\tau, h)\\in B:=\\left\\{(\\tau, h)\\in \\mathbb R^2_+ \\,: \\, \\tau\\geq 4, \\, {4\\over \\tau}\\leq h\\leq {\\tau^3\\over 8(\\tau^2-8)}\\right\\}$$ has the following solutions\n\t\\[a_6=\\frac{\\tau-\\sqrt{\\tau^2-4h\\tau-4\\sqrt{h\\tau(h\\tau-4)}}}{h\\tau+\\sqrt{h\\tau(h\\tau-4)}}, \\ \\ a_7=\\frac{\\tau+\\sqrt{\\tau^2-4h\\tau-4\\sqrt{h\\tau(h\\tau-4)}}}{h\\tau+\\sqrt{h\\tau(h\\tau-4)}}. \\]\n\tUsing (\\ref{b}) we get $b_6=a_7$ and $b_7=a_6$. Clearly all of these solutions are positive.\n\nDenote by $\\mu_i$ the gradient Gibbs measure corresponding to solution $(a_i, b_i)$, $i=1,2,\\dots,7$.\n\nThus depending on the values $(\\tau, h)$ related to $\\Delta$ and sets $A$, $B$ we have the following result.\n\n\\begin{thm} For the SOS model with 4-periodic external field there are up to seven 4-periodic gradient Gibbs measures $\\mu_i$, $i=1,\\dots,7$.\n\n\\end{thm}\n\n\n\\section*{ Acknowledgements}\n\n The author thanks C. K\\\"ulske for helpful discussions.\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA classical kind of problem in algebraic geometry is to consider vanishing conditions on a linear system $\\mathcal L$, \nand to ask if the dimension of the resulting linear system is what one would expect based on the dimension of $\\mathcal L$ \nand the specific conditions imposed. That is, one asks if the desired vanishing imposes the {\\em expected number} of \nconditions on $\\mathcal L$. For example, if $\\mathcal L$ is the complete linear system of conics in $ \\ensuremath{\\mathbb{P}}^2$ \n(which is 5-dimensional) and $P$ is a point, then vanishing to multiplicity 2 at $P$ imposes three conditions on $\\mathcal L$; \nthat is, there is a 2-dimensional linear system of conics double at $P$ as expected. However, if we impose vanishing to \nmultiplicity 2 at each of two points $P_1$ and $P_2$, we expect $3+3 = 6$ conditions, i.e. we expect there to be no such \nconic, while in fact the double line passing through $P_1$ and $P_2$ is such a conic. Continuing in this direction leads \nto the well-known Segre-Harbourne-Gimigliano-Hirschowitz (SHGH) Conjecture \\cite{segre,Harb,Gi,Hi}. \nA conjecture of Laface and Ugaglia addresses the corresponding situation in $ \\ensuremath{\\mathbb{P}}^3$ \\cite{LU}, while results of \nAlexander and Hirschowitz \\cite{AH} partially address the situation in $ \\ensuremath{\\mathbb{P}}^n$ for all $n\\geq2$, but much remains unknown.\n\nIt is not only vanishing conditions imposed by points that is of interest.\nGiven a general set of $r$ lines in $ \\ensuremath{\\mathbb{P}}^n$, one could ask if vanishing on all of these lines with multiplicity 1 \nimposes the expected number of conditions on the complete linear system of hypersurfaces of given degree $d$, \nand an affirmative answer was given by Hartshorne and Hirschowitz \\cite{HaHi}. Their paper led to much other \nwork, such as research in which lines of higher multiplicity are allowed (for just two recent examples, see \\cite{DHRST} and \\cite{BDSSS}). The work of Hartshorne and Hirschowitz also led to the paper \\cite{CCG}, which can be viewed as a direct precursor to the study of unexpected hypersurfaces.\n\n\n\nThus in recent years, a flurry of activity has emerged on this kind of problem. Some of it grew out of a striking \nexample in \\cite{DIV}, later formalized in \\cite{CHMN} and since then branching off in many different directions \n(for some examples, see \\cite{CM, DMO, DHRST, FGST, HaHa, HMNT, HMT, S1, S2, Tr}). The path that led to \nthis paper began \nby our looking for conditions on a variety $X$ that either automatically force the existence of unexpected hypersurfaces, \nor else force the conclusion that no unexpected hypersurfaces exist. The results we describe below give examples of \nsuch conditions. But more specifically we can ask: are there conditions on the Hilbert function that force either of \nthese outcomes? It turns out that if the Hilbert function forces $X$ to be degenerate in $ \\ensuremath{\\mathbb{P}}^n$ then $X$ \ndoes not admit any unexpected hypersurfaces of any kind in $ \\ensuremath{\\mathbb{P}}^n$ (see Corollary \\ref{degenerate}), \nregardless of whether it does in the smallest linear space containing it. Beyond that, additional conditions involving the \ngeometry of $X$ seem to be involved. Indeed, with a \nminor assumption on the geometry of $X$, there are such Hilbert functions (see Theorem \\ref{force unexp}),\nbut in the setting of non-degenerate, finite sets of points, we conjecture that there \nare no Hilbert functions that force any kind of unexpectedness (see Conjecture \\ref{conj about existence}).\n\nWe now describe our results in more detail.\nGiven a subscheme $X$ of $ \\ensuremath{\\mathbb{P}}^n_K$, \nits defining saturated homogeneous ideal $I_X\\subseteq R=K[ \\ensuremath{\\mathbb{P}}^n_K]=K[x_0,\\ldots,x_n]$ (where $K$ is a field)\nand integers $t\\ge m\\ge 1$, we define three numbers associated \nto $(X,t,m)$ (see Notation \\ref{dim not}). The {\\em actual dimension}, ${\\ensuremath{\\mathrm{adim}}}(X,t,m)$, is the dimension \nof the vector space of the forms in $I_X$ of degree $t$ vanishing at a general point $P$ with multiplicity $m$. That is,\n\\[\n{\\ensuremath{\\mathrm{adim}}}(X,t,m) = \\dim [I_X \\cap I_P^m]_t.\n\\]\nNext, the {\\em virtual dimension}, ${\\ensuremath{\\mathrm{vdim}}}(X,t,m)$, is the dimension of the linear system of the forms of \ndegree $t$ in $I_X$ minus the expected number of conditions imposed by taking $P$ with multiplicity $m$. That is,\n\\[\n{\\ensuremath{\\mathrm{vdim}}}(X,t,m) = \\dim [I_X]_t - \\binom{m-1+n}{n}.\n\\] \nFinally, the {\\em expected dimension} ${\\ensuremath{\\mathrm{edim}}}(X,t,m)$ is the maximum of ${\\ensuremath{\\mathrm{vdim}}}(X,t,m)$ and 0.\n\nOf course, ${\\ensuremath{\\mathrm{adim}}}(X,t,m) \\ge {\\ensuremath{\\mathrm{edim}}}(X,t,m) \\geq {\\ensuremath{\\mathrm{vdim}}}(X,t,m)$. We say \nthat $X$ {\\em admits an unexpected hypersurface of degree $t$ vanishing at a general point $P$ with multiplicity $m$} when \n${\\ensuremath{\\mathrm{adim}}}(X,t,m)>{\\ensuremath{\\mathrm{edim}}}(X,t,m)$, i.e., when\n${\\ensuremath{\\mathrm{adim}}}(X,t,m)>0$ and ${\\ensuremath{\\mathrm{adim}}}(X,t,m)>{\\ensuremath{\\mathrm{vdim}}}(X,t,m)$. \n\nThe purpose of this paper is to get a better feel for when unexpected hypersurfaces are forced to occur \n(hence ``expecting the unexpected\"). We relate unexpectedness to algebraic and geometric properties (see for example \nProposition \\ref{p.lex segments are bad for unexpectedness} for the former and \nCorollary \\ref{degenerate} and Theorem \\ref{uct} for the latter)\nand we bring to bear methods not previously applied to unexpectedness (such as the use of generic \ninitial ideals and partial elimination ideals) to clarify when unexpectedness can and when it cannot be \nexpected. When it can, we also study to what extent\nit can, which we do by introducing AV sequences \nmeasuring the gap between the actual dimension and the virtual dimension (see Definition \\ref{def of AV}),\nwhich also allows us to frame our work in terms of persistence (i.e., how long does the gap remain\npositive?).\n\nIn Section \\ref{s.Background} we define these AV sequences. \nSpecifically, for a given subscheme $X$ of $ \\ensuremath{\\mathbb{P}}^n$ and a non-negative integer $j\\ge 0$, we define \n$AV_{X,j}: \\ensuremath{\\mathbb{Z}}_{>0}\\to \\ensuremath{\\mathbb{Z}}_{\\geq0}$ where\n$AV_{X,j}(m):={\\ensuremath{\\mathrm{adim}}}(X,m+j,m)-{\\ensuremath{\\mathrm{vdim}}}(X,m+j,m)$. Studying the difference ${\\ensuremath{\\mathrm{adim}}} - {\\ensuremath{\\mathrm{vdim}}}$ is \nnot new (see for instance \\cite{HMT}), but considering it as a sequence is novel. This sequence has interesting \nproperties and leads to compact formulas. \nSometimes we will consider the general case, but even the cases $j=0$ and $j=1$ are interesting \n(see Proposition \\ref{AV_X,0(alpha)} and Sections \\ref{s.irreducible ACM curve in P3} -- \\ref{s.Unmixed curves and unions with finite sets in P3}).\n\nIn Section \\ref{s. gin and unexp}, for any subscheme $X$ and integer $j\\ge 0$, we prove (Theorem \\ref{t. AV is an O-sequence}) \nthat the sequence $AV_{X,j}$ is actually an $O$-sequence, up to a shift. Indeed, it is the \nHilbert function of the $K$-algebra $R\/(\\ensuremath{\\mathrm{gin}}(I_X):x^{j+1})$, where $\\ensuremath{\\mathrm{gin}}(I_X)$ denotes the \ngeneric initial ideal with respect to the lexicographic order.\nWe use this fact to obtain results which ensure the non-existence of unexpected hypersurfaces. \nIn particular, if $X$ lies on a hyperplane or if $\\ensuremath{\\mathrm{gin}}(I_X)$ is a lex-segment ideal then $X$ does not admit {\\em any} unexpected hypersurfaces of any type (Corollary \\ref{degenerate} and Proposition \\ref{p.lex segments are bad for unexpectedness}), and hence \n${\\ensuremath{\\mathrm{adim}}}(X,t,m)$ {\\it always} has the expected value.\n\nThe fact that the AV sequence is actually an $O$-sequence raises many other related questions. \nIn Section \\ref{s.irreducible ACM curve in P3} we use a cohomological interpretation of the AV sequence, \ndescribed in Remark \\ref{cohom interp}, to prove that if $X$ is an irreducible, arithmetically Cohen-Macaulay \n(ACM) curve in $ \\ensuremath{\\mathbb{P}}^3$, then $AV_{X,1}(m)$ is unimodal and the first part is differentiable \n(see Theorem \\ref{part of conj}). Moreover, we have a great deal of experimental evidence which\nsuggests that if $X$ is smooth then this sequence is also finite and symmetric (see Conjecture \\ref{SI conj}).\n\nIn Section \\ref{ci section} we apply a new method to study the question of unexpected hypersurfaces, \nnamely the theory of {\\em partial elimination ideals} introduced by Green \\cite{Gr}. We consider the \ncase of a general codimension 2 complete intersection $C\\subset \\ensuremath{\\mathbb{P}}^n$ and we show the \nnon-vanishing of $[I_C \\cap I_P^m]_t$ for prescribed values of $t$ and $m$ (Proposition \\ref{p. codimension 2 complete intersection}). \nWe apply this to the case of $n=3$ to show for $t = (a-1)(b-1) + 1$ and $m = (a-1)(b-1)$, that a \ngeneral complete intersection curve $C$ of type $(a,b)$ with \n$2 k_{d-1} > \\dots > k_j \\geq j \\geq 1$ are integers.\n\\end{lemma}\n\n\\noindent Given the above $d$-Macaulay representation of $a$, we define\n\\[\na^{\\langle d \\rangle} = \\binom{k_d+1}{d+1} + \\binom{k_{d-1}+1}{d} + \\dots + \\binom{k_j+1}{j+1}\n\\] \nand set $0^{\\langle d \\rangle} = 0$.\n\n\\begin{theorem}[\\cite{BH} Theorem 4.2.10] \\label{macaulay thm}\nLet $K$ be a field and let $h : \\ensuremath{\\mathbb{Z}}_{\\geq0} \\rightarrow \\ensuremath{\\mathbb{Z}}_{\\geq0}$ be a numerical function. \nThen $h$ is the Hilbert function of some standard graded $K$-algebra if and only if\n\\[\nh(0) = 1 \\ \\ \\ \\hbox{ and } \\ \\ \\ h(d+1) \\leq h(d)^{\\langle d \\rangle}\n\\]\nfor all $d \\geq 1$.\n\\end{theorem}\n\n\\noindent An infinite sequence $a_0,a_1,a_2,\\ldots$ of non-negative integers with $a_i=h(i)$ for an $h$ \nsatisfying the conditions of \nTheorem~\\ref{macaulay thm} is called an {\\em $O$-sequence}. \nWe will regard a finite sequence $a_0,a_1,a_2,\\ldots, a_r$ as an infinite sequence\nby setting $a_i=0$ for $i>r$.\n\nHereafter, let $R = K[x_0,\\dots, x_n]$ be a polynomial ring over a field $K$.\nOur default assumption will be that $K$ is algebraically closed,\nbut sometimes we will need the characteristic to be zero, \nand sometimes we will need only that $K$ is infinite;\nin these cases we will say so explicitly.\n \n\\begin{notation}\nFor any subvariety (or subscheme) $V \\subseteq \\ensuremath{\\mathbb{P}}^n$ we write $I_V \\subseteq R$ for the saturated ideal of $V$\nand $\\mathcal{I}_V$ for the sheaf on $ \\ensuremath{\\mathbb{P}}^n$ corresponding to $I_V$. For a standard graded algebra $R\/I$ \nwe write $h_{R\/I}(t)$ for the {\\em Hilbert function} of $R\/I$, i.e. $h_{R\/I}(t) = \\dim_K [R\/I]_t$. When $I = I_V$ for \nsome subscheme $V$, we sometimes write $h_V(t)$ for $h_{R\/I_V}(t)$. We say that $V$ is {\\em arithmetically \nCohen-Macaulay (ACM)} if $R\/I_V$ is a Cohen-Macaulay ring.\n\nFor any integer function $h : \\ensuremath{\\mathbb{Z}}_{\\geq 0} \\to \\ensuremath{\\mathbb{Z}}$, the first difference $\\Delta h$ is the backward difference,\ndefined by setting $\\Delta h(0) = h(0)$ and $\\Delta h(t) = h(t)-h(t-1)$ for $t>0$. \nWhen $X$ is a finite set of points, it is well-known and easy to see that\n$h_X$ is strictly increasing until it becomes constant, hence there is a $j$ such that\n$1=h_X(0)<\\cdots {\\ensuremath{\\mathrm{edim}}}(X,t,m)$, we say that $X$ {\\em admits an unexpected \nhypersurface of degree $t$ for multiplicity $m$}. (In this case, note that ${\\ensuremath{\\mathrm{adim}}} (X,t,m)>0$,\nhence $t\\geq m$.) If $X \\subset \\ensuremath{\\mathbb{P}}^2$ is a finite set of \npoints which admits an unexpected hypersurface of degree $t$ for multiplicity $m=t-1$,\nthen following \\cite{CHMN} we say simply that {\\em $X$ admits an unexpected curve of degree $t$}.\n\\end{definition}\n\n\\begin{remark} \\label{equiv to unexp}\nAn equivalent condition for $X$ to admit an unexpected hypersurface of degree $t$ for \nmultiplicity $m$ is ${\\ensuremath{\\mathrm{adim}}}(X,t,m) > 0$ and ${\\ensuremath{\\mathrm{adim}}}(X,t,m) > {\\ensuremath{\\mathrm{vdim}}}(X,t,m)$.\n\\end{remark}\n\n\\begin{remark}\nAny hypersurface of degree $t$ with an isolated singularity of multiplicity $t$ must be a \ncone (by Bezout's theorem). Thus ${\\ensuremath{\\mathrm{adim}}}(X,t,t)$ is the dimension of the vector space \nof cones over $X$ of degree $t$ with vertex at~$P$. If ${\\ensuremath{\\mathrm{adim}}} (X,t,t) > {\\ensuremath{\\mathrm{edim}}}(X,t,t)$, \nwe say that $X$ {\\em admits an unexpected cone of degree $t$}. See \\cite{HMNT, CM, HMT} for \nmore on unexpected cones. In particular, if $X$ has codimension two and is reduced, \nequidimensional and non-degenerate then the cone $S_P$ over $X$ with vertex $P$ is \nan unexpected cone of degree $t=\\deg X$ (\\cite{HMNT} Proposition 2.4).\n\\end{remark}\n\n\n\\begin{definition} \\label{def of AV}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^n$ be a closed subscheme. Fixing a non-negative integer $j$, \nwe define the sequence $AV_{X,j}$ as follows:\n\\[\nAV_{X,j}(m) = \n\\begin{array}{cc}\n{\\ensuremath{\\mathrm{adim}}}(X,m+j,m) - {\\ensuremath{\\mathrm{vdim}}}(X,m+j,m), & m\\ge 1.\n\\end{array}\n\\] \n\\end{definition}\n\n\\begin{remark}\nRephrasing Remark \\ref{equiv to unexp}, if ${\\ensuremath{\\mathrm{adim}}} (X,t,m) > 0$ then $X$ admits \nan unexpected hypersurface of degree $t$ for multiplicity $m$ if and only if $AV_{X,j}(m) > 0$ for $j = t-m$.\n\n\\end{remark}\n\n\\begin{notation} \\label{Pm vs mP}\nLet $P \\in \\ensuremath{\\mathbb{P}}^n$ be a general point, with defining ideal $I_P$. We will \ndenote the scheme defined by $I_P^m$ in $ \\ensuremath{\\mathbb{P}}^n$ by $P^m$. We will \nsometimes consider the hyperplane section of $P^m$ by a hyperplane $H$ \ncontaining $P$, and we will denote the corresponding subscheme of $H$ by $mP$, thus $mP=P^m\\cap H$.\n\\end{notation}\n\n\nWe now give an interpretation of the sequence $AV_{X,j}$. Notice that, in the \nfollowing lemma, the ideal in the dimension of the quotient on the right changes with $m$. \n\n\\begin{lemma}\\label{l.AV seq 1} \nLet $X \\subset \\ensuremath{\\mathbb{P}}^n$ be a subscheme. Then\n\t\\[\n\tAV_{X,j}(m) = \\dim\\left[ \\faktor{R}{(I_X + I_P^m)}\\right]_{m+j} .\n\t\\] \n\\end{lemma}\n\\begin{proof}\n\tSet $t:=m+j$. From the short exact sequence\n\t\\[\n\t0\\to R\/(I_X \\cap I_P^m) \\to R\/I_X \\oplus R\/ I_P^m \\to R\/(I_X + I_P^m)\\to 0\n\t\\]\n\twe get the relation between the dimension of the modules in degree $t$\n\t\\[\n\t\\dim[R]_t- {\\ensuremath{\\mathrm{adim}}} (X,t,m) -\\dim[R]_t+\\dim[I_X]_t - \\dim[R]_t+ \\dim [I_P^m]_t +\\dim[ R\/(I_X + I_P^m)]_t= 0.\n\t\\]\n\tTherefore\n\t\\[\n\t\\begin{array}{rcl}\n\t\\dim[ R\/(I_X + I_P^m)]_t&=& {\\ensuremath{\\mathrm{adim}}} (X,t,m) -\\dim[I_X]_t + \\dim[R]_t-\\dim [I_P^m]_t =\\\\\n\t&=& {\\ensuremath{\\mathrm{adim}}} (X,t,m) -\\dim[I_X]_t + h_{P^m}(t) .\n\t\\end{array}\n\t\\]\n\tSince $t\\ge m$ the Hilbert function of the fat point $P^m$ in degree $t$ is \n\t$h_{P^m}(t)={m+n-1\\choose n}$. (It reaches the degree $\\deg(P^m)$ in degree $m-1$.) So we get\n\t\\[\\dim[ R\/(I_X + I_P^m)]_t = {\\ensuremath{\\mathrm{adim}}} (X,t,m) - {\\ensuremath{\\mathrm{vdim}}} (X,t,m) \\] \n\tas desired.\n\\end{proof}\n\n\n\\begin{remark} \\label{cohom interp}\n\tWe can also give a cohomological interpretation for the sequence $AV_{X,j}(m)$. \n\t Let $X$ be an ACM subscheme in $ \\ensuremath{\\mathbb{P}}^n$ of dimension $\\geq 1$. Assume $m \\geq 0$ and $j \\geq 0$. Consider the exact sequence of sheaves\n\t\\[\n\t0 \\rightarrow \\mathcal I_{X \\cup P^m} \\rightarrow \\mathcal I_X \\rightarrow \\mathcal O_{P^m} \\rightarrow 0.\n\t\\]\n\tTwisting by $m+j$ and taking cohomology gives the exact sequence\n\t\\[\n\t0 \\rightarrow [I_{X \\cup P^m}]_{m+j} \\rightarrow [I_X]_{m+j} \\rightarrow H^0(\\mathcal O_{P^m}(m+j)) \\rightarrow H^1(\\mathcal I_{X \\cup P^m}(m+j)) \\rightarrow 0.\n\t\\]\n\t(Exactness on the right is because $X$ is ACM of dimension $\\geq 1$; see \\cite[pp. 9-11]{migbook}.) This gives\n\t\\[\n\th^1(\\mathcal I_{X \\cup P^m}(m+j)) = \\binom{(m-1)+n}{n} - \\dim [I_X]_{m+j} + \\dim [I_{X \\cup P^m}]_{m+j} = AV_{X,j}(m).\n\t\\]\n\\end{remark}\n\n\\begin{remark}\nGiven a subscheme $X \\subset \\ensuremath{\\mathbb{P}}^n$, it is natural to ask about the persistence \nof the unexpectedness imposed by $X$. For example, in \\cite[Corollary 2.12]{HMNT}, it is \nshown that a nondegenerate curve $C \\subset \\ensuremath{\\mathbb{P}}^3$ of degree $d=\\deg C$ admits \nan unexpected hypersurface of degree $t$ for multiplicity $t$ at a general point for all $t \\geq d$. \nThus fixing $0 = j = t-m$, and fixing $C$, we have the persistence of unexpectedness as long as $t \\geq d$.\n\nMany of the results in this paper give formulas for the sequences $AV_{X,j} (m)$. Leaving \naside the issue of whether ${\\ensuremath{\\mathrm{adim}}} (X,t,m) > 0$, this sequence can be interpreted both \nas a measure of unexpectedness (how much bigger is the actual dimension than what one \nwould expect?) and as a measure of persistence (how long is $AV_{X,j}(m)$ positive?). \nThe fact that these sequences are represented by simple formulas, as we will see in the coming sections, is a pleasant bonus.\n\\end{remark}\n\n\n\n\n\\section{Generic initial ideals and unexpectedness}\\label{s. gin and unexp}\nIn this section we relate the study of unexpected hypersurfaces of a subscheme $X\\subseteq \\ensuremath{\\mathbb{P}}^n$ \nto the generic initial ideal of $I_X$ with respect to the lexicographic order. Then, we prove that the $AV_{X,j}$ \nsequence, up to a shift, is an $O$-sequence. As a consequence of this result we are able to ensure the \nnon-existence of unexpected hypersurfaces in several cases.\n \nLet $R = K[x_0,x_1,\\ldots,x_n]$ be a standard graded polynomial ring. In this section we only \nrequire $K$ to be infinite. We assume the monomials of $R$ are ordered by $>_{lex}$, the \nlexicographic monomial order which satisfies $x_0> x_1 > \\cdots > x_n$.\t\nWe recall that a set $M\\subseteq R$ of monomials is a {\\em lex-segment} \nif the monomials have the same degree and they satisfy the condition that whenever $u,v$ \nare monomials with $u\\geq v$ and $v\\in M$, then $u\\in M$ \\cite{V}.\nIt is convenient to also refer to a vector subspace $W\\subseteq R$\nas a lex-segment if $W$ is spanned by a lex-segment in the previous sense.\nWe also recall that a homogeneous ideal $I\\subseteq R$ is a {\\em lex-segment ideal} if \nfor each degree $d$ the the homogeneous component $I_d$ of $I$ of degree $d$\nis a lex segment \\cite{V}; see also \\cite{Hu}.\n\nFor a graded ideal $I\\subseteq R$, we will denote by $\\ensuremath{\\mathrm{gin}}(I)$ the generic initial ideal of $I$ with respect\nto the monomial order $>_{lex}$. For an introduction to generic initial ideals, see for instance \\cite{Gr} and Section 15.9 in \\cite{E}.\nThe next lemma relates the actual and virtual dimensions of a scheme in terms of the generic initial ideal of its ideal. \n\n\n\\begin{lemma}\\label{l. adim vdim and Gin} Let $X\\subseteq \\ensuremath{\\mathbb{P}}^n$ be a subscheme. For any non-negative integers $t$ and $m$, we have \n\t\\begin{itemize}\n\t\t\\item[\\em (i)] ${\\ensuremath{\\mathrm{adim}}} (X,t,m)=\\dim [\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^m]_t$, where $Q = (1,0,\\dots,0)$.\n\t\t\\item[\\em (ii)] ${\\ensuremath{\\mathrm{vdim}}} (X,t,m)={\\ensuremath{\\mathrm{vdim}}} (\\ensuremath{\\mathrm{gin}}(I_X),t,m)$.\n\t\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\n(i) Let $X'$ be the image of $X$ under a general linear change of variables, so $\\hbox{gin}(I_{X})=\\hbox{in}(I_{X'})$\nand so the point $Q$ is general for $X'$.\t\nMoreover, if $\\mu_1,\\mu_2$ are monomials of the same degree with $\\mu_1\\in[I_Q^m]_t$ and $\\mu_1>_{lex}\\mu_2$,\nthen $\\mu_2\\in I_Q^m$, hence $\\hbox{in} (I_{X'}\\cap I_Q^m)=\\hbox{in} (I_{X'})\\cap I_Q^m$. \n(To see this note that both sides of the equality are monomial ideals, and \nthat $\\hbox{in} (I_{X'}\\cap I_Q^m)\\subseteq \\hbox{in} (I_{X'})\\cap I_Q^m$ is clear.\nSo suppose that $w\\in \\hbox{in} (I_{X'})\\cap I_Q^m$ is a monomial. \nThen there is a form $W\\in I_{X'}$ with $w=\\hbox{in}(W)$. \nIt follows that the other terms of $W$ have lex order less than $w$, so\neach is in $I_Q^m$, hence $W\\in I_Q^m$, and we have $W\\in I_{X'}\\cap I_Q^m$\nso $w\\in \\hbox{in} (I_{X'}\\cap I_Q^m)$ giving $\\hbox{in} (I_{X'}\\cap I_Q^m)\\supseteq \\hbox{in} (I_{X'})\\cap I_Q^m$.)\nThus we have\n${\\ensuremath{\\mathrm{adim}}}(X,t,m)=\\dim [I_X\\cap I_P^m]_t \n\t\t\t = \\dim [I_{X'}\\cap I_Q^m]_t \n\t\t\t = \\dim [\\hbox{in}(I_{X'}\\cap I_Q^m)]_t \n\t\t\t = \\dim [\\hbox{in}(I_{X'})\\cap I_Q^m]_t \n\t\t\t = \\dim [\\hbox{gin}(I_{X})\\cap I_Q^m]_t$.\n\n(ii) This is a consequence of $h_X=h_{R\/(\\ensuremath{\\mathrm{gin}}(I_X))}$.\n\\end{proof}\n\n\n\\begin{remark} \\label{r: partial elimination ideal} \n\tLet $I=\\oplus_k I_k$ be a homogeneous ideal in $R$. \nThen we make the following definition (see \\cite[Definition 6.1]{Gr}):\n\t\t\\[\n\t\t\\widetilde K_d(I) =\\bigoplus_k[I\\cap I_Q^{k-d}]_k,\n\t\t\\]\nwhere $Q = (1,0,\\dots,0)$, so\n\t $\\widetilde K_d(I)$ is a graded module over $K[x_1,\\ldots,x_n]$. \n\t In particular, $\\widetilde K_0(I)$ is an ideal: it is obtained from $I$ by \\textit{eliminating} the variable $x_0$. \n\t Geometrically, it corresponds to the linear space of the cones having vertex at $Q$. Then Lemma \\ref{l. adim vdim and Gin} (i) can be rephrased as follows: \t\n\t \\[\n\t \\dim (I_X \\cap I_P^m)_t=\\dim \\left[\\widetilde K_{t-m}\\left(\\ensuremath{\\mathrm{gin}}(I_X) \\right)\\right]_t.\n\t \\]\n\\end{remark}\n\n\\begin{example}\\label{Fig1Fig2Example}\nAssume $\\hbox{\\rm char}(K)=0$. \nHere, given only the Hilbert function $h_X$ of a set of points $X\\subset \\ensuremath{\\mathbb{P}}^2$,\nwe show how information about $\\ensuremath{\\mathrm{gin}}(I_X)$ relates to whether or not $X$ has\nan unexpected curve.\nSo let $X\\subseteq \\ensuremath{\\mathbb{P}}^2$ be a set of 13 points with \n$h_X=(1, 3, 6, 10, 12, 13, 13, \\ldots)$,\nso $X$ lies on three independent quartics but no cubics (such examples exist, as we see later in this example). Then \n$\\dim [I_X]_6=15$, so ${\\ensuremath{\\mathrm{vdim}}} (X,6,5)=0$, hence $X$ admits an unexpected curve \nof degree~$6$ if and only if ${\\ensuremath{\\mathrm{adim}}}(X,6,5)>0$. \nFrom Lemma \\ref{l. adim vdim and Gin}(i), \n${\\ensuremath{\\mathrm{adim}}}(X,6,5)>0$ if and only if $\\dim [\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^5]_6>0.$ In this case, the ideal defining the point $Q$ is $I_Q=(y,z)\\subseteq K[x,y,z]$, so the monomials of degree 6 contained in $[I_Q^5]_6$ are exactly those\nlexicographically less than or equal to $xy^5$.\nBut (in characteristic 0) generic initial ideals are (by \\cite{CaS}) strongly stable \n(see \\cite{AL} for the definition and properties of strong stability),\nhence if any monomial of degree 6 less than or equal to $xy^5$ is in $\\ensuremath{\\mathrm{gin}}(I_X)$,\nthen $xy^5$ is in $\\ensuremath{\\mathrm{gin}}(I_X)$ too. \nThus $X$ admits an unexpected curve of degree 6 if and only if $xy^5\\in \\ensuremath{\\mathrm{gin}}(I_X)$.\n\nNow we determine all the monomials in $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}$, assuming that $X$ admits no unexpected curves in degrees strictly lower than 6. With this assumption and using the fact that unexpected curves have degree at least 4 \n\\cite{A, FGST}, from Lemma \\ref{l. adim vdim and Gin}(i) we have \n\\begin{itemize}\n\t\\item $[\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^3]_4=[\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^4]_5=(0)$, so if $x^ay^bz^c \\in [\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}$ then $a\\ge 2$, and\n\\smallskip\n\t\\item ${\\ensuremath{\\mathrm{edim}}}(X,4,2)={\\ensuremath{\\mathrm{adim}}}(X,4,2)=0$ (by Bertini's Theorem), hence $[ \\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^2]_4=(0)$, so if $x^ay^bz^c \\in [\\ensuremath{\\mathrm{gin}}(I_X)]_{4}$ then $a\\ge 3.$\n\\end{itemize}\nCollecting this information, we get only one strongly stable ideal through degree 5 (with the given Hilbert function), that is \n$(x^4, x^3y, x^3z, \nx^2y^3, x^2y^2z)$. So, $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 5}$ is a lex segment.\n\nTaking generators up to degree 6 of the lex-segment ideal with Hilbert function $h_X$ we get\n\\[ L = (x^4, x^3y, x^3z, x^2y^3, x^2y^2z, x^2yz^3, x^2z^4), \\]\nthus $X$ has an unexpected curve of degree 6 with a general multiple point of multiplicity 5 if and only if the component $[\\ensuremath{\\mathrm{gin}}(I_X)]_6$ \nfails to be equal to the degree 6 component of the lex-segment ideal with the same Hilbert function.\n\nAs promised, we now show that sets $X$ do arise.\nWe first give an example of a set $X$ of 13 points with no unexpected curves of degree 6 or less.\nWe get 12 points of the points of $X$\nas the complete intersection of a general cubic and quartic; add to this a general point $Q$\nto obtain $X$. It is easy to see that $X$ has the Hilbert function as specified above,\nand by direct computation with Macaulay2 \\cite{GS}, we find that\n$X$ has no unexpected curves of degree 6 or less. In this case, we find from Macaulay2 that\n$\\ensuremath{\\mathrm{gin}}(I_X)=(x^4, x^3y, x^3z,\nx^2y^3, x^2y^2z,\nx^2yz^3, x^2z^4,\nxy^6, xy^5z,\nxy^4z^3,\nxy^3z^5,\nxy^2z^7,\nxyz^9,\nxz^{11},\ny^{13})$, hence $xy^5$, as claimed, does not occur.\n\n\\begin{comment}\nMacaulay2 code for running the previous example\n\nR=QQ[x,y,z];\nC=random(3,R)\nQ=random(4,R)\nI=intersect(ideal(C,Q),ideal(x,y));\nfor i from 0 to 6 do print {i,hilbertFunction(i,I)}\nP=ideal(random(1,R),random(1,R));\nJ3=intersect(I,P^3);\nfor i from 0 to 6 do print {i,hilbertFunction(i,J3)\n J4=intersect(I,P^4);\nfor i from 0 to 6 do print {i,hilbertFunction(i,J4)}\nJ5=intersect(I,P^5);\nfor i from 0 to 6 do print {i,hilbertFunction(i,J5)}\nloadPackage \"GenericInitialIdeal\"\nlexgin(I)\n\nNote 1: To check the claim about the HF of G_2 use:\nK=ideal(x^4, x^3*y, x^3*z, x^2*y^3, x^2*y^2*z, x*y^5, x*y^4*z);\n\\end{comment}\n\nWe now give two examples of an $X$ with the specified Hilbert function which do have unexpected sextics.\nReturning to the specfied Hilbert function, we see that two monomials of degree 6 are needed in the minimal set of generators of $\\ensuremath{\\mathrm{gin}}(I_X)$. If $X$ admits an unexpected curve of degree 6 with a general multiple point of multiplicity 5, we have already noticed that $xy^5\\in \\ensuremath{\\mathrm{gin}} I_X$.\nThus, the strongly stable property forces $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}$ to be either\n\\[\n\\mathcal G_1 = (x^4, x^3y, x^3z, x^2y^3, x^2y^2z,x^2yz^3, xy^5)\n\\]\nor \n\\[\n\\mathcal G_2 = (x^4, x^3y, x^3z, x^2y^3, x^2y^2z, xy^5, xy^4z).\n\\]\nBy direct computation we see that a set of points $X$ with $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}= \\mathcal G_2$ would have ${\\ensuremath{\\mathrm{adim}}}(X,6,5)=2$,\nbut by \\cite[Corollary 5.5]{CHMN} this would mean $X$ has an unexpected quintic, contrary to our assumption\nthat $X$ admits no unexpected curves in degrees strictly lower than 6.\nThus $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}= \\mathcal G_2$ cannot occur if $X$ is a reduced set of 13 points.\n\nHowever, $[\\ensuremath{\\mathrm{gin}}(I_X)]_{\\le 6}= \\mathcal G_1$ can occur; we give two examples.\nSpecifically, the following two sets of points $X_1$ and $X_2$ have $h_{X_1}=h_{X_2}=h_X$ \nand $[\\ensuremath{\\mathrm{gin}}(I_{X_i})]_{\\le 6}= \\mathcal G_1$. The lines (see Figure \\ref{f.line config X }) dual to the points $X_1$\ngive what \\cite{DMO} refers to as a $(1,1)$ tic-tac-toe arrangement:\n \n\\[\\begin{array}{ccc}\nX_1&:=&\\{(1,0,0), (0,1,0), (0,0,1), (1,1,0),(0,1,1),(1,0,1), (-1,1,0),(0,-1,1),(-1,0,1),\\\\\n&&(1,1,1), (-1,1,1),(-1,1,-1),(1,1,-1)\\}.\\\\\n\\end{array}\n\t\\]\nThe ideal defining $X_1$ is\n\\[\nI_{X_1}=(y^3z - yz^3, x^3z - xz^3, x^3y - xy^3).\n\\]\n\n \\begin{figure}[!ht]\n \t\\centering\n \t\\begin{tikzpicture}[scale=0.7]\n \\draw (-4,-1) -- (4,-1);\n \\draw (-4,0) -- (4,0);\t\n \\draw (-4,1) -- (4,1);\n \n \\draw (1,-4) -- (1,4);\n \\draw (0,-4) -- (0,4);\t\n \\draw (-1,-4) -- (-1,4); \t\n\n \\draw (-4,-3) -- (3,4);\n \\draw (-4,-4) -- (4,4);\n \\draw (-3,-4) -- (4,3);\n\n \\draw (-4,3) -- (3,-4);\n \\draw (-4,4) -- (4,-4);\n \\draw (-3,4) -- (4,-3);\n \n \t\\end{tikzpicture}\n \t\\caption{A sketch of the line configuration dual to the points of $X_1$ from Example \\ref{Fig1Fig2Example}. \n\t(The line at infinity, corresponding to the point $(0,0,1)$, is not shown).}\n \t\\label{f.line config X } \n \\end{figure}\n\nThe second set of points is\n\\[\\begin{array}{ccc}\nX_2&:=&\\{(1,0,0), (0,1,0), (0,0,1), (1,1,0),(0,1,1),(1,0,1), (-1,1,0),(0,-1,1),(-1,0,1),\\\\\n&&(2,1,1), (-2,1,1),(-2,1,-1),(2,1,-1)\\};\\\\\n\\end{array}\n\\]\nit is defined by the ideal\n\\[I_{X_2}=(y^3z - yz^3, x^3z - xz^3 - 3xy^2z, x^3y - xy^3 - 3xyz^2).\\]\n(The lines dual to $X_2$ are shown in Figure \\ref{f.line config X'}.)\n \n \\begin{figure}[!ht]\n \t\\centering\n \t\\begin{tikzpicture}[scale=0.7]\n \\draw (-4,-1) -- (4,-1);\n \\draw (-4,0) -- (4,0);\t\n \\draw (-4,1) -- (4,1);\n \n \\draw (1,-4) -- (1,4);\n \\draw (0,-4) -- (0,4);\t\n \\draw (-1,-4) -- (-1,4); \t\n \t\n \t\\draw (-2.5,-4) -- (1.5,4);\n \t\\draw (-4,-4) -- (4,4);\n \t\\draw (-1.5,-4) -- (2.5,4);\n \t\n \t\\draw (-2.5,4) -- (1.5,-4);\n \t\\draw (-4,4) -- (4,-4);\n \t\\draw (-1.5,4) -- (2.5,-4);\n \t\n \t\\end{tikzpicture}\n \t\\caption{A sketch of the line configuration dual to the points of $X_2$ from Example \\ref{Fig1Fig2Example}. \n\t(The line at infinity, corresponding to the point $(0,0,1)$, is not shown).}\n \t\\label{f.line config X'} \n \\end{figure}\n\\end{example}\n\n\nWe now show that the sequence $AV_{X,j}$ is the Hilbert \nfunction of a fixed standard graded algebra; since $AV_{X,j}(m)$ is defined for $m\\ge 1$ we need to shift it by 1.\n\n\\begin{theorem}\\label{t. AV is an O-sequence}\nFor any non-negative integer $j$, the sequence $AV_{X,j}$ \nshifted to the left by $1$ is an $O$-sequence. In particular, setting $J:=\\ensuremath{\\mathrm{gin}}(I_X)\\ :\\ x_0^{j+1},$ \nthe sequence $AV_{X,j}$ shifted to the left by $1$ coincides with the Hilbert function of $R\/J$, i.e.,\n\t\\[\n\tAV_{X,j}(d+1)=h_{R\/J}(d), \\ d\\ge 0.\n\t\\]\n\n\\end{theorem}\n\\begin{proof}Set $\\mathfrak q= (x_1,\\cdots, x_n)$ the ideal defining the point $Q=(1,0,\\ldots,0)$.\nFor any $m\\ge 1$ and for any non-negative integer $j$, from Lemma \\ref{l.AV seq 1}, reasoning as in the proof of Lemma \\ref{l. adim vdim and Gin}, we have\n$$AV_{X,j}(m)=\\dim\\left[ \\faktor{R}{(I_X + I_P^m)}\\right]_{m+j}=\\dim\\left[ \\faktor{R}{( \\ensuremath{\\mathrm{gin}}(I_X) + \\mathfrak q^m)}\\right]_{m+j} .$$\n\nFor a monomial ideal $T$, it is easy to show that we can write $\\left[T\\right]_{m+j}$ as the following direct sum (and so the summands have only 0 in common):\n\\[ \\left[T\\right]_{m+j}= \\left[x_0^{j+1}\\cdot (T : x_0^{j+1}) \\right]_{m+j}\\bigoplus\\left[ T\\cap \\mathfrak q^m\\right]_{m+j}.\\]\nIn particular, \n$$\\left[R\\right]_{m+j}= \\left[x_0^{j+1}\\cdot (R : x_0^{j+1}) \\right]_{m+j}\\bigoplus\\left[ R\\cap \\mathfrak q^m\\right]_{m+j}=\\left[x_0^{j+1}R\\right]_{m+j}\\bigoplus\\left[\\mathfrak q^m\\right]_{m+j},$$ \nso $\\dim \\left[{R\/\\mathfrak q^m}\\right]_{m+j}=\\dim \\left[x_0^{j+1}R\\right]_{m+j}$.\nSimilarly, \n\\begin{align*}\n\\left[\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m\\right]_{m+j}&=\\left[x_0^{j+1}\\cdot ((\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m) : x_0^{j+1}) \\right]_{m+j}\\bigoplus\\left[(\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m)\\cap \\mathfrak q^m\\right]_{m+j}\\\\\n&=\\left[x_0^{j+1}\\cdot (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1}) \\right]_{m+j}\\bigoplus\\left[\\mathfrak q^m\\right]_{m+j},\n\\end{align*}\nso $\\dim \\left[ {(\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m)\/\\mathfrak q^m}\\right]_{m+j}=\\dim \\left[x_0^{j+1}\\cdot (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1}) \\right]_{m+j}$.\nThus\n\t\\[\\begin{array}{rcl}\n\t\\dim\\left[\\dfrac{R}{\\ensuremath{\\mathrm{gin}}(I_{X}) +\\mathfrak q^m}\\right]_{m+j}&=& \\dim \\left[\\dfrac{R\/\\mathfrak q^m}{(\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m)\/\\mathfrak q^m}\\right]_{m+j}\\\\\n\t& & \\\\\n\t&=& \\dim \\left[{R\/\\mathfrak q^m}\\right]_{m+j} -\\dim \\left[ {(\\ensuremath{\\mathrm{gin}}(I_{X}) + \\mathfrak q^m)\/\\mathfrak q^m}\\right]_{m+j}\\\\\n\t& & \\\\\n\t&=& \\dim \\left[x_0^{j+1}R \\right]_{m+j} -\\dim \\left[ x_0^{j+1}\\cdot (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1})\\right]_{m+j}\\\\\n\t& & \\\\\n\t&=& \\dim \\left[R \\right]_{m-1} -\\dim \\left[\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1}\\right]_{m-1}\\\\\n\t& & \\\\\n\t&=& \\dim \\left[R \/ (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1})\\right]_{m-1}.\\\\\n\t\\end{array}\n\t\\]\t \n\\end{proof}\n\nAs a consequence of Theorem \\ref{t. AV is an O-sequence} we get a new criterion\n(in this case, geometric) for \nthe non-existence of unexpected hypersurfaces. \n\n\\begin{corollary} \\label{degenerate}\n\tIf $X$ is a reduced subscheme of $ \\ensuremath{\\mathbb{P}}^n$ contained in a hypersurface of degree $d+1\\geq1$, \n\tthen for any $t\\geq d+m$ and $m\\ge 1$ we have \n\\[\n{\\ensuremath{\\mathrm{adim}}} (X,t,m) = {\\ensuremath{\\mathrm{vdim}}} (X,t,m).\n\\]\nIn particular, if $X$ is degenerate (meaning, $X$ is contained in a hyperplane, or, equivalently, $d=0$), then\n$X$ admits no unexpected hypersurfaces for multiplicity $m$ in degrees $t\\geq m$,\nand hence no unexpected hypersurfaces of any kind.\n\\end{corollary}\n\n\\begin{proof}\n\tSince $I_X$ has an element of degree $d+1$, $x_0^{d+1}\\in \\ensuremath{\\mathrm{gin}}(I_X)$. This implies \n\t$1\\in (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{t-m+1})$ when $t\\geq d+m, m\\geq 0$. \n\tFrom Theorem \\ref{t. AV is an O-sequence} and Definition \\ref{def of AV},\n\tfor $t=m+j, j\\geq d, m\\geq 1$, we have $0=AV_{X,j}(m)={\\ensuremath{\\mathrm{adim}}}(X,t,m) - {\\ensuremath{\\mathrm{vdim}}}(X,t,m)$.\n\tThe last part, about unexpected hypersurfaces, follows from Definition \\ref{unexpDef}.\n\\end{proof}\n\nThe next corollary compares the sequences $AV_{X,j} (m) $ and $ AV_{X,j+1}(m)$ for a subscheme $X$.\n\n\\begin{corollary} \\label{ineq for AV}\n\t\tLet $X$ be a subscheme of $ \\ensuremath{\\mathbb{P}}^n$. Then\n\t\t\\[AV_{X,j} (m) \\ge AV_{X,j+1}(m).\\] \n\\end{corollary}\n\n\\begin{proof}From Theorem \\ref{t. AV is an O-sequence} we have, for any $i$,\n\\[\nAV_{X,i} (m) = \\dim \\left[R \/ (\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{i+1})\\right]_{m-1}.\n\\]\nSo the statement is equivalent to proving that\n\\[\n\\dim \\left[\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1} \\right]_{m-1}\\le \\dim \\left[\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+2} \\right]_{m-1},\n\\]\nand this is trivial since we always have $\\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+1} \\subseteq \\ensuremath{\\mathrm{gin}}(I_{X}) : x_0^{j+2}.$ \n\\end{proof}\n\n\\begin{lemma} \\label{subscheme}\nLet $P$ be a general point in $ \\ensuremath{\\mathbb{P}}^n$. \nLet $Y_1$ be the zero-dimensional subscheme of $ \\ensuremath{\\mathbb{P}}^n$ defined by $I_P^m$ and let $Y_2$ be the subscheme of $Y_1$ defined by $I_P^{m-1}$. Fix a positive integer~$t \\geq m-1$ and consider the component $ [I_X]_t$, where $X$ is some subvariety of $ \\ensuremath{\\mathbb{P}}^n$. If $Y_1$ imposes $\\binom{m-1+n}{n}$ independent conditions on $[I_X]_t$ then $Y_2$ imposes $\\binom{m-2+n}{n}$ independent conditions on $[I_X]_t$.\n\\end{lemma}\n\n\\begin{proof}\nWe know that $\\deg(Y_1) = \\binom{m-1+n}{n}$ and $\\deg(Y_2) = \\binom{m-2+n}{n}$, and \n that $Y_1$ and $Y_2$ impose independent conditions on $[R]_t$ since $t \\geq m-1$ (by a regularity argument). In particular we know that \n\\[\n\\dim [I_{Y_1}]_t = \\dim [R]_t - \\binom{m-1+n}{n} \\ \\ \\hbox{ and } \\ \\ \\dim [I_{Y_2}]_t = \\dim [R]_t - \\binom{m-2+n}{n}.\n\\]\nConsider first the exact sequence\n\\[\n0 \\rightarrow I_{Y_1} \\rightarrow I_{Y_2} \\rightarrow A \\rightarrow 0,\n\\]\nwhere $A$ is the quotient, and is supported on $P$. We have\n\\[\n\\dim[A]_t = \\binom{m-1+n}{n} - \\binom{m-2+n}{n} = \\binom{m-2+n}{n-1}.\n\\]\nLet $\\mathcal A$ be the sheafification of $A$. From the above exact sequence and the fact that $t \\geq m-1$ (so $h^1(\\mathcal I_P^m(t)) = h^1(\\mathcal I_{Y_1}(t)) = 0$) and the fact that $I_{Y_1}$ and $I_{Y_2}$ are saturated, we get $h^0(\\mathcal A(t)) = \\dim [A]_t$. \n\nRemembering that $X$ and $P$ are disjoint, consider the exact sequence of sheaves\n\\[\n0 \\rightarrow \\mathcal I_{X \\cup Y_1}(t) \\rightarrow \\mathcal I_{X \\cup Y_2}(t) \\rightarrow \\mathcal A(t) \\rightarrow 0.\n\\]\nTaking cohomology we obtain\n\\[\n0 \\rightarrow [I_X \\cap I_P^m]_t \\rightarrow [I_X \\cap I_P^{m-1}]_t \\rightarrow [A]_t \\rightarrow \\cdots\n\\]\nHence\n\\[\n\\begin{array}{rcl}\n\\dim [I_X \\cap I_{Y_2}]_t & = & \\dim [I_X \\cap I_P^{m-1}]_t \\\\ \\\\\n& \\leq & \\dim [I_X \\cap I_P^m]_t + \\dim [A]_t \\\\ \\\\\n& = & \\dim [I_X]_t - \\binom{m-1+n}{n} + \\binom{m-2+n}{n-1} \\\\ \\\\\n& = & \\dim [I_X]_t - \\binom{m-2+n}{n}.\n\\end{array}\n\\]\nBut $Y_2$ cannot impose more conditions in degree $t$ than its degree, so we must have equality.\n\\end{proof}\n\nLike Corollary \\ref{degenerate}, the next result gives a criterion (again basically geometric), \nbased on one single piece of information,\nfor a subvariety $X \\subset \\ensuremath{\\mathbb{P}}^n$ to admit no unexpected hypersurfaces of any degree or multiplicity at a general point.\n\n\\begin{proposition} \\label{AV_X,0(alpha)}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^n$ be a subscheme. Let $\\alpha:= \\alpha(I_X)$ be the least degree $t$ such that $[I_X]_t\\neq 0$. If $AV_{X,0} (\\alpha)=0$, then $X$ does not admit unexpected hypersurfaces, for any degree and multiplicity at a general point. \n\\end{proposition}\n\n\\begin{proof}\nBy Remark \\ref{equiv to unexp}, $X$ fails to admit an unexpected hypersurface \nof degree $t$ for multiplicity $m$ if and only if $T_{t,m}=\\min({\\ensuremath{\\mathrm{adim}}}(X,t,m), AV_{X,t-m}(m))=0$.\nConsider the table $T$ of values of $T_{t,m}$ for $t,m\\geq1$. It suffices to show we always have\n$T_{t,m}=0$. We will divide the table into three regions as follows:\n\\[\n\\begin{array}{cc|ccccc|ccccccccccccccccc}\n& t & 1 & 2 & 3 & \\dots & \\alpha-1 & \\alpha & \\alpha +1 & \\alpha+2 & \\alpha+3 & \\dots\\\\ \nm & &&&&&&&&& \\\\ \\hline\n1 & &&&&&&&&& \\\\ \n2 & &&&&&&&&& \\\\\n\\vdots & &&&&&&& & {\\rm\\Large III}& \\\\\n\\alpha-1 & &&&&&&&&& \\\\ \\cline{8-12}\n\\alpha & && && {\\rm\\Large I}&&&&& \\\\\n\\alpha +1 & &&&&&&&&& \\\\\n\\vdots & &&&&&&&& {\\rm\\Large II} & \\\\\n\n\\end{array}\n\\]\nAs always, $P$ denotes a general point.\n\nFor $t < \\alpha$ it is clear that $T_{t,m}={\\ensuremath{\\mathrm{adim}}}(X,t,m)=0$. \nThis takes care of region I.\n\nWe now show that all entries in region II are 0. The condition $AV_{X,0}(\\alpha) = 0$ gives us that $T_{\\alpha, \\alpha}=0$;\nthis the top left point of region II.\nIt also gives us that $Y_1$, defined by $I_P^\\alpha$, imposes $\\binom{\\alpha-1+n}{n}$ independent conditions on $[I_X]_\\alpha$.\nBut then by Corollary \\ref{ineq for AV} we have $T_{j+\\alpha,\\alpha}=AV_{X,j}(\\alpha) = 0$ for all $j \\geq 0$ \n(this gives the top row of region~II). \n\nIn the portion below the main diagonal of region II we have $m > t$, so $T_{t,m}={\\ensuremath{\\mathrm{adim}}}(X,t,m)$ is 0,\nsince there there can be no hypersurfaces with degree $t$ and multiplicity $m$ (unexpected or not). \nFor the portion on and above the main diagonal of region II, \nwe apply Theorem \\ref{t. AV is an O-sequence}, which says that $AV_{X,j}(m)$ is an $O$-sequence. \nRecall that the top row of region II gives us $AV_{X,j}(\\alpha) = 0$ for $j \\geq 0$.\nThen it follows for $i>0$ that $T_{\\alpha+i+j,\\alpha+i}=AV_{X,j}(\\alpha+i) = 0$ as well, since a Hilbert function that attains a value 0 cannot subsequently become non-zero. Thus starting from each entry on the top row of region II (all of which are 0), the entries \ndescending diagonally and to the right are all 0. Thus all entries in region II above the main diagonal are also 0.\n\nWe now show all entries in region III are 0.\nBecause $AV_{X,j}(\\alpha) = 0$, the scheme $Y_1$ defined by the (saturated) ideal $I_P^\\alpha$ \nimposes independent conditions on $[I_X]_t$ for all $t\\geq\\alpha$. Hence, applying Lemma \\ref{subscheme}\niteratively, so does the scheme $Y_2$ defined by $I_P^k$ for $\\alpha-1 \\geq k \\geq 1$.\nThus $T_{t,k}=AV_{X,t-k}(k) = 0$ for $t\\geq\\alpha$. Thus all entries in region III are 0, and we are done.\n\\end{proof}\n\nThe next proposition shows that if a subscheme $X$ admits an unexpected hypersurface of \ndegree $t$ vanishing with multiplicity $m$ at a general point, then $[\\ensuremath{\\mathrm{gin}}(I_X)]_t$ is not a lex-segment. \nThis gives a criterion (this time algebraic), again based on a single piece of information \n(admittedly more difficult to verify), for the non-existence of any unexpected hypersurfaces.\n\n\\begin{proposition}\\label{p.lex segments are bad for unexpectedness}\n\tLet $X$ be a subscheme of $ \\ensuremath{\\mathbb{P}}^n$. Assume there exists $j\\ge 0$ such that \n\t$[\\ensuremath{\\mathrm{gin}}(I_X)]_{m+j}$ is a lex-segment and $\\hbox{adim(X,m+j,m)}>0.$ Then \n\t $AV_{X,j}(m)=0.$ In particular, if $\\ensuremath{\\mathrm{gin}}(I_X)$ is a lex-segment ideal, then $X$ does not admit any unexpected hypersurfaces.\n\\end{proposition}\n\n\\begin{proof}\n Let $j\\ge 0$ be an integer such that $X$ admits a hypersurface of degree $m+j$ \n vanishing with multiplicity at least $m$ at a general point $P$. Then by Lemma \\ref{l. adim vdim and Gin}(i) \n we have ${\\ensuremath{\\mathrm{adim}}} (X,t,m)=\\dim [\\ensuremath{\\mathrm{gin}}(I_X)\\cap I_Q^m]_t>0$, so\n there is a monomial $x_0^sM\\in \\ensuremath{\\mathrm{gin}}(I_X)$ where $M]\\in I_P^m$ so has degree $\\deg M=m+j-s\\geq m$, \n but $j\\geq s\\geq0$.\n Now, since $[\\ensuremath{\\mathrm{gin}}(I_X)]_{m+j}$ is a lex-segment, we have $x_0^sM \\in \\ensuremath{\\mathrm{gin}}(I_X)$. \n In order to prove the statement, it is enough by Theorem \\ref{t. AV is an O-sequence} to show that\n $\\dim [R\/(\\ensuremath{\\mathrm{gin}}(I_X):x_0^{j+1})]_{m-1}=0$. So, take any monomial \n $F\\in R_{m-1}$. Then $x_0^{j+1}F\\in[R]_{m+j}$ and $x_0^{j+1}F>_{lex} x_0^sM$. Thus, $x_0^{j+1}F\\in \\ensuremath{\\mathrm{gin}}(I_X)$,\n so $F\\in \\ensuremath{\\mathrm{gin}}(I_X):x_0^{j+1}$, hence $[R\/(\\ensuremath{\\mathrm{gin}}(I_X):x_0^{j+1})]_{m-1}=0$.\n\\end{proof}\n\n It is natural to ask if the converse of Proposition \\ref{p.lex segments are bad for \n\tunexpectedness} is true, namely if it is true for a finite set of points $X$ that if \n$\\ensuremath{\\mathrm{gin}}(I_X)$ is {\\em not} a lex-segment ideal then $X$ must admit some sort of \nunexpected hypersurface. Recalling that one must take with a grain of salt the \ngeneric initial ideal produced by a computer algebra program (is the change of \nvariables ``general enough?\"), a counterexample is given in Example \\ref{ex.root A_n}. \nIt would be interesting to have a theoretical procedure to determine if the generic initial \nideal of a finite set of points is a lex-segment ideal or not. \n\n\\begin{example}\\label{ex.root A_n} \nAssume the characteristic of the field $K$ is $0$. Let $X_n\\subseteq \\ensuremath{\\mathbb{P}}^n$ be the set of $\\binom{n+2}{2}$ points obtained from the root system $A_{n+1}$ as described in \\cite{HMNT}, section 3.1. Specifically, $X_n$ consists of the $\\binom{n+1}{2}$ points having one entry equal to 1, one equal to $-1$ and the rest 0, together with the $n+1$ coordinate points.\n\t \t\n\t\tThe initial degree of $I_{X_n}\\subseteq K[x_0, x_1, \\ldots, x_n]$ is $\\alpha(I_{X_n})=3$ for $n\\ge 2$, because the product of any 3 indeterminates vanishes at $X_n$ and $[I_{X_n}]_2= (0)$. To see that $X_n$ does not lie on a quadric we take $F:=\\sum c_{ab}x_ax_b\\in I_{X_n}$ and we show that $F=0$. Indeed, since $F$ vanishes at the coordinate points, $c_{ab}=0$ if $a= b$; also $c_{ab}=0$ for $a\\neq b$ because $F$ vanishes at the point having no zero entries at the positions $a$ and $b$. \n\t\t\n\t\tA computer calculation, by {\\hbox{C\\kern-.13em o\\kern-.07em C\\kern-.13em o\\kern-.15em A}}\\ \\cite{cocoa}, showed that $AV_{X_n,0}(3)=0$ for $2\\le n\\le 12$. \n\t\tTherefore, by Proposition \\ref{AV_X,0(alpha)}, $X_n$ does not admit any unexpected hypersurface of any sort, for $2\\le n\\le 12$.\n\t\t\n\t\tThis result is consistent with \\cite{HMNT}, where a computer search did not turn up any unexpected hypersurfaces\n\t\tfor $X_n$ in the cases $2\\leq n\\leq 6$, $2 \\leq d \\leq 6$, $2 \\leq m \\leq d$.\n\t\t\n\t\tFurthermore, we checked with {\\hbox{C\\kern-.13em o\\kern-.07em C\\kern-.13em o\\kern-.15em A}}\\ that, for $4\\le n\\le 12$, the ideal defining the set of points $X_n\\subseteq \\ensuremath{\\mathbb{P}}^n$ has a generic initial ideal that is not a lex-segment ideal. In particular, in the cases $5 \\le n\\le 12$ we noticed that $\\left[\\ensuremath{\\mathrm{gin}}(I_{X_n})\\right]_3$ fails to be a lex-segment because $x_1x_n^2$ fails to belong to $\\ensuremath{\\mathrm{gin}}(I_{X_n})$.\n\t\t\n\t\tInterestingly, we have checked for $3\\le n\\le 12$ that the set of points $Y_n\\subseteq \\ensuremath{\\mathbb{P}}^n$, constructed from $X_n$ by replacing in the coordinates of its points all the ``-1\" with ``+1,\" admits an unexpected cone of degree 3, and $x_0x_n^2$ fails to belong to $\\ensuremath{\\mathrm{gin}}(I_{Y_n})$.\n\\end{example}\n\n\n\\section{On the sequence $AV_{X,1}(m)$ when $X$ is an irreducible ACM curve in $ \\ensuremath{\\mathbb{P}}^3$}\\label{s.irreducible ACM curve in P3}\n\nIn this section we will assume that $K$ has characteristic zero. We have already seen in \nTheorem \\ref{t. AV is an O-sequence} that for a subscheme $X \\subset \\ensuremath{\\mathbb{P}}^n$, the \nsequence $AV_{X,j}$ shifted to the left by 1 is an $O$-sequence. In this section we will abuse \nterminology and just say that the sequence is an $O$-sequence, often (but not always) suppressing \nthe shift. Furthermore, if the positive part of the sequence $AV_{X,j}$ is finite, we will ignore the terms \nthat are zero and just say that $AV_{X,j}$ is finite. In this section our focus is on the case $j=1$.\n\nRecall that an SI-sequence is a finite, non-zero, symmetric $O$-sequence such that the first half is a \ndifferentiable $O$-sequence (i.e., also the first difference of the first half is an $O$-sequence). \nThe significance of SI-sequences is that they characterize the $h$-vectors of arithmetically \nGorenstein subschemes of projective space whose artinian reductions have the Weak Lefschetz Property. \nIn codimension three they characterize the $h$-vectors of all arithmetically Gorenstein subschemes. \nNote that SI-sequences are automatically unimodal. See \\cite{Hrm} for properties of SI-sequences.\n\nWe have produced a great deal of experimental evidence for the following conjecture\nconcerning SI-sequences. We will shortly prove part of it.\nWe recall that an ACM scheme is always connected \\cite[Theorem 18.12]{E}, hence a smooth\nACM scheme is irreducible.\n\n\\begin{conjecture}\\label{SI conj}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^3$ be a smooth ACM curve not lying on a quadric surface. \nThen the sequence $AV_{X,1}$ is an SI-sequence (shifted by 1). The last non-zero term in this sequence \nis $AV_{X,1}(\\deg X - 5)$, so the SI-sequence ends in degree $\\deg X - 6$.\n\\end{conjecture}\n\n \\begin{remark}\n We point out two things. First, \nwe have not yet seen a direct connection to Gorenstein algebras; we have only this numerical conjecture. It would be very interesting to tie Gorenstein algebras to the study of unexpected hypersurfaces in some way. Second, \n sometimes the SI-sequences that we obtain have codimension three and sometimes codimension four.\n \\end{remark}\n \n The case where $X$ {\\em does} lie on a quadric surface is contained in Theorem \\ref{part of conj}.\nWe first give examples to show that all of the other assumptions are needed in this conjecture. \n\n\\begin{example} \\label{hyp not hold}\nThe symmetry of the sequence $AV_{X,1}(m)$ requires all of the given assumptions. \nThe following examples, which were run in either {\\hbox{C\\kern-.13em o\\kern-.07em C\\kern-.13em o\\kern-.15em A}}\\ \\cite{cocoa} or Macaulay2 \\cite{GS}, show that dropping a hypothesis can\nresult in the AV sequence either not being finite, or, if finite, not being symmetric.\n\n\\begin{itemize}\n\n\\item[(a)] ({\\em $X$ satisfies all assumptions.}) Assume that $X_1$ is a line in $ \\ensuremath{\\mathbb{P}}^3$ \nand $X$ is obtained as the residual to $X_1$ by two general cubic surfaces containing $X_1$. \nThen $X$ is a smooth ACM curve of degree 8 and genus 7, and the positive part of the sequence \n$AV_{X,1}$ is $(1,2,1)$. This satisfies Conjecture \\ref{SI conj}, and in particular it is symmetric\nwith its last value being in degree $8-6=2$.\n\n\\item[(b)] ({\\em $X$ is not ACM}.) Assume that $X_1$ is the disjoint union of two lines in \n$ \\ensuremath{\\mathbb{P}}^3$ and $X$ is linked to $X_1$ by a general choice of two cubic surfaces. \nThen $X$ is a smooth, non-ACM curve of degree 7 and genus 4, and the positive part of \nthe sequence $AV_{X,1}$ is $(1,2)$, which is not symmetric.\n\n\\item[(c)] ({\\em $X$ is not in $ \\ensuremath{\\mathbb{P}}^3$}.) Assume that $X$ is a smooth surface of \ndegree 8 in $ \\ensuremath{\\mathbb{P}}^4$ obtained from a plane by linking using two general hypersurfaces of \ndegree 3. (The example in (a) is a hyperplane section of this one.) Then the positive part of \nthe sequence $AV_{X,1}$ is $(1,3,4,4,\\dots)$, which is not finite. (Note that its first difference is the sequence in (a).)\n\n\\item[(d)] ({\\em $X$ is not equidimensional, but the curve part is ACM}.) Assume that $X$ is the \nresidual in $ \\ensuremath{\\mathbb{P}}^3$ of a line inside the complete intersection of two cubics (as in \n(a)). Let $Y$ be the union of $X$ with a general point. Then the positive part of $AV_{Y,1}$ is the sequence $(1,3,2)$,\nwhich is not symmetric.\n\n\\item[(e)] ({\\em $j = 0$ instead of $j=1$}.) Assume that $X$ is the curve in (a) but take $j=0$. \nThen the positive part of $AV_{X,0}$ is the sequence $(1, 4, 8, 11, 13, 14, 14, 14, \\dots)$,\nwhich is not finite.\n\n\\item[(f)] ({\\em $j = 2$ instead of $j = 1$}.) Assume that $X$ is the curve in (a) but take $j=2$. \nThen $AV_{X,2}$ is the sequence $(0,0,\\dots)$, and so is finite and vacuously symmetric, but does not end in the conjectured degree. However, linking the line \n$X_1$ from (a) using two surfaces of degree 4 gives a curve $X$ of degree 15 and genus 28, with the positive part of $AV_{X,2}$ being the sequence $(1,2,2)$,\nwhich is not symmetric.\n\n\\item[(g)] ({\\em $X$ is zero-dimensional}.) Assume that $X$ is the complete intersection in $ \\ensuremath{\\mathbb{P}}^3$ \nof three general quartic surfaces. Then the positive part of the sequence $AV_{X,1}$ is $(1, 4, 7, 8, 5)$\nwhich is not symmetric.\n\n\\item[(h)] ({\\em $X$ is ACM but not irreducible, 1}.) Let $Z \\subset \\ensuremath{\\mathbb{P}}^2$ be the set of 9 points \ncoming from the $B_3$ root system (see \\cite[Figure 2]{DIV} and \\cite{HMNT}). Let $X$ be the cone over $Z$ with vertex \nat a general point $Q$. Then $X$ is an ACM union of lines (whose general hyperplane section admits an \nunexpected quartic curve), and the positive part of the sequence $AV_{X,1}$ is $(1,3,4,4,4,\\dots)$, which is not finite.\n\n\\item[(i)] ({\\em $X$ is ACM but not irreducible, 2}). Let $X$ be as in (h), except now let $Z \\subset \\ensuremath{\\mathbb{P}}^2$ be a general \nset of 9 points. The Hilbert function of $Z$ (and hence $X$) is the same as it was in (h), but now one computes that \nthe positive part of the sequence $AV_{X,1}$ is $(1,3,3,3,\\dots)$, which is not finite. (Thus the AV sequence depends on\ngeometry beyond the Hilbert function.)\n\n\\item[(j)] ({\\em $X$ is ACM and irreducible but not smooth}). Let $Q$ be a general point and \nlet $X$ be the complete intersection of two general quartic surfaces in $I_Q^3$. Then $X$ is \nACM and we have verified that $X$ is irreducible (using Macaulay2), but the positive part of the \nsequence $AV_{X,1}$ is $(1, 4, 8, 12, 15, 16, 15, 12, 8, 4, 2, 2, 2, \\dots)$, which is not finite. (Thus we need \nsmoothness and not simply irreducibility in the statement of the conjecture.)\n\n\\end{itemize}\n\\end{example}\n\nWhile we are not able to prove the full conjecture, we at least show unimodality and the \ndifferentiability of the increasing part, using only irreducibility of $X$ and not necessarily \nsmoothness. What is missing is the finiteness, the symmetry and the degree of the last \npositive term in the sequence when $X$ is smooth. Notice that nothing about our \nargument fails for (i) or (j) in Example \\ref{hyp not hold}, so something more will be \nneeded to prove the rest of the conjecture.\n\n\\begin{theorem} \\label{part of conj} Let $X \\subset \\ensuremath{\\mathbb{P}}^3$ be an irreducible ACM curve. \n\n\\begin{itemize}\n\n\\item[(a)] If $X$ lies on a quadric surface, then for each $j\\geq1$ the sequence $AV_{X,j}(m)$ is zero.\n\n\\item[(b)] If $X$ does not lie on a quadric surface then the sequence $AV_{X,1}(m)$ is \nnon-zero and unimodal. Furthermore, the increasing part is a differentiable $O$-sequence.\n\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\t\n(a) This follows by Corollary \\ref{degenerate}.\n\n(b) We assume that \n\t\n\t\\begin{itemize}\n\t\t\\item $P \\in \\ensuremath{\\mathbb{P}}^3$ is a general point with defining ideal $I_P$, \n\t\t\n\t\t\\item $H$ is a general plane in $ \\ensuremath{\\mathbb{P}}^3$ containing $P$,\n\t\t\n\t\t\\item $L {\\color{blue} \\in I_P}$ is a linear form defining $H$,\n\t\t\n\t\t\\item $Z = X \\cap H$.\n\t\t\n\t\\end{itemize}\n\t\n\t\\noindent \n\tNote that $P$ may be taken to be a general point in $H$ with respect to the set $Z$. \n\tIn addition to the notation introduced in Remark~\\ref{cohom interp}, recall from Notation \\ref{Pm vs mP} that\n\t\\begin{quotation}\n\t\t{\\em we will denote the scheme defined by $I_{P|H}^m$ by $mP$, a fat point in the \n\t\tplane, to distinguish it from the fat point scheme $P^m$ in $ \\ensuremath{\\mathbb{P}}^3$ defined by $I_{P}^m$.} \n\t\\end{quotation}\n\tNotice that \n\t\\[\n\tI_{X \\cup P^{m+1}} : L = I_{X \\cup P^m} \\ \\ \\ \\hbox{ and } \\ \\ \\ (I_{X\\cup P^m} + (L))^{sat} = I_{Z \\cup mP}.\n\t\\]\n\tSince $Z = X \\cap H$ is a general hyperplane section of $X$, and $X$ is irreducible, $Z$ is \n\ta set of points in linearly general position in $H$ (in this case meaning no three points of $Z$ \n\tare collinear). Hence by \\cite{CHMN} Corollary 6.8, $Z$ does not admit any unexpected \n\tcurves in the plane. Considering the exact sequence of sheaves \n\t\\[\n\t0 \\rightarrow \\mathcal I_{Z \\cup (m+1)P} (m+2) \\rightarrow \\mathcal I_Z (m+2) \\stackrel{r_{m+2}}{\\longrightarrow} \\mathcal O_{(m+1)P} (m+2) \\rightarrow 0\n\t\\]\n\tfrom Remark \\ref{cohom interp} (where the ideal sheaves are on $H = \\ensuremath{\\mathbb{P}}^2$), the fact that $Z$ \n\tdoes not admit any unexpected curves means that $r_{m+2}$ has maximal rank on \n\tglobal sections. (For some values of $m$ it will be injective, and eventually it will be surjective.)\n\t\n\tNow consider the commutative diagram of sheaves (the rows are exact since $P\\not\\in X$ and the columns are exact since $X$ is irreducible):\n\t\\[\n\t\\begin{array}{cccccccccccccccc}\n\t&& 0 && 0 && 0 \\\\\n\t&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t0 & \\rightarrow & \\mathcal I_{X \\cup P^m}(m+1) & \\rightarrow & \\mathcal I_X (m+1) & \\rightarrow & \\mathcal O_{P^m}(m+1) & \\rightarrow & 0 \\\\\n\t&& \\phantom{{ \\times L}} {\\Big \\downarrow} {{ \\times L}} && \\phantom{{ \\times L}} {\\Big \\downarrow} {{ \\times L}} && \\phantom{{ \\times L}} {\\Big \\downarrow} {{ \\times L}} \\\\\n\t0 & \\rightarrow & \\mathcal I_{X \\cup P^{m+1}}(m+2) & \\rightarrow & \\mathcal I_X (m+2) & \\rightarrow & \\mathcal O_{P^{m+1}}(m+2) & \\rightarrow & 0 \\\\\n\t&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t0 & \\rightarrow & \\mathcal I_{Z \\cup (m+1)P}(m+2) & \\rightarrow & \\mathcal I_Z(m+2) & \\rightarrow & \\mathcal O_{(m+1)P}(m+2) & \\rightarrow & \\phantom{.} 0 .\\\\\n\t&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t&& 0 && 0 && 0 \n\t\\end{array}\n\t\\]\n\tIn cohomology we obtain\n\t{\\footnotesize\n\t\t\\[\n\t\t\\begin{array}{cccccccccccccccc}\n\t\t&& 0 && 0 && 0 \\\\\n\t\t&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t\t0 & \\rightarrow & [ I_{X \\cup P^m}]_{m+1} & \\rightarrow & [ I_X ]_{m+1}& \\rightarrow & H^0(\\mathcal O_{P^m}(m+1)) & \\rightarrow & H^1(\\mathcal I_{X \\cup P^m}(m+1)) & \\rightarrow & 0 \\\\\n\t\t&& {\\big \\downarrow} && {\\big \\downarrow} && {\\big \\downarrow} && \\phantom{\\alpha_{m+1}} {\\big \\downarrow} \\alpha_{m+1} \\\\\n\t\t0 & \\rightarrow & [ I_{X \\cup P^{m+1}}]_{m+2} & \\rightarrow & [ I_X ]_{m+2}& \\rightarrow & H^0(\\mathcal O_{P^{m+1}}(m+2)) & \\rightarrow & H^1(\\mathcal I_{X \\cup P^{m+1}}(m+2)) & \\rightarrow & 0 \\\\\n\t\t&& \\downarrow && \\downarrow* && \\downarrow && \\downarrow \\\\\n\t\t0 & \\rightarrow & [ I_{Z \\cup (m+1)P}]_{m+2} & \\rightarrow & [ I_Z ]_{m+2}& \\stackrel{r_{m+2}}{\\longrightarrow} & H^0(\\mathcal O_{(m+1)P}(m+2)) & \\rightarrow & \n\t\t\\coker (r_{m+2}) & \\rightarrow & \\phantom{.} 0 \\\\\n\t\t&&&& \\downarrow && \\downarrow && \\downarrow \\\\\n\t\t&&&& 0 && 0 && 0\n\t\t\\end{array} \n\t\t\\] }\nusing the fact that $X$ is ACM for the first vertical surjection (marked by an asterisk).\n\t\t\nTo begin, we focus only on the first line of the above commutative diagram. For $m=0$ we obtain\n\\[\nh^1(\\mathcal I_{X\\cup P^0} (1)) = h^1(\\mathcal I_{X} (1)) = 0\n\\]\nsince $X$ is ACM. For $m=1$ we obtain\n\\[\nh^1(\\mathcal I_{X \\cup P}(2)) = \n\\left \\{\n\\begin{array}{ll}\n0 & \\hbox{if $\\dim [I_X]_2 \\geq 1$} \\\\\n1 & \\hbox{if $\\dim [I_X]_2 = 0$.}\n\\end{array}\n\\right.\n\\]\nSince, by Remark \\ref{cohom interp} we have $h^1(\\mathcal I_{X \\cup P^m}(m+1)) = AV_{X,1}(m)$, \nand we know the latter is an $O$-sequence shifted by one thanks to Theorem \\ref{t. AV is an O-sequence}, \nthis gives that the start of the shifted $O$-sequence (the value 1) corresponds to $h^1(\\mathcal I_{X \\cup P}(2))$. \nThus the sequence is non-zero since $X$ does not lie on a quadric surface, so we have the first part of (b).\n(If we had merely assumed that $X$ is not degenerate, this also shows that the sequence is non-zero if and only if $X$ \ndoes not lie on a quadric surface.)\n\nSo for the rest of the proof we assume that $X$ does not lie on a quadric surface.\nApplying the Snake Lemma to the above commutative diagram, the fact that $r_{m+2}$ has \nmaximal rank means that also $\\alpha_{m+1}$ has maximal rank (in the right-hand column of \nthe diagram). Then applying Remark \\ref{cohom interp} as $m$ varies, since $r_{m+2}$ is initially \ninjective and then surjective, the same is true for the map $\\alpha_{m+1}$. This shows that the \nsequence $AV_{X,1}(m)$ is unimodal.\n\nFinally, we argue that the increasing part of the sequence $\\{h^1(\\mathcal I_{X \\cup P^m}(m+1))\\}$ is a \ndifferentiable $O$-sequence (shifted by 2).\n\nThanks to Theorem \\ref{t. AV is an O-sequence} and Remark \\ref{cohom interp}, we know \nthat the sequence $\\{ h^1(\\mathcal I_{X \\cup P^m}(m+1)) \\}$ is an $O$-sequence, and we have \njust seen that the map $\\alpha_{m+1}$ is injective as long as $\\coker (r_{m+2})$ is non-zero. So \nwe have only to show that $\\dim ( \\coker (r_{m+2}))$ is an $O$-sequence (shifted by~2). But \nbecause $r_{m+2}$ has maximal rank, when $\\coker (r_{m+2})$ is non-zero we have from the \nexactness of the bottom row of the above diagram that\n\\[\n\\begin{array}{rcl} \n\\displaystyle \\dim [ \\coker (r_{m+2}) ]_{m+2} & = & \\displaystyle \\binom{(m+1)-1+2}{2} - \\dim [I_Z]_{m+2} \\\\ \\\\\n& = & h_{R\/I_Z}(m+2) - [ 2(m+2) + 1 ] \n\\end{array}\n\\]\nas long as this number is positive. \n\nSo for $t \\geq 2$ {\\em we want to show that \n\\[\nk_{t-2} := h_{R\/I_Z}(t) - (2t+1) \n\\]\nis an $O$-sequence as long as it is positive}. Let $s = t-2$, so we want to show that \n$k_s$ is an $O$-sequence for $s \\geq 0$.\nSince $X$ does not lie on a quadric surface, $Z$ does not lie on a conic (since $X$ is ACM). \nThus $k_0 = \\dim [ \\coker (r_2) ]_2 = 6-5 = 1$. \n\nNow assume that $t \\geq 3$, so $s \\geq 1$. As long as $h_{R\/I_Z}(t) = \\binom{t+2}{2}$ \n(i.e., before $I_Z$ begins) this difference is $k_s = \\binom{s+2}{2}$, which is an $O$-sequence. So \nassume $h_{R\/I_Z}(t) < \\binom{t+2}{2}$. Consider the $t$-binomial expansion of $h_{R\/I_Z}(t)$. Since\n\\[\n2t+1 < h_{R\/I_Z}(t) < \\binom{t+2}{2},\n\\]\nwe have \n\\[\nh_{R\/I_Z}(t) = \\binom{t+1}{t} + \\binom{t}{t-1} + (\\hbox{terms in degrees $\\leq t-2$})\n\\]\nand \n\\[\n(2t+1) = \\binom{t+1}{t} + \\binom{t}{t-1}.\n\\]\nHence $k_s = h_{R\/I_Z}(t) - (2t+1)$ has an $s$-binomial expansion coming directly from the \n$t$-binomial expansion of $h_{R\/I_Z}(t)$, by removing the first two binomial coefficients. \nBut $h_{R\/I_Z}$ is an $O$-sequence, so it obeys Macaulay's bound (Theorem \\ref{macaulay thm}). \nThus $k_s$ does as well, and so is an $O$-sequence.\n\\end{proof}\n\n\n\\section{Codimension 2 complete intersections in $ \\ensuremath{\\mathbb{P}}^n$} \\label{ci section}\n\nIn this section we apply results of Section \\ref{s. gin and unexp} to the case of complete \nintersections of codimension 2 in $ \\ensuremath{\\mathbb{P}}^n$. In particular, as introduced in \nRemark \\ref{r: partial elimination ideal}, we get partial information on the actual dimension \nfor a complete intersection of codimension 2 in a certain degree by using the theory of \\textit{partial elimination ideals}. \n(See \\cite{Gr} for background on partial elimination ideals and for Sylvester matrices, which appear\nin the proof of Proposition \\ref{p. codimension 2 complete intersection}.)\nComplete intersections of codimension 2 have been investigated in several papers. Indeed, \nthe following result can be deduced from Proposition 6.8 in \\cite{Gr} and Corollary 3.9 in \\cite{CoS}. \nWe include here a proof for completeness of the exposition and to show explicitly how the \npartial elimination theory affects the existence of unexpected hypersurfaces. \n \nRecall that $R = K[x_0,x_1,\\ldots,x_n]$ denotes a standard graded polynomial ring and the monomials of $R$ are ordered by $>_{lex}$, the \nlexicographic monomial order which satisfies $x_0> x_1 > \\cdots > x_n$.\t\n\n\n\\begin{proposition}\\label{p. codimension 2 complete intersection}\n\tLet $C\\subseteq \\ensuremath{\\mathbb{P}}^n$ be a codimension 2 complete intersection. \n\tAssume $C$ is defined by two, sufficiently general, forms of degree $a,b$ respectively. Say $a\\le b$.\n\tSet, for any integer $0\\le j< a$,\n\\[\n\\begin{array}{rcl}\nt & := & (a-j)(b-j)+j \\hbox{ and } \\\\ \nm& := & (a-j)(b-j).\n\\end{array}\n\\]\nThen\n\t\\[\n\t{\\ensuremath{\\mathrm{adim}}} (X,t,m)> 0.\n\t\\]\n\\end{proposition}\n\\begin{proof}\n\tAfter a general change of coordinates, say\n\t\\[\n\t\\begin{array}{ccc}\n\tF &=& f_a+f_{a-1}x_0+f_{a-2}x_0^2+\\cdots +f_{1}x_0^{a-1}+ f_0 x_0^a\\\\\n\tG &=& g_b+g_{b-1}x_0+g_{b-1}x_0^2+\\cdots+ g_1x_0^{b-1}+ g_0 x_0^b\\\\\n\t\\end{array}\n\t\\]\n\tand define $J:=(F,G)$ to be the ideal generated by $F$ and $G$.\n\tLet $M$ be the following matrix of size $(a+b-2j)\\times (a+b-2j)$: \n\t{\\tiny\\[M:=\\left(\\begin{array}{ccccccccccccc}\n\t\tf_{a-j} & f_{a-j-1} & \\cdots & f_1 & f_{0} & 0 & \\cdots & 0 \\\\ \n\t\tf_{a-j+1} & f_{a-j} & \\cdots & f_{2} & f_{1} & f_0 & \\cdots & 0\\\\ \n\t\t\\vdots & \\vdots& \\ddots \\\\\n\t\tf_{a} & f_{a-1} & \\cdots & & & & \\\\\n\t\t0 & f_a& \\\\\n\t\t\\vdots& \\vdots \\\\\n\t\t0 & 0 & \\cdots &&&&\\cdots & f_0\\\\\n\t\tg_{b-j} & g_{b-j-1} & \\cdots & & & & \\cdots & 0 \\\\ \n\t\tg_{b-j+1} & g_{b-j} & \\cdots & & & & \\cdots & 0\\\\ \n\t\t\\vdots & \\vdots & \\ddots\\\\\n\t\tg_{b } & g_{b-1} & \\cdots & & & & \\\\\n\t\t0 & g_b& \\cdots \\\\\n\t\t\\vdots & \\vdots\\\\\n\t\t0 & 0 & \\cdots &&&& \\cdots & g_0\\\\\\\n\t\t\\end{array}\\right).\n\t\t\\]} \n(Note that the determinant of the matrix $M$ is one of the minors of the Sylvester matrix $Syl(F,G,x_0)$ of \n$F$ and $G$, and these minors belong to the partial elimination ideal $K_{j}(J)$. See in \nparticular Corollary 3.9 in \\cite{CoS} and Proposition 6.9 and the following Remark in \\cite{Gr}.) \nOne can check that $\\det(M)$ is homogeneous and it has degree $m$.\nLet $M_i$ denote the cofactor of $M$ corresponding to $(-1)^{i+1}$ times the determinant of the matrix that results from deleting the $i$-th row and the first column of $M$.\n\tConsider the following $a+b-2j$ polynomials\n\t\\[\n\t\\begin{array}{cccccccrrcccccc}\n\tF & =& f_a &+& f_{a-1}x_0 &+& \\cdots +& f_{a-j}x_0^j &+& f_{a-j-1}x_0^{j+1} &+& \\cdots\\\\\n\tx_0F &=& & & f_ax_0 & +&\\cdots +& f_{a-j+1}x_0^j & +& f_{a-j}x_0^{j+1} &+& \\cdots\\\\ \n\t\\vdots & & & & & & & & & & \\\\\n\tx_0^{j-1}F &=& & & & & & f_{a}x_0^j & +& f_{a-1}x_0^{j+1} &+& \\cdots\\\\ \n\t\\vdots & & & & & & & & & & \\\\\n\tx_0^{b-j-1}F &=& \\cdots & & & & & \\\\\n\tG &=& g_b &+& g_{b-1}x_0 &+& \\cdots +& g_{b-j}x_0^j & +& g_{b-j-1}x_0^{j+1} &+& \\cdots\\\\\n\tx_0G &=& & & g_bx_0 & +&\\cdots +& g_{b-j+1}x_0^j & +& g_{b-j}x_0^{j+1} &+& \\cdots\\\\ \n\t\\vdots & & & & & & & & & & \\\\\n\tx_0^{a-j-1}G &=& \\cdots & & & & & \\\\\n\t\\end{array}\n\t\\] \t\n\tMultiplying the above $a+b-2j$ polynomials respectively by $M_1, M_2, \\ldots, M_{a+b-2j}$ and taking the sum of them, we get\n\t\\[\\begin{array}{ccl}\n\tT:=\\left(\\sum\\limits_{i=1}^{b-j} M_ix_0^{i-1} \\right) F&+& \\left(\\sum\\limits_{i=1}^{a-j} M_{b-j+i}x_0^{i-1} \\right) G=\\\\ &=& f_aM_1+g_bM_{b-j+1} +x_0 \\left(\\cdots \\right) +\\cdots+ x_0^{j}(\\cdots) +x_0^{j+1}(\\cdots)+\\cdots\\\\\n\t\\end{array}\n\t\\] \t\n\tNote that in the form $T\\in K[x_1, \\ldots x_n][x_0]$ the coefficient of $x_0^j$ is $\\det(M)$. Also, note that all the coefficients \n\tof the powers of $x_0$ greater than $j$ are zero; for instance the coefficient of $x_0^{j+1}$ \n\tis the sum of the entries of the second column in $M$ multiplied by the cofactors of the entries \n\tin the first column. Of course $T$ belongs to the ideal generated by $F$ and $G$, therefore, since we performed a general change of variables, its leading term belongs to $\\ensuremath{\\mathrm{in}}\\hspace{1pt}(J)=\\ensuremath{\\mathrm{gin}} (I_C)$. As noted above, the form $T$ can be written as $T=T_{m+j}+x_0 \\left(T_{m+j-1} \\right) +\\cdots+ x_0^{j}(T_m)$ where $T_i\\in K[x_1, \\ldots x_n]_i$ and $T_m=\\det(M),$ thus\n\t$\\ensuremath{\\mathrm{in}}\\hspace{1pt}(T)=x_0^j\\cdot \\ensuremath{\\mathrm{in}}\\hspace{1pt}(\\det(M)) \\in \\ensuremath{\\mathrm{gin}}(I_C)$. \n\tSince $\\ensuremath{\\mathrm{in}}\\hspace{1pt}(T)$ vanishes with multiplicity $m$ at the point $Q:=(1,0,\\ldots, 0)$, by Lemma \\ref{l. adim vdim and Gin}, we are done.\n\\end{proof}\n\n\\begin{remark}\\label{r. compute edim CI} As a consequence of Proposition \n\\ref{p. codimension 2 complete intersection}, a general codimension 2 complete intesection \n$C\\subseteq \\ensuremath{\\mathbb{P}}^n$ admits an unexpected hypersurface of degree $t$ for multiplicity $m$\n\twhenever $\\mbox{edim}\\left(C,t,m \\right)=0$, i.e. $\\mbox{vdim}\\left(C,t,m \\right)\\le 0$.\n\tIn order to compute $\\mbox{edim}\\left(C,t,m \\right)$, where $C$ is a complete \n\tintersection of type $(a,b)$ in $ \\ensuremath{\\mathbb{P}}^n$, take the short exact sequence\n\t\\[\n\t0\\to R(-a-b)\\to R(-a)\\oplus R(-b) \\to I_{C} \\to 0. \n\t\\]\n\tSo, computing the dimension of the graded pieces of degree $t$, we get\n\t\\[\\dim [I_C]_t= {t-a +n\\choose n}+{t-b +n\\choose n}- {t-a-b +n\\choose n}. \\]\n\tTherefore, we have\n\t\\[\n\t\\mbox{edim}\\left(C,t,m \\right)=\\max\\left\\{\\dim [I_C]_t-{m-1+n \\choose n},0\\right\\}.\n\t\\]\n\\end{remark}\n\n\n\\begin{remark}The case $j=0$ is covered by \\cite[Proposition 2.4]{HMNT}. The cone \nwith vertex $P$ over a codimension 2 complete intesection $C\\subseteq \\ensuremath{\\mathbb{P}}^n$ \nof type $(a,b)$ is an unexpected hypersurface for $C$ of degree $ab$ and multiplicity $ab$ at $P$. \n\\end{remark}\n\nIf $j\\ge 1$ it is enough to show that ${\\ensuremath{\\mathrm{vdim}}}\\big(C, (a-j)(b-j)+j, (a-j)(b-j)\\big)\\le 0$ to \nensure the existence of an unexpected hypersurface.\nIn the next proposition we deal with the case $j=1$ in $ \\ensuremath{\\mathbb{P}}^3$. Theorem \\ref{part of conj} \nshows that a general complete intersection of type $(2,b)$ never admits an unexpected hypersurface \nwith $j=1$. The following shows that when $a>2$ we do obtain unexpected hypersurfaces.\n\n\\begin{proposition} \\label{unexpected CI}\tLet $C\\subseteq \\ensuremath{\\mathbb{P}}^3$ be a general \ncodimension 2 complete intersection defined by two forms of degree $a,b$ respectively. \nSay $a\\le b$. \tSet\\ $t:=(a-1)(b-1)+1$ and $m:=(a-1)(b-1)$.\n\tIf $a >2$ then $C$ admits an unexpected hypersurface of degree $t$ for multiplicity $m$. \n\\end{proposition}\n\\begin{proof}\n\nFrom Remark \\ref{r. compute edim CI} we have\n\t\\[\\mbox{vdim}\\left(C,t,m \\right)={t-a +3\\choose 3}+{t-b +3\\choose 3}- {t-a-b +3\\choose 3}-{t-2+3 \\choose 3}.\n\t\\]\n\tIf $a=b=3$ then $t=5$ and\n\t\\[\\mbox{vdim}\\left(C,t,m \\right)={5\\choose 3}+{5\\choose 3}-{6 \\choose 3}\\le 0.\n\t\\]\n\tA similar computation follows if $a=3$ and $b=4$. Note that, in this case $t+3-a-b=3$. \n\tSince the integer $t+3-a-b$ increases with $a$ and $b$, in particular if $(a,b)>(3,4)$ \n\twe have $t+3-a-b > 3$, so the binomial coefficient ${t +3-a-b\\choose 3}$ is not zero. Then, \n\tassuming $(a,b)>(3,4)$, after a standard computation we get\n\t\\[\n\t\\begin{array}{rl}\n\t\\mbox{vdim}\\left(C,t,m \\right)& = -\\dfrac{1}{2}a^2b-\\dfrac{1}{2}ab^2+a^2+b^2+4ab-6a-6b+9=\\\\\n\t&\\\\\n\t&=-\\dfrac{1}{2}(a-2)(b-2)(a+b-4)+1\\le 0.\\\\\n\t\\end{array}\n\t\\]\n\tThus $\\mbox{vdim}\\left(C,t,m \\right)\\le 0$.\n\\end{proof}\n\n\\begin{remark} \\label{ans question}\n\nQuestion 2.11 of \\cite{HMNT} asks the following: Let $Z$ be a non-degenerate set of points in linear general position in $ \\ensuremath{\\mathbb{P}}^n$, $n \\geq 3$. Is it true that there does not exist an unexpected hypersurface of any degree $t$ and multiplicity $m = t - 1$ at a general point? (All other possible combinations of $(n,t,m)$ are settled.)\n\nThe work in this section allows us to give a negative answer to this question. Indeed, let $X$ be a general complete intersection of type $(3,3)$ in $ \\ensuremath{\\mathbb{P}}^n$ ($n \\geq 3$). Then let $t = (3-1)(3-1)+1 = 5$ and $m = t-1 = 4$. From Remark \\ref{r. compute edim CI} we have \n\\[\n\\hbox{vdim } (X,t,m) = \\binom{n+2}{n} + \\binom{n+2}{n} - \\binom{n-1}{n} - \\binom{n+3}{n} \\leq 0\n\\]\nfor all $n \\geq 3$. On the other hand, Proposition \\ref{p. codimension 2 complete intersection} gives (with $a=b=3$ and $j=1$) that $X$ does lie on a hypersurface of degree 5 and multiplicity 4 at a general point, so this hypersurface is unexpected. Now take a set $Z$ consisting of sufficiently many general points on $X$, so that $[I_X]_5 = [I_Z]_5$. Then $Z$ is in linear general position since $X$ is irreducible, and so $Z$ admits an unexpected hypersurface and gives a negative answer to the question.\n\nWe believe that the natural extension of Proposition \\ref{unexpected CI} to $ \\ensuremath{\\mathbb{P}}^n$ is also true, but we do not have a proof.\n\\end{remark}\n\n\\begin{remark}\nOne could make the following objection to Remark \\ref{ans question} as an answer to Question 2.11 of \\cite{HMNT}. That is that the degree 5 component of $I_Z$ is the same as the degree 5 component of $I_X$, so the base locus of $[I_Z]_5$ is $X$ and the geometry is really only about $X$ and not about $Z$.\n\nTo respond to this, we make the following tweak. Take $n = 4$. We choose the same $X$ as above (now a surface in $ \\ensuremath{\\mathbb{P}}^4$), and we still take $t=5$, $m=4$. One checks that the Hilbert function of $R\/I_X$ is\n\\[\n1, 5, 15, 33, 60, 96, 141 ...\n\\]\nand we still have \n\\[\n\\hbox{vdim }(X,t,m) = \\binom{4+2}{4} + \\binom{4+2}{4} - \\binom{4-1}{4} - \\binom{4+3}{4} = -5.\n\\]\nBut $X$ contains a set $Z$ of 225 points giving a $(3,3,5,5)$ complete intersection, hence\n\\[\n\\dim [I_{Z}]_5 = 32 = \\dim [I_X]_5 + 2,\n\\]\nso $\\hbox{vdim}(Z,t,m) = -3$, hence all quintic hypersurfaces containing $Z$ with a general point of multiplicity 4\nare unexpected, and there is certainly at least one (this being the one that contains $X$).\nThere still remains an objection: computations show in this case that $X$ and $Z$ \nboth have a unique unexpected quintic\nfor multiplicity 4, so the one for $Z$ is the same one that we already got for $X$.\nNevertheless, what we have gained is that the component of the ideal for $Z$ in degree 5 is no longer the same as that of $X$, and indeed the base locus now is finite while the base locus in the original example was $X$ itself.\n\\end{remark}\n\n\\begin{comment}\nBut $X$ contains a set $Z''$ of 225 points giving a $(3,3,5,5)$ complete intersection, hence\n$Z''$ imposes 94 independent conditions on quintics, so $Z''$ contains a subset \n$Z'$ of 91 points imposing 91 conditions on quintics. We then have\n\\[\n\\dim [I_{Z'}]_5 = \\dim [I_X]_5 + 5 = 35\n\\]\nLet $Z$ be the base locus of these 35 quintics; thus $Z'\\subseteq Z\\subset Z''\\subset X$.\nBut now $\\hbox{vdim}(Z,t,m) = 0$, so the hypersurface is still unexpected.\n\nR=QQ[x,y,z,w,v];\nF=random(3,R);\nG=random(3,R);\nI=ideal(F,G);\nP=ideal(x,y,z,w);\nA=random(5,R);\nB=random(5,R);\nJ=ideal(F,G,A,B);\nJ4P=intersect(J,P^4);\nfor i from 0 to (numgens J4P)-1 do print {i, degree J4P_i}\nfor i from 0 to 10 do print {i,hilbaertFunction(i,J)}\nI4P=intersect(I,P^4);\n\\end{comment}\n\n\n\n\\section{Cones, unmixed curves and unions with finite sets of points in \n$ \\ensuremath{\\mathbb{P}}^3$} \\label{s.Unmixed curves and unions with finite sets in P3}\n\nIn this section, for the most part we restrict our attention to the case of subvarieties of $ \\ensuremath{\\mathbb{P}}^3$. \nWe recall that\n\\[\nAV_{X,0} (t) = {\\ensuremath{\\mathrm{adim}}} (X,t,t) - {\\ensuremath{\\mathrm{vdim}}} (X,t,t).\n\\]\nWe will determine the $AV_{Z,0}$ sequence for the cases where $Z=C$ is an \nequidimensional curve in $ \\ensuremath{\\mathbb{P}}^3$ and where $Z=C\\cup X$ is the union of a \ncurve $C$ and a finite set of points $X$. In particular, since $j=0$, we are focusing on the case of unexpected {\\em cones}.\nIn both cases the geometric information on $C$ provides a description of the {\\em persistence} of \nunexpected hypersurfaces. We recall the following result, which reflects the persistence of \nunexpectedness for cones for a non-degenerate curve in $ \\ensuremath{\\mathbb{P}}^3$.\n\n\\begin{theorem}[\\cite{HMNT} Corollary 2.12] \\label{HMNT cone}\nLet $C \\subset \\ensuremath{\\mathbb{P}}^3$ be a reduced, equidimensional, non-degenerate curve \nof degree $d \\geq 2$ ($C$ may be reducible, singular, and\/or disconnected). Let \n$P \\in \\ensuremath{\\mathbb{P}}^3$ be a general point. Let $k \\geq d$ be a positive integer. Then \n$C$ admits an unexpected hypersurface of degree $k$ with multiplicity $k$ at $P$. When $k = d$, this hypersurface is unique.\n\\end{theorem}\n\nOur next result not only reproduces the persistence \ngiven by the result of Theorem~\\ref{HMNT cone} but also gives a measure of the unexpectedness in each degree. \n\n\\begin{theorem} \\label{uct}\nLet $C \\subset \\ensuremath{\\mathbb{P}}^3$ be a reduced, equidimensional curve of degree $e$ and arithmetic genus $g$. Then\n\\[\nAV_{C,0}(t) = \\binom{e-1}{2} - g\n\\]\nfor all $t \\geq e$. Moreover, $AV_{C,0}(e) = 0$ if and only if $AV_{C,0}(t)=0$ for $t\\geq e$ if and only if $C$ lies in a plane.\n\\end{theorem}\n\n\\begin{proof}\nLet $P$ be a general point. We are considering hypersurfaces of degree $t$ with multiplicity $t$ at \n$P$, which are cones with vertex $P$ that contain $C$. Since the projection of $C$ from \n$P$ is contained in the hyperplane section of such a cone, we must have $t \\geq e$.\nWe first compute the virtual dimension of $I_{C \\cup P^t}$:\n\\[\n\\begin{array}{rcl}\n\\dim [I_C]_t - \\binom{t+2}{3} & = & \\binom{t+3}{3} - h_C(t) - \\binom{t+2}{3} =\\\\\n& = & \\binom{t+2}{2} - [te - g + 1].\n\\end{array}\n\\] \n(The fact that for $t \\geq e$ we have $h_C(t) = te - g + 1$ follows from the main \nresult of \\cite{GLP}.)\nOn the other hand, if $S$ is such a cone of degree $t$ and if $Q$ is a point of $C$, \nthen the line joining $P$ and $Q$ meets $S$ with multiplicity at least $t+1$, so it \nmust lie on $S$. That is, $S$ contains as a component the cone over $C$ with vertex \n$P$. Then the actual dimension of $I_{C \\cup P^t}$ is the (vector space) dimension of the \nlinear system of plane curves of degree $t-e$, i.e. it is $\\binom{t-e+2}{2}$. Then we obtain\n\\[\nAV_{C,0}(t) = \\binom{e-1}{2} - g\n\\]\nafter a simple calculation. The last part follows since $g = \\binom{e-1}{2}$ if and only if $C$ is a plane curve, \\cite[Proposition 2.1, Claim 1]{HMNT}.\n\\end{proof}\n\n\nEven if your interest is for the case when $Z$ is a finite set of points (which we consider in the next section),\nsituations involving curves (such as we are looking at in this section) \nsometimes force themselves into the picture in subtle ways,\nas the next example shows.\n\n\\begin{example}\nLet $X_1 , X_2 \\subset \\ensuremath{\\mathbb{P}}^3 $ be finite sets of points with $h$-vectors, respectively,\n\\[\n(1,3,6,5,3,3,2) \\ \\ \\ \\hbox{ and } \\ \\ \\ (1,3,6,6,3,3,2).\n\\]\nIn both cases, the two 3's constitute maximal growth, viewing the $h$-vector as a Hilbert function,\nand force ``many\" of the points to lie on a curve of degree 3 \\cite{BGM}. For the sake of this \nexample, let us assume that in both cases this curve is a twisted cubic, that it contains 18 points \nof each of $X_1$ and $X_2$ (with $h$-vector $(1,3,3,3,3,3,2)$ in both cases), and that the \nremaining 5 points of $X_1$ and the remaining 6 points of $X_2$ are chosen generically. \nThen in the first case there is a unique unexpected cone of degree 5 with vertex at a general \npoint, while in the second case there is no unexpected cone of degree 5. \nWe omit details here since we will study this \nkind of situation in the next section (see especially Example \\ref{1366332}).\n\\end{example}\n\nOur next result can again be viewed as a measure of unexpectedness for cones in \neach degree, and a statement about the persistence of unexpected cones.\n\n\n\\begin{theorem}\\label{t.C U X}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^3$ be a finite set of points. Let $C$ be a reduced, \nequidimensional curve of degree $e$ and arithmetic genus $g$. Assume that $X$ \nis disjoint from $C$. \nLet $t$ be the smallest integer such that \n\\begin{enumerate}\n\\item[(i)] $|X| < \\binom{t+2}{2}$, and \n\\item[(ii)] $X$ imposes independent conditions on forms of degree $t$.\n\\end{enumerate}\nThen \n\\[\nAV_{X \\cup C,0}(t+e) = AV_{X,0}(t) + \\left [ \\binom{e-1}{2} - g \\right ].\n\\]\n\\end{theorem}\n\n\\begin{proof}\n\nLet $P$ be a general point. Let $S_P$ be the cone over $C$ with vertex $P$; we \nknow $\\deg S_P = e$, $S_P$ is reduced, and $S_P$ has multiplicity $e$ at $P$. Let $T_P$ \nbe a surface (unmixed) of degree $t+e$ containing $X \\cup C$ and having multiplicity $t+e$ at \n$P$. ($T_P$ is a cone over a suitable plane curve.) Note that $S_P$ is a component of $T_P$, \nand note that since $P$ is general, no point of $X$ lies on $S_P$. \nWrite $U_P$ for the residual to $S_P$ in $T_P$. Note that $\\deg U_P = t$, $U_P$ has \nmultiplicity $t$ at $P$ and $U_P$ contains $X$. We also know that ${\\ensuremath{\\mathrm{adim}}} (C,e,e) = 1$ \n(Theorem \\ref{HMNT cone}). These observations imply\n\\[\n\\dim [I_X \\cap I_P^t]_t = \\dim [I_{X \\cup C} \\cap I_P^{t+e}]_{t+e}.\n\\]\nWe also remark that since $X$ imposes independent conditions on forms of degree $t$ and \n$C$ imposes independent conditions on forms of degree $e$ (by \\cite{GLP}), we can conclude \n\\begin{equation} \\label{indep cond}\n\\hbox{\\em $X$ also imposes independent conditions on $[I_{C}]_{t+e}$.}\n\\end{equation}\n\nBy definition we have\n\\[\nAV_{X,0}(t) = \\dim [I_X \\cap I_P^t]_t - \\left [ \\dim [I_X]_t - \\binom{t+2}{3} \\right ].\n\\]\nNow we compute (using the observation about independent conditions)\n\\small{\n\\[\n\\begin{array}{lcl}\nAV_{X \\cup C,0}(t+e) & = \\displaystyle & \\displaystyle \\dim [I_{X \\cup C} \\cap I_P^{t+e}]_{t+e} - \\left [ \\dim [I_{X \\cup C}]_{t+e} - \\binom{t+e+2}{3} \\right ] \\\\ \\\\\n& = & \\displaystyle \\dim [I_X \\cap I_P^t]_t - \\left [ \\dim [I_{X \\cup C}]_{t+e} - \\binom{t+e+2}{3} \\right ] \\\\ \\\\\n& = & \\displaystyle \nAV_{X,0}(t) + \\left [ \\dim [I_X]_t - \\binom{t+2}{3} \\right ] - \\left [ \\dim [I_{X \\cup C}]_{t+e} - \\binom{t+e+2}{3} \\right ] \\\\ \\\\\n& = & \\displaystyle\nAV_{X,0}(t) + \\left [ \\binom{t+3}{3} - |X| - \\binom{t+2}{3} \\right ] - (\\dim [ I_{C} ]_{t+e} - |X|) + \\binom{t+e+2}{3} \\\\ \\\\\n& = & \\displaystyle \nAV_{X,0}(t) + \\binom{t+2}{2} - \\dim [I_C]_{t+e} + \\binom{t+e+2}{3} \\\\ \\\\\n& = & \\displaystyle \nAV_{X,0}(t) + \\binom{t+2}{2} - \\left [ \\binom{t+e+3}{3} - (e(t+e) - g + 1) \\right ] + \\binom{t+e+2}{3} \\\\ \\\\\n& = & \\displaystyle \nAV_{X,0}(t) + \\binom{e-1}{2} - g\n\\end{array}\n\\] }\n(the fourth line uses (\\ref{indep cond}) and the last line comes after a routine calculation).\n\\end{proof}\n\n\n\n\\begin{corollary} \\label{X U plane curve}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^3$ be a finite set of points. Let $C$ be a reduced plane curve in \n$ \\ensuremath{\\mathbb{P}}^3$ of degree $d$ disjoint from $X$. Then $X$ has an unexpected cone of degree \n$t$ if and only if $X \\cup C$ has an unexpected cone of degree $t+d$. Furthermore, \n$AV_{X,0} (t) = \nAV_{X\\cup C,0}(t+d)$ for all $t$.\n\\end{corollary}\n\n\n\\begin{proof}\nThe arithmetic genus of a plane curve of degree $e$ is $\\binom{e-1}{2}$.\n\\end{proof}\n\n\n\n\\section{Finite sets of points in $ \\ensuremath{\\mathbb{P}}^3$} \\label{s.Finite sets of points in P3}\n\nIn this section we assume that $K$ has characteristic zero. \nOne of the original motivations for this paper was to determine if there are any Hilbert \nfunctions for non-degenerate sets of points that {\\em force} the existence of unexpected hypersurfaces of some sort. \n(A consequence of \nCorollary \\ref{degenerate} is that given a finite $O$-sequence $(1,a_1,\\dots,a_r)$, \none can trivially find a set of points in some projective space, with this $h$-vector, that does \nnot admit any unexpected hypersurfaces. One simply produces a set $X$ in $ \\ensuremath{\\mathbb{P}}^n$ for $n > a_1$ \nhaving this $h$-vector. Then $X$ is degenerate, hence admits no unexpected hypersurfaces. \nThus it is more interesting to consider non-degenerate sets.)\n\nIt now seems plausible that in a strict sense there are no such Hilbert functions. Indeed,\nwe make the following conjecture (but see Theorem \\ref{force unexp} and Corollary \\ref{LGP}, which show that such Hilbert functions\ndo arise when combined with a little geometric information).\n\n\\begin{conjecture} \\label{conj about existence}\nFor every possible $h$-vector $(1,n,a_2,\\dots,a_r)$ for a non-degenerate, finite set of \npoints in $ \\ensuremath{\\mathbb{P}}^n$, there is a set of points $X$ with that $h$-vector such that $X$ \ndoes not admit any unexpected hypersurfaces of any degree and multiplicity.\n\\end{conjecture}\n\nIn trying to prove Conjecture \\ref{conj about existence} we made the following observations. \nRecall that a {\\em distraction} is a construction that converts, in particular, an artinian monomial \nideal in $K[x_0,\\dots,x_{n-1}]$ to the ideal of a reduced set of points in $ \\ensuremath{\\mathbb{P}}^n$. It was \nintroduced in \\cite{hartshorne}. See also \\cite{MN2} for related constructions and results.\n\nOne way of constructing a reduced set of points with a given $h$-vector is \nto start with the artinian lex-segment ideal with Hilbert function $h$ and perform a distraction to \nproduce a set of points $X$. Experimentally, it seems that very often $\\hbox{gin}(I_X)$ is a \nlex-segment ideal. If this were always the case, we would be done by Corollary \\ref{recall gin}:\n\n\\begin{corollary} \\label{recall gin}\nLet $h= (1,n, a_2,\\dots,a_r)$ be a finite $O$-sequence. Let $X \\subset \\ensuremath{\\mathbb{P}}^n$ be a \nset of points with this $h$-vector. If $\\ensuremath{\\mathrm{gin}}(I_X)$ is a lex-segment ideal in \n$R = K[x_0,x_1,\\dots,x_n]$, then $X$ does not admit any unexpected hypersurfaces, \nfor any degree and multiplicity.\n\\end{corollary}\n\n\\begin{proof}\nThis follows immediately from Proposition \\ref{p.lex segments are bad for unexpectedness}.\n\\end{proof}\n\nUnfortunately, we have verified that the $h$-vector {\\tt (1,3,6,10,5,5,2)} results in a set of \npoints for which $\\ensuremath{\\mathrm{gin}}(I_X)$ is not a lex-segment ideal, and in fact one can check on {\\hbox{C\\kern-.13em o\\kern-.07em C\\kern-.13em o\\kern-.15em A}}\\ \nthat it admits an unexpected hypersurface of degree 4 with multiplicity 3. However, we were able to \nconfirm that the union in $ \\ensuremath{\\mathbb{P}}^3$ of 22 general points on a plane curve of degree 5 and 10 \ngeneral points in $ \\ensuremath{\\mathbb{P}}^3$ results in a set of points with the desired $h$-vector, which \ndoes not admit any unexpected hypersurfaces, so the conjecture is still true for this $h$-vector \neven though the distraction does not produce the desired set of points. This should be \ncontrasted with the end of Example \\ref{1366332}; in this case we verified that the \ndistraction does produce a set of points whose gin is a lex-segment. Thus the conjecture remains open.\n\nWe recall a result from \\cite{BGM}, modified to fit our context. For a set of points $X$ we \ndenote by $\\langle [I_X]_{\\leq d} \\rangle$ the ideal generated by the polynomials in $I_X$ of \ndegree $\\leq d$. Recall also that for a subscheme $Y$ of $ \\ensuremath{\\mathbb{P}}^n$ we denote by $h_Y (t)$ its Hilbert function.\n\n\\begin{proposition}[\\cite{BGM} Theorem 3.6] \\label{BGM thm}\nLet $X \\subset \\ensuremath{\\mathbb{P}}^3$ be a reduced, finite set of points with $h$-vector\n\\[\n(1,3,a_2,a_3,\\dots,a_k,d,d, a_{k+3},\\dots,a_r)\n\\]\nwhere $d \\leq k+1$. Then\n\\begin{itemize}\n\n\\item[\\rm (a)] $\\langle [I_X]_{\\leq k+1} \\rangle$ is the saturated ideal of a reduced curve, $V$, of \ndegree $d$ (not necessarily unmixed). Also, $I_X$ has no minimal generators in degree $k+2$, \nso $\\langle [I_X]_{\\leq k+1} \\rangle = \\langle [I_X]_{\\leq k+2} \\rangle$.\n\n\\item[\\rm (b)] Let $C$ be the unmixed, one-dimensional part of $V$. Let $X_1$ be the subset \nof $X$ on $C$ and let $X_2$ be the subset of $X$ not on $C$; note $X = X_1 \\cup X_2$. \nThen $\\langle [I_{X_1}]_{\\leq k+1} \\rangle = I_C$, and $V = C \\cup X_2$.\n\n\\item[\\rm (c)] $h_{X_1} (t) = h_X(t) - |X_2|$ for all $t \\geq k$.\n\n\\item[\\rm (d)] \n\\[\n\\Delta h_{X_1} (t) = \n\\left \\{\n\\begin{array}{ll}\n\\Delta h_C (t) & \\hbox{for } t \\leq k+2; \\\\\n\\Delta h_X (t) & \\hbox{ for } t \\geq k+1.\n\\end{array}\n\\right.\n\\]\n\n\\end{itemize}\n\n\n\\end{proposition}\n\n\nFrom now on we focus on the special case $j=0$, i.e. when the degree of the unexpected \nhypersurface is equal to the multiplicity at the general point. The following example shows \nthat knowing the $h$-vector of a finite set of points, and taking into account the fact that the \nbase locus of some component of $I_X$ contains a curve, is not enough\nto ensure that $X$ has an unexpected surface. \nWe generally have to know something more about the curve.\n\n\\begin{example} \\label{1366332}\nConsider the $h$-vector\n\\[\n(1,3,6,6,3,3,2).\n\\]\nThe values in degrees 4 and 5 force the existence of a cubic curve of some sort in the \nbase locus of $[I_X]_4$ and of $[I_X]_5$ for any finite set $X$ with this $h$-vector. We \nwill look at a few different kinds of cubic curves to see how the differences in the geometry \nof the curves gives different behavior with respect to unexpected hypersurfaces (specifically cones). \nOur goal is not to give an exhaustive list of possible sets of points \nwith this $h$-vector, but rather to highlight a few to see how they differ.\nSo consider the following sets of points sharing this $h$-vector. \n\n\\begin{itemize}\n\\item Let $X$ consist of 18 points on a twisted cubic $C$ (note that the \n$h$-vector of these 18 points is (1,3,3,3,3,3,2)) plus 6 general points. \nThen we claim that $X$ admits no unexpected cone of any degree. \n\nNotice that \n\\[\n\\dim [I_X ]_t = \\left \\{\n\\begin{array}{ll}\n4 & \\hbox{if } t = 3; \\\\\n16 & \\hbox{if } t = 4; \\\\\n34 & \\hbox{if } t = 5; \\\\\n\\binom{t+3}{3} - 24 & \\hbox{if } t \\geq 6 .\n\\end{array}\n\\right.\n\\]\nWe first consider the case $t \\leq 5$, and we notice ${\\ensuremath{\\mathrm{edim}}} (X,t,t) = 0$ in this case.\nSince any cone of degree $\\leq 5$ containing $X$ must also contain $C$, it also contains \nthe cone over $C$ (which is a surface of degree 3) as a component. But the projection \nfrom $P$ of the 6 general points gives 6 general points in the plane, and there is no conic \nthrough 6 general points. Thus $\\dim [I_{X \\cup P^t}]_t = 0$ for $t \\leq 5$ so ${\\ensuremath{\\mathrm{adim}}}(X,t,t) = {\\ensuremath{\\mathrm{edim}}}(X,t,t)=0$.\n\nNow let $t \\geq 6$. We have\n\\[\n{\\ensuremath{\\mathrm{vdim}}} (X,t,t) = \\binom{t+3}{3} - 24 - \\binom{t+2}{3} = \\binom{t+2}{2} - 24 > 0.\n\\]\nNow, the projection of the points on $C$ gives a set of points with \n$h$-vector \\linebreak $(1,2,3,3,3,3,3)$ so adding six general points gives a set with \n$h$-vector $(1,2,3,4,5,6,3)$, hence the general projection imposes independent \nconditions on plane curves of degree $t$. Thus the vector space dimension of this \nlinear system (hence the vector space dimension of the family of cones of degree $t$ with vertex $P$) is the expected one.\n\nNotice that if the $h$-vector had been $(1,3,6,6,3,3,3)$ then the projection would \nhave $h$-vector $(1,2,3,3,3,3,3,1)$ and the above argument would not work for $t=6$. \nIndeed, Theorem \\ref{force unexp} gives an unexpected sextic cone.\n\n\\medskip\n\n\\item Let $C$ be a set of three disjoint lines. Let $X$ consist of 6 points on one of the \nlines and 7 points on each of the remaining lines, chosen generally, together with 4 \ngeneral points in $ \\ensuremath{\\mathbb{P}}^3$. (The $h$-vector of the points on $C$ is (1,3,5,3,3,3,2), \nand the $h$-vector of $X$ is again $(1,3,6,6,3,3,2)$.) Then the expected dimension in \ndegree 5 is 0 as before, but since there is a pencil of conics through four general points in \nthe plane we obtain ${\\ensuremath{\\mathrm{adim}}}(X,5,5) = 2$, i.e. there is a pencil of unexpected cones of degree 5. \n\n\\medskip\n\n\\item Let $C$ be a smooth plane cubic curve and let $X$ consist of 17 points on $C$ \n(with $h$-vector (1,2,3,3,3,3,2)) plus a set $X_1$ of 7 general points in $ \\ensuremath{\\mathbb{P}}^3$. \nOne can check that $1 = AV_{X,0}(5) = AV_{C \\cup X_1,0} (5) = AV_{X_1,0}(2)$ in \naccordance with Corollary \\ref{X U plane curve} but that there is no unexpected hypersurface because ${\\ensuremath{\\mathrm{adim}}}(X_1,2,2) = 0$.\n\n\\medskip\n\n\\item Let $C$ be a smooth plane cubic curve in $ \\ensuremath{\\mathbb{P}}^3$, and let $\\lambda_1$ \nand $\\lambda_2$ be general lines in $ \\ensuremath{\\mathbb{P}}^3$. Let $X$ consist of 17 general \npoints on $C$, plus a subset $X_1$ of four general points on $\\lambda_1$ and three \ngeneral points on $\\lambda_2$. One can check that $X$ also has the $h$-vector \n$(1,3,6,6,3,3,2)$ so we expect no surface of degree 5 with a point of multiplicity 5 \nat a general point $P$. However, the cone over $C \\cup \\lambda_1 \\cup \\lambda_2$ \nis such a surface. But notice that the one-dimensional \ncomponent of the base locus of $[I_X]_5$ is only the plane cubic.\n\n\\end{itemize}\n\n\\noindent Thus the $h$-vector $(1,3,6,6,3,3,2)$ may or may not force an unexpected \ncone, depending mostly, but not entirely, on the cubic curve that is forced by the $h$-vector. \n\nIt is worth noting that the $h$-vector $(1,3,6,5,3,3,2)$ (analyzed as above) admits \nan unexpected cone even when the cubic curve is a twisted cubic, and $(1,3,6,6,3,3,2)$ \nadmits an unexpected cone even when the cubic is a plane cubic.\n\n\\end{example}\n\nAs mentioned at the beginning of this section, we do not believe that any finite \n$O$-sequence forces the existence of unexpected hypersurfaces for non-degenerate \nsets of points. However, some sequences force the existence of a curve in the base \nlocus of at least some components of the ideal, and if this curve is not a plane curve \nthen we {\\em can} find finite $O$-sequences that force unexpected hypersurfaces. \nThis idea is elementary, but a bit technical. Thus we will first look at an example, to \nmake the proof of Theorem \\ref{force unexp} clearer. \nWe will refer to the notation of Theorem \\ref{force unexp} in this example.\n\n\\begin{example}\nConsider sets of points $X$ with the $h$-vector\n\\[\n(1,3,6,9,8,7,\\underbrace{6,6,\\dots,6}_\\ell)\n\\]\nwhere $\\ell \\geq 6$. In the notation of Theorem \\ref{force unexp} we have $k = 5$, $d = 6$, $N = (6-4) + (9-5) + (8-6) + (7-6) = 9$ and $m = 3$. Because of the values of this $h$-vector, Proposition \\ref{BGM thm} applies. We get that for $6 \\leq t \\leq \\ell+5$, $(I_X)_{\\leq t}$ has a 1-dimensional base locus, $C$, of degree 6, together with a finite set $X_2$, which imposes independent conditions on hypersurfaces of degree $\\geq 5$. In fact\n\\[\n(I_X)_{\\leq t} = (I_{C \\cup X_2})_{\\leq t}\n\\]\nfor $6 \\leq t \\leq \\ell +5$.\nFurthermore, using Lemma \\ref{bound X2} as in the proof of Theorem \\ref{force unexp}, we see that $|X_2| \\leq N = 9$. {\\em Assume that $C$ is not a plane curve.}\n\nNow we look in degree 11, which is in the range $6 \\leq t \\leq \\ell+5$.\nApplying Theorem \\ref{t.C U X} we obtain\n\\[\nAV_{X,0}(11) = AV_{X_2 \\cup C,0}(11) = AV_{X_2,0} (5) + \\left [ \\binom{6-1}{2} - g \\right ] > 0, \n\\]\nwhere $g$ is the arithmetic genus of $C$. The fact that this is positive follows since $C$ is not a plane curve. \n\nWe remark that this works because the degree 11 is such that the value of the $h$-vector is still $6$ in that degree. Beyond degree $k+\\ell$ there is no longer a curve, and Theorem \\ref{t.C U X} no longer applies.\n\nNow let $P$ be a general point in $ \\ensuremath{\\mathbb{P}}^3$ and consider the projection from $P$ to a general $ \\ensuremath{\\mathbb{P}}^2$. The image of $X_2$ is thus a set of $\\leq 9$ points in the plane, and as such it lies on a plane curve of degree $m=3$ and hence also a plane curve of degree $k = 5$. The cone over this curve is a surface of degree 5 containing $X_2$ with multiplicity 5 at $P$. Together with the cone over $C$ (which has degree 6), we have a surface of degree 11 having multiplicity 11 at $P$. This means $\\hbox{adim} (X,11,11) > 0$. Since also $AV_{X,0}(11) >0$, $X$ admits an unexpected cone of degree 1.\n\\end{example}\n\n\\begin{lemma}[\\cite{BGM} Lemma 3.1] \\label{BGM lemma} \nLet $I \\subset R$ be an ideal satisfying $h_{R\/I}(t) = \\binom{t+m}{t}$ and $h_{R\/I}(t+1) = \\binom{t+1+m}{t+1}$. \nThen $[I]_t$ is the degree $t$ component of the saturated ideal of an $m$-dimensional linear space in $ \\ensuremath{\\mathbb{P}}^n$ (and similarly for $[I]_{t+1}$).\n\\end{lemma}\n\n\\begin{lemma} \\label{bound X2}\nLet $C$ be a reduced, unmixed, non-degenerate curve in $ \\ensuremath{\\mathbb{P}}^3$ of degree $d$. \nLet $h_C(t)$ be its Hilbert function. Then $\\Delta h_C(t)$ \nhas a sharp lower bound as follows: If $d=2$ or 3 then the lower bound is (respectively) \n\\[\n\\begin{array}{c|ccccccccccccccc}\n\\hbox{deg } t & 0 & 1 & 2 & 3 & 4 & 5 & \\dots \\\\ \\hline\n& 1 & 3 & 2 & 2 & 2 & 2 & \\dots \n\\end{array}\n\\ \\ \\ \\hbox{ and } \\ \\ \\ \n\\begin{array}{c|ccccccccccccccc}\n\\hbox{deg } t & 0 & 1 & 2 & 3 & 4 & 5 & \\dots \\\\ \\hline\n& 1 & 3 & 3 & 3 & 3 & 3 & \\dots \n\\end{array}\n\\]\nIf $d \\geq 4$ then the lower bound is\n\\[\n\\begin{array}{c|ccccccccccccccc}\n\\hbox{deg } t & 0 & 1 & 2 & 3 & 4 & \\dots & d-3 & d-2 & d-1 & d & d & \\dots \\\\ \\hline\n& 1 & 3 & 4 & 5 & 6 & \\dots & d-1 & d & d & d & d & \\dots \n\\end{array}\n\\]\n\\end{lemma}\n\n\\begin{proof}\nIn all cases, since $C$ is non-degenerate we must have $\\Delta h_C(1) = 3$.\nIf $d=2$ and $C$ is non-degenerate then $C$ must be a pair of disjoint lines, and the first \ngiven $\\Delta h_C$ is its Hilbert function. (So this is precisely the Hilbert function and not a lower bound.) If $d=3$ and $\\Delta h_C(t) \\leq 2$ for any $t \\geq 2$ then by Macaulay it can \nnever grow to 3, which it must do since $\\deg C = 3$. Thus the first two cases are done.\n\nNotice that the given sequence is $\\Delta h_C$ for the curve $C$ consisting of the union of a \nplane curve of degree $d-1$ and a line, meeting at one point. Thus this sequence occurs.\n\nLet $I_C$ be the saturated ideal of $C$. Since $R\/I_C$ has depth $\\geq 1$, if $L$ is a \ngeneral linear form then the first difference of $h_{C}(t)$ is the Hilbert function of $R\/(I_C,L)$ \nand so is an $O$-sequence. We know $\\Delta h_{C}(t) = d$ for $t \\gg 0$. If $\\Delta h_{C}(2) \\leq 2$ then by \nMacaulay's theorem it can never grow to $d$, so we must have $\\Delta h_{C}(2) \\geq 3$. \n\nSuppose $\\Delta h_C(2) = 3$. In order to eventually reach $d$, by Macaulay's theorem \nwe must have $\\Delta h_C(t) = t+1$ for $2 \\leq t \\leq d-1$. Then by Lemma \\ref{BGM lemma} \n(taking $m=1$), $[(I_C + (L))\/(L)]_t$ is the degree $t$ component of a line in $K[x,y,z]$. \nThus since $I_C$ is saturated, $[I_C]_t$ is the degree $t$ component of a plane, i.e. $C$ \nis a plane curve of degree $d$. This is impossible since $C$ is non-degenerate.\n\nThe same argument applies for all degrees $3 \\leq t \\leq d-2$: we must have $\\Delta h_C(t) \\geq t+1$ \nin order to reach $d$, and if we have equality then $C$ must be a plane curve. Thus the stated Hilbert function is the smallest possible.\n\\end{proof}\n\n If Conjecture \\ref{conj about existence} is true, the following kind of result is the best that one can \n hope for, in terms of finding a Hilbert function that forces unexpectedness (but we do \n not claim that this result is optimal in any way). It says that for a certain class of Hilbert \n functions (which we define specifically via some numerical conditions) for which a curve \n is forced in some component of the ideal because of maximal growth, if you {\\em assume} \n that this curve is not a plane curve, then {\\em any} set of points with this Hilbert function \n must admit unexpected cones. Conjecture \\ref{conj about existence} thus implies that if, however, you allow the curve \n to be a plane curve then a set of points can be found for which there is no unexpected cone. \n\n\\begin{theorem}\n \\label{force unexp}\nLet $X$ be a set of points in $ \\ensuremath{\\mathbb{P}}^3$ with $h$-vector\n\\[\n(1,3,a_2, a_3, \\dots, a_k, \\underbrace{d, d , \\dots, d}_\\ell),\n\\]\nwhere $k \\geq 2$ and $a_k > d$. Assume\n\\[\n2 \\leq d \\leq \\min \\{ k+1, \\ell \\}.\n\\]\nIn case $d \\geq 4$, let \n\\[\nb_i = \n\\left \\{\n\\begin{array}{lll}\na_i - (i+2) & \\hbox{for $2 \\leq i \\leq d-2$} \\\\\na_i - d & \\hbox{for $d-1 \\leq i \\leq k$}. \\\\\n\\end{array}\n\\right.\n\\]\nIf $d = 2$ or $d=3$, we replace $i+2$ by the bounds given in the first two parts of Lemma \\ref{bound X2}.\n\nSet \n\\[\nN = \\sum_{i=2}^k b_i \n\\]\nand\n\\[\nm = \\min \\left \\{ i \\ | \\ \\binom{i+2}{2} >~N \\right \\}. \n\\]\nWe also assume $m \\leq k$.\n\nLet $C$ be the equidimensional curve of degree $d$ guaranteed by Proposition~\\ref{BGM thm}. \nIf $C$ is not a plane curve then $X$ admits an unexpected cone of degree $d+k$.\n\n\\end{theorem}\n\n\\begin{proof}\nSince $d \\leq k+1$, Proposition \\ref{BGM thm} applies in degree $k$.\nIn particular, we get from Proposition \\ref{BGM thm} (c) that $X_2$ \nimposes independent conditions on $[I_X]_s$ for any $s \\geq k$, hence it also imposes \nindependent conditions on the complete linear system of forms of degree $s$; we will use \nthe case $s=k$. \n\nNow we look in degree $d+k$. By Theorem \\ref{t.C U X} we have \n\\[\nAV_{C \\cup X_2} (d+k) = AV_{X_2,0}(k) + \\left [ \\binom{d-1}{2} - g \\right ] > 0 \n\\]\nsince $C$ is not a plane curve.\n\n\nNote that $N$ is an upper bound for $|X_2|$, thanks to Lemma \\ref{bound X2}. If we \ndenote by $\\pi_P$ the projection from a general point $P$ to a general plane, the assumption $m \\leq k$\nguarantees that $\\pi_P(X_2)$ lies on a curve of degree $k$. This means that $X_2$ lies on a \ncone of degree $k$ with vertex at $P$. If $S_P$ is the cone over $C$ with vertex $P$, the \nunion of these cones is a surface of degree $d+k$ with multiplicity $d+k$ at $P$. \nThus ${\\ensuremath{\\mathrm{adim}}} (C \\cup X_2,d+k,d+k) > 0$, so we have an unexpected cone of degree $d+k$ for \n$C \\cup X_2$. \nBut $k+1 \\leq d+k \\leq k+\\ell$ so $[I_X]_{d+k} = [I_{C \\cup X_2}]_{d+k}$, \nso also $X$ admits an unexpected cone of degree $d+k$.\n\\end{proof}\n\n\\begin{example}\nAs mentioned above, the preceding result is not meant to be optimal. \nConsider for instance the $h$-vector $(1,3,6,5,3,3,3)$. We have $d=3$, $\\ell = 3$, $N = 3 + 2 = 5$, $m = 2$, $k = 3$. \nThe cubic curve $C$ guaranteed in the base locus of $[I_X]_t$ for $t = 4,5,6$ is either a twisted cubic, \nthe union of a line and a conic (meeting in 0 or 1 points), or the union of three lines (meeting in a total of $<3$ points). \nConsidering possible Hilbert functions of such curves, the given $h$-vector forces a set $X_2$ of at most $N = 5$ points off the curve (as a result of Lemma~\\ref{bound X2}).\n\nThe theorem guarantees an unexpected cone of degree 6. Indeed, \n\\[\n\\dim [I_X]_6 - \\binom{6-1+3}{3} = 60 - 56 = 4\n\\]\nand since the projection of $\\leq 5$ points to $ \\ensuremath{\\mathbb{P}}^2$ lies on at least a 5-dimensional vector space of plane cubics, the cones over these cubics (with vertex at the general point $P$) together with the cone over $C$ confirm the conclusion that there is an unexpected sextic. \n\nHowever, these projected points also lie on at least one conic, so there is a quadric cone containing $X_2$, and together with the cone over $C$ we get an unexpected quintic cone (since $\\dim [I_X]_5 - \\binom{5-1+3}{3} = 0$), which is not covered by the theorem. \n\nIf we had allowed $C$ to be a plane cubic curve, Lemma \\ref{bound X2} would no longer hold: the lower bound in this case would be given by the sequence $(1,2,3,3,3 ,\\dots)$ so $|X_2|$ could also be 6.\n\\end{example}\n\nThe following result gives a geometric property for a set of points that is enough to find $h$-vectors that force unexpected cones.\n\n\\begin{corollary}\n\\label{LGP}\nLet $X$ be a set of points in $ \\ensuremath{\\mathbb{P}}^3$ in linear general position, and assume \nthat $X$ has $h$-vector given by the numerical conditions in Theorem \\ref{force unexp}. Then $X$ admits an unexpected cone.\n\\end{corollary} \n\n\\begin{proof}\nThe assumption of linear general position forces $C$ to be non-degenerate.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nIn \\cite{wein}, Weinstein introduced the notion of coisotropic submanifold of a Poisson manifold generalizing the notion of Lagrangian submanifold\nof symplectic manifold. A submanifold $C$ of a Poisson manifold $(P, \\pi)$ is called coisotropic, if $\\pi^{\\sharp}(TC)^0 \\subset TC,$ or, equivalently\n$\\pi (\\alpha, \\beta) = 0$, for all $ \\alpha, \\beta \\in (TC)^0$, where $(TC)^0$ is the conormal bundle of $C$. Moreover, Weinstein proved the following results.\n\\begin{enumerate}\n\\item A map $\\phi : P_1 \\rightarrow P_2$ between Poisson manifolds is a Poisson map if and only if its graph is a coisotropic submanifold of\n$P_1 \\times P_2^{-},$ where $P_2^{-}$ stands for the manifold $P_2$ with opposite Poisson structure.\n\\item If $\\phi : P \\rightarrow Q$ is a surjective submersion from a Poisson manifold $P$ to some manifold $Q$, then $Q$\nhas a Poisson structure for which $\\phi$ is a Poisson map if and only if\n$$\\{(x,y)| \\phi(x) = \\phi(y)\\} \\subset P \\times P$$ is a coisotropic submanifold of $P \\times P^{-}.$\n\\end{enumerate}\n\nTo define the coisotropic submanifold of a Poisson manifold, one does not require the Poisson tensor to be closed, that is, $[\\pi, \\pi] = 0,$\nwhere $[~,~]$ denotes the Schouten bracket on multivector fields. Therefore, the notion of coisotropic submanifolds make sense for any bivector field, or more generally, for any multivector field.\nExplicitly, if $M$ is a smooth manifold and $\\Pi \\in \\mathcal{X}^n(M) = \\Gamma \\bigwedge^n TM$ be an $n$-vector field on $M$, then a submanifold $C \\hookrightarrow M$ is called coisotropic with respect to $\\Pi$\nif \n$$\\Pi^{\\sharp} ({\\bigwedge}^{n-1}(TC)^0) \\subset TC ~~\\Leftrightarrow ~~\\Pi (\\alpha_1, \\ldots, \\alpha_n) = 0,~~ \\mbox{for all}~~ \\alpha_1, \\ldots, \\alpha_n \\in (TC)^0,$$\nwhere $\\Pi^\\sharp : \\bigwedge^{n-1}T^*M \\rightarrow TM$ is the bundle map induced by $\\Pi$.\n\nNambu-Poisson manifolds are generalization of Poisson manifolds. Recall that a Nambu-Poisson manifold of order $n$ is a manifold $M$ equipped with an $n$-vector field $\\Pi$ such that the induced bracket on functions satisfies the Fundamental identity (Definition \\ref{nambu-poisson}). The $n$-vector field $\\Pi$ of a Nambu-Poisson manifold is referred to as the associated Nambu tensor. Coisotropic submanifolds of a Nambu-Poisson manifold $M$ are those submanifolds which are coisotropic with respect to the Nambu tensor $\\Pi$. \n \nIn the present paper, we study some basic properties of coisotropic submanifolds of a manifold with respect to a given multivector field\nand generalize the results of Weinstein to the case of multivector field. More precisely, we prove the following results (Propositions \\ref{nambu map-coiso} and \\ref{coinduced-coiso}).\n\\begin{enumerate}\n\\item Let $(M, \\Pi_M)$ and $(N, \\Pi_N)$ be two manifolds with $n$-vector fields and $\\phi: M \\rightarrow N$ be a smooth map. Then $\\phi_* \\Pi_M = \\Pi_N$ if and only if its graph\n$$\\text{Gr}(\\phi) := \\{(m, \\phi(m))| m \\in M\\}$$\nis a coisotropic submanifold of $M \\times N$ with resoect to $\\Pi_M \\oplus (-1)^{n-1} \\Pi_N$.\n\\item Let $(M, \\Pi_M)$ be a manifold with an $n$-vector field and $\\phi: M \\rightarrow N$ be a surjective submersion. Then $N$ has an (unique) $n$-vector field $\\Pi_N$ such that $\\phi_* \\Pi_M = \\Pi_N$ if and only if \n$$R(\\phi) := \\{(x,y) \\in M \\times M|\\phi(x)=\\phi(y)\\}$$\nis a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M \\oplus (-1)^{n-1}{\\Pi}_M$.\n\\end{enumerate}\n\nPoisson Lie group is a Lie group equipped with a Poisson structure such that the group multiplication map is a Poisson map. Equivalently, a Lie group equipped with a Poisson structure is a Poisson Lie group if the Poisson bivector field is multiplicative \\cite{lu-wein}. These definitions have no natural extension when one wants to define Poisson groupoid. Nevertheless, the notion of coisotropic submanifolds of Poisson manifolds was used by Weinstein \\cite{wein} to introduce the notion of Poisson groupoid. Recall that a Poisson groupoid is a Lie groupoid $G \\rightrightarrows M$ with a Poisson structure on $G$ such that the graph of the groupoid (partial) multiplication map is a coisotropic submanifold of $G \\times G \\times G^{-}.$ \n\nIn \\cite{xu}, P. Xu gave an equivalent formulation of Poisson groupoid which generalizes the multiplicativity condition for Poisson Lie group. More generally, In \\cite{pont-geng-xu}, the authors introduced the notion of multiplicative multivector fields on a Lie groupoid. Given a Lie groupoid $G \\rightrightarrows M$, an $n$-vector field $\\Pi \\in \\mathcal{X}^{n}(G)$ is called multiplicative, if the graph of the groupoid multiplication\nis a coisotropic submanifold of $G \\times G \\times G$ with respect to $\\Pi \\oplus \\Pi \\oplus (-1)^{n-1} \\Pi$. In this terminology, a Poisson groupoid is a Lie groupoid equipped with a\nmultiplicative Poisson tensor. \n\nIn the present paper, we extend this approach to the case of Lie groupoid with a Nambu structure. We introduce the notion of a Nambu-Lie groupoid as a \nLie groupoid with a Nambu structure $\\Pi$ such that the Nambu tensor $\\Pi$ is multiplicative (Definition \\ref{nambu-lie groupoid}).\nWhen $G$ is a Lie group, this definition coincides with the definition of Nambu-Lie group given by Vaisman \\cite{vais}. Using results proved in \\cite{pont-geng-xu} for mutiplicative multivector fields on Lie groupoid, we deduce the following facts which are parallel to the case of Poisson groupoid. \nSuppose $(G \\rightrightarrows M, \\Pi)$ is a Nambu-Lie groupoid, then \n\\begin{enumerate}\n\\item $M \\hookrightarrow G$ is a coisotropic submanifold of $G$;\n\\item the groupoid inversion map $i: G \\rightarrow G$ is an anti Nambu-Poisson map; \n\\item there is a unique Nambu-Poisson structure $\\Pi_M$ on $M$ for which the source map is a Nambu-Poisson map (Proposition \\ref{inverse-basenambu}).\n\\end{enumerate}\n\nIt is well known that for a Nambu-Poisson manifold $M$ of order $n$, the space of $1$-forms admits a\nskew-symmetric $n$-bracket which satisfies the Fundamental identity modulo some restriction (\\cite{vais , gra-mar , bas-bas-das-muk}). Moreover, the bracket on forms and the de-Rham differential of the manifold satisfy a compatibility condition similar to that of a Lie bialgebroid. This motivates the authors \\cite{bas-bas-das-muk} to introduce a notion of weak Lie-Filippov bialgebroid of order $n$. If $M$ is a Nambu-Poisson manifold of order $n$, then $(TM, T^*M)$ provides such an example. Roughly speaking, a weak Lie-Filippov bialgebroid of order $n$ ($n > 2$) over $M$ is a Lie algebroid $A \\rightarrow M$ together with a skew-symmetric $n$-ary bracket\non the space of sections of the dual bundle $A^* \\rightarrow M$ and a bundle map $\\rho : \\bigwedge^{n-1}A^* \\rightarrow TM$ satisfying some conditions (cf. Definition \\ref{lie-fill-defn}). Moreover it is proved in \\cite{bas-bas-das-muk} that, if $(A, A^*)$ is a weak Lie-Filippov bialgebroid of order $n$ over $M$, then\nthere is an induced Nambu-Poisson structure of order $n$ on the base manifold $M$.\n\nIn the present paper, we prove that weak Lie-Filippov bialgebroids are infinitesimal form of Nambu-Lie groupoids. \nExplicitly, if $C$ is a coisotropic submanifold of a Nambu-Poisson manifold $(M, \\Pi)$, then we show that the $n$-ary bracket on the space of $1$-forms\non $M$ restricts to the sections of the conormal bundle $(TC)^0 \\rightarrow C$ and the induced bundle map $\\Pi^{\\sharp} : \\bigwedge^{n-1}T^*M \\rightarrow TM$ \nmaps $\\bigwedge^{n-1}(TC)^0$ to $TC$ (Proposition \\ref{coiso-n-bracket}). Therefore, if $G \\rightrightarrows M$ is a Nambu-Lie groupoid of order $n$ whose Lie algebroid is \n$AG \\rightarrow M,$ then as $M$ is a coisotropic submanifold of $G$, the space of sections of the dual bundle $A^*G \\cong (TM)^0 \\rightarrow M$ is equipped with a skew-symmetric $n$-ary bracket. \nMoreover, there is a bundle map $\\bigwedge^{n-1} A^*G \\rightarrow TM$ so that the pair $(AG, A^\\ast G),$ with the above data, satisfies the conditions of a weak Lie-Filippov bialgebroid. Thus we prove the following (cf. Theorem \\ref{nambu-grpd-bialgbd}). \n\\begin{thm}\n Let $(G \\rightrightarrows M , \\Pi) $ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$. Then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ over $M$.\n\\end{thm}\n\nFinally, we compare the Nambu-Poisson structures on the base manifold $M$ induced from the Nambu-Lie groupoid $G \\rightrightarrows M$ and the weak Lie-Filippov bialgebroid $(AG, A^*G)$ (cf. Proposition \\ref{compare-two-NP-structure}).\n\nNext, we introduce the notion of a coisotropic subgroupoid $H \\rightrightarrows N$ of a Nambu-Lie groupoid $(G \\rightrightarrows M, \\Pi).$\nTo study the infinitesimal form of a coisotropic subgroupoid, we introduce the notion of coisotropic subalgebroid of a weak Lie-Filippov bialgebroid.\nThen we show that the Lie algebroid of a coisotropic subgroupoid $H \\rightrightarrows N$ is a coisotropic subalgebroid of the corresponding weak Lie-Filippov bialgebroid $(AG, A^*G)$ (cf. Proposition \\ref{coiso-subgrpd-coiso-subalgbd}). \n\n{\\bf Organization.} The paper is organized as follows. In section 2, we recall basic definitions and conventions. In section 3, we study some properties of coisotropic\nsubmanifolds of a manifold with respect to a given multivector field and in section 4 we introduce Nambu-Lie groupoids and study some of its basic properties.\nIn section 5, we show that the infinitesimal object corresponding to Nambu-Lie groupoid is weak Lie-Filippov bialgebroid and in section 6 we introduce coisotropic\nsubgroupoids of a Nambu-Lie groupoid and study their infinitesimal.\\\\\n{\\bf Acknowledgements.} The author would like to thank his Ph.D supervisor Professor Goutam Mukherjee for his guidance and carefully reading the manuscript.\n\\section{Preliminaries}\nIn this section, we recall some basic preliminaries from \\cite{duf-zung , mac-book , xu} and fix the notations that will be used throughout the paper .\n\nNambu-Poisson manifolds are $n$-ary generalizations of Poisson manifolds introduced by Takhtajan \\cite{takhtajan}.\n\\begin{defn}\\label{nambu-poisson}\nLet $M$ be a smooth manifold. A {\\it Nambu-Poisson structure} of order $n$ on $M$ is a skew-symmetric $n$-multilinear bracket\n\\begin{align*}\n \\{~, \\ldots, ~\\} : C^\\infty (M) \\times \\stackrel{(n)}{\\ldots} \\times C^\\infty (M) \\rightarrow C^\\infty (M)\n\\end{align*}\nsatisfying the following conditions:\n\\begin{enumerate}\n \\item {\\it Leibniz rule}:\\\\\n $\\{f_1, \\ldots, f_{n-1}, fg\\} = f \\{f_1, \\ldots, f_{n-1}, g \\} + \\{f_1, \\ldots, f_{n-1}, f \\} g; $\n \\item {\\it Fundamental identity}:\\\\\n $\\{f_1, \\ldots ,f_{n-1}, \\{g_1,\\ldots, g_n\\}\\} = \\sum_{k=1}^n \\{g_1,\\ldots,g_{k-1},\\{f_1,\\ldots,f_{n-1},g_k\\},\\ldots, g_n\\}; $\n\\end{enumerate}\nfor all $f_i, g_j, f, g \\in C^\\infty (M).$ A manifold together with a Nambu-Poisson structure of order $n$ is called a {\\it Nambu-Poisson manifold of order $n$}. Thus the space of smooth functions with this bracket forms a Nambu-Poisson algebra. A Nambu-Poisson manifold of order $2$ is nothing but a Poisson manifold \\cite{vais-book}. Since the bracket above is skew-symmetric and satisfies Leibniz rule, there exists an $n$-vector field $\\Pi \\in \\mathcal{X}^{n}(M)$, such that\n$$ \\{f_1, \\ldots, f_n\\} = \\Pi (df_1, \\ldots, df_n),$$\nfor all $ f_1, \\ldots, f_n \\in C^\\infty (M)$. Given any $(n-1)$ functions $ f_1, \\ldots, f_{n-1} \\in C^\\infty (M)$, the {\\it Hamiltonian vector field}\nassociated to these functions is denoted by $X_{f_1\\ldots f_{n-1}}$ and is defined by\n$$ X_{f_1\\ldots f_{n-1}} (g) = \\{f_1, \\ldots ,f_{n-1}, g\\}.$$\nNote that the Fundamental identity, in terms of Hamiltonian vector fields, is equivalent to the condition\n$$ [X_{f_1\\ldots f_{n-1}}, X_{g_1\\ldots g_{n-1}}] = \\sum_{k=1}^{n-1} X_{g_1\\ldots \\{f_1, \\ldots, f_{n-1},g_k\\}\\ldots g_{n-1}}.$$\nfor all $f_1, \\ldots, f_{n-1}, g_1, \\ldots, g_{n-1} \\in C^\\infty(M).$ The Fundamental identity can also be rephrased as\n$$ \\mathcal{L}_{X_{f_1\\ldots f_{n-1}}} \\Pi = 0,$$\nfor all $f_1, \\ldots, f_{n-1} \\in C^\\infty(M),$ which shows that every Hamiltonian vector field preserves the Nambu tensor. A Nambu-Poisson manifold is often denoted by $(M, \\{~, \\ldots, ~\\})$ or simply by $(M, \\Pi).$\n\\end{defn}\n\n\\begin{exam}\n\\begin{enumerate}\n\\item Let $M$ be an orientable manifold of dimension $n$ and $\\nu$ be a volume form on $M$. Define an $n$-bracket $\\{~, \\ldots, ~\\}$ on\n$C^\\infty(M)$ by the following identity\n$$ df_1 \\wedge \\cdots \\wedge df_n = \\{f_1, \\ldots, f_n\\} \\nu.$$\nThen $\\{~, \\ldots, ~\\}$ defines a Nambu-Poisson structure of order $n$ on $M$. Let $\\Pi_\\nu \\in \\Gamma{\\bigwedge^n TM}$ denotes the associated Nambu-Poisson\ntensor. If $\\Pi \\in \\Gamma{\\bigwedge^n TM}$ is any Nambu-Poisson structure of order $n$ such that $\\Pi \\neq 0$ at every point, then there\nexists a volume form $\\nu'$ on $M$ such that $\\Pi = \\Pi_{\\nu'}$. If $M = \\mathbb{R}^n$ and $\\nu = dx_1 \\wedge \\cdots \\wedge dx_n$ is the standard volume form, then one recovers the Nambu structure on $\\mathbb{R}^n$ originally discussed by Y. Nambu \\cite{nambu}.\n\\item Let $\\Pi$ be any $n$-vector field on an oriented manifold $M$ of dimension $n$. Then $\\Pi$ defines a Nambu structure of order $n$ on $M$ (see \\cite{ibanez1}).\n\\item Let $M$ be a manifold of dimension $m$ and $X_1, \\ldots, X_n$ be linearly independent vector fields such that $[X_i, X_j] = 0$ for all\n$i, j = 1, \\ldots, n$. Then the $n$-vector field $\\Pi = X_1 \\wedge \\cdots \\wedge X_n$ defines a Nambu structure of order $n$.\n\\item Let $(M, \\{~, \\ldots, ~\\})$ be a Nambu-Poisson manifold of order $n$. Suppose $k \\leqslant n-2$ and $F_1, \\ldots, F_k \\in C^\\infty(M)$ be any fixed functions on $M$. Define a $(n-k)$-bracket\n$\\{~, \\ldots, ~\\}'$ on $C^\\infty(M)$ by\n\\begin{align*}\n \\{f_1, \\ldots, f_{n-k}\\}' = \\{F_1, \\ldots, F_k, f_1, \\ldots, f_{n-k}\\}\n\\end{align*}\nfor $f_1, \\ldots, f_{n-k} \\in C^\\infty(M)$. Then $\\{~, \\ldots, ~\\}'$ defines a Nambu-structure of order $(n-k)$ on $M$. This Nambu structure\nis called the {\\it subordinate Nambu structure} of $(M, \\{~, \\ldots, ~\\})$ with subordinate function $F_1, \\ldots, F_k$.\n \\item If $\\Pi_i$ is a Nambu structure of order $n_i$ on a manifold $M_i$ $(i=1,2)$, then $\\Pi = \\Pi_1 \\wedge \\Pi_2$ is a Nambu structure of order $n_1 + n_2$ on $M_1 \\times M_2$ \\cite{duf-zung}.\n\\end{enumerate}\nMore examples of Nambu structures can be found in \\cite{ibanez2, vais}.\n\\end{exam}\n\n\nLet $(M, \\Pi)$ be a Nambu-Poisson manifold of order $n$. For each $m \\in M$, let $\\mathcal{D}_mM \\subset T_mM$ be the subspace of the tangent space at $m$ generated by all Hamiltonian vector fields at $m$. Since the Lie bracket of two Hamiltonians is again a Hamiltonian, therefore $\\mathcal{D}$ defines a {\\it (singular) integrable distribution} whose leaves are either $n$-dimensional submanifolds endowed with a volume form or just singletons \\cite{duf-zung}.\n\n\n\\begin{defn}\nLet $(M, \\Pi_M)$ and $(N, \\Pi_N)$ be two manifolds with $n$-vector fields. A smooth map $\\phi: M \\rightarrow N$ is called {\\it $(\\Pi_M, \\Pi_N)$-map} if the induced brackets on functions satisfies:\n\\begin{align*}\n \\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M = \\phi^* \\{f_1, \\ldots, f_n\\}_N\n\\end{align*}\nfor all\n$f_1, \\ldots, f_n \\in C^\\infty(N),$ or equivalently, $\\phi_* \\Pi_M = \\Pi_N.$ The map $\\phi$ is called an {\\it anti $(\\Pi_M, \\Pi_N)$-map} if \n\\begin{align*}\n \\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M = (-1)^{n-1} \\phi^* \\{f_1, \\ldots, f_n\\}_N\n\\end{align*}\nfor all $f_1, \\ldots, f_n \\in C^\\infty(N)$. A $(\\Pi_M, \\Pi_N)$-map $\\phi : (M, \\Pi_M) \\longrightarrow (N, \\Pi_N)$ between Nambu-Poisson manifolds of the same order $n$ is called a {\\it Nambu-Poisson map} or a $N$-$P$-map.\n\\end{defn}\n\\begin{remark}\nThe condition for a $(\\Pi_M, \\Pi_N)$-map can also be expressed in terms of the induced bundle maps as\n\\begin{align*}\n\\Pi^{\\sharp}_{N, {\\phi(m)}} = T_m \\phi \\circ \\Pi^{\\sharp}_{M, {m}} \\circ T^*_m \\phi ~ ~~\\mbox{~~ for each}~~ m \\in M,\n\\end{align*}\nwhere $\\Pi^{\\sharp}_M :\\bigwedge^{n-1}T^*M \\rightarrow TM$ is the induced bundle map and is given by\n\\begin{align*}\n \\langle \\beta, \\Pi^{\\sharp}_M (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}) \\rangle = \\Pi_M (\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta)\n\\end{align*}\nfor all $\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta \\in T_x^* M,$ $x \\in M$.\n\\end{remark}\n\n\\begin{defn}\n A {\\it Lie groupoid} over a smooth manifold $M$ is a smooth manifold $G$ together with the following structure maps:\n\\begin{enumerate}\n \\item two surjective submersions $\\alpha, \\beta : G \\rightarrow M $, called the {\\it source} map and the {\\it target} map respectively;\n \\item a smooth {\\it partial multiplication} map\n$$ G_{(2)} = \\{(g,h) \\in G \\times G | \\beta(g) = \\alpha (h) \\} \\rightarrow G , ~~ (g,h)\\mapsto gh ;$$\n \\item a smooth {\\it unit} map $\\epsilon : M \\rightarrow G$, $x \\mapsto \\epsilon_x ;$\n \\item and a smooth {\\it inverse} map $i : G \\rightarrow G$, $g \\mapsto g^{-1}$ with $\\alpha (g^{-1}) = \\beta (g)$ and $\\beta (g^{-1}) = \\alpha (g)$\n\\end{enumerate}\nsuch that, the following conditions are satisfied\\\\\n$\\hspace*{1.5cm}$(i) $\\alpha (gh) = \\alpha (g)$ and $\\beta(gh) = \\beta (h)$;\\\\\n$\\hspace*{1.5cm}$(ii) $(gh)k = g(hk),$ whenever the multiplications make sense;\\\\\n$\\hspace*{1.5cm}$(iii) $\\alpha(\\epsilon_x) = \\beta (\\epsilon_x) = x$, $\\forall x \\in M$;\\\\\n$\\hspace*{1.5cm}$(iv) $\\epsilon_{\\alpha(g)} g = g$ and $ g \\epsilon_{\\beta(g)} = g$, $\\forall g \\in G$;\\\\\n$\\hspace*{1.5cm}$(v) $g g^{-1} = \\epsilon_{\\alpha(g)}$ and $g^{-1} g = \\epsilon_{\\beta(g)}$, $\\forall g \\in G$.\n\\end{defn}\nA Lie groupoid $G$ over $M$ is denoted by $G \\rightrightarrows M$ when all the structure maps are understood.\n\n\\begin{remark}\nNote that the smooth structure on $G_{(2)}$ comes from the fact that \n$$G_{(2)} = (\\beta \\times \\alpha)^{-1}(\\Delta_M),$$ where $\\beta \\times \\alpha : G \\times G \\rightarrow M \\times M,~~(g,h) \\mapsto (\\beta(g), \\alpha (h))$\nand $ \\Delta_M = \\{(m,m)| m \\in M\\} \\subset M \\times M$ is the diagonal submanifold of $M \\times M.$ Then these conditions imply that the inverse\nmap $i : G \\rightarrow G,$ $g \\mapsto g^{-1}$ is also smooth \\cite{mac-book}. Moreover, $\\alpha$-fibers and $\\beta$-fibers are submanifolds of $G$ as both $\\alpha$ and $\\beta$ are surjective submersions.\n\\end{remark}\n\n\\begin{defn}\n Given a Lie groupoid $G \\rightrightarrows M$, define an equivalence relation $'\\sim'$ on $M$ by the following: two points $x, y \\in M$ are said to be\nequivalent, written as $x \\sim y$, if there exists an element $g \\in G$ such that $\\alpha(g) = x$, $\\beta(g) = y$. The quotient $M\/\\sim$ is called the {\\it orbit set} of $G$.\n\\end{defn}\n\n\\begin{defn}\n Given two Lie groupoid $G_1 \\rightrightarrows M_1$ and $G_2 \\rightrightarrows M_2$, a {\\it morphism} between Lie groupoids is a pair $(F, f)$ of smooth maps\n$F : G_1 \\rightarrow G_2$ and $f: M_1 \\rightarrow M_2$ which commute with all the structure maps of $G_1$ and $G_2$. In other words,\n$$ \\alpha_2 \\circ F = f \\circ \\alpha_1, ~ ~ \\beta_2 \\circ F = f \\circ \\beta_1, ~ ~ \\text{and} ~ ~ F(g_1 h_1) = F(g_1)F(h_1)$$\nfor all $(g_1,h_1) \\in (G_1)_{(2)}.$\n\\end{defn}\n\n\\begin{defn}\n Let $G \\rightrightarrows M$ be a Lie groupoid. A {\\it Lie subgroupoid} of it is a Lie groupoid $ H \\rightrightarrows N$ together with injective immersions $i : H \\rightarrow G$ and\n$i_0 : N \\rightarrow M$ such that $(i, i_0)$ is a Lie groupoid morphism.\n\\end{defn}\n\n\n\\begin{defn}\n Let $G \\rightrightarrows M$ be a Lie groupoid. A submanifold $\\mathcal{K}$ of $G$ is called a {\\it bisection} of the Lie groupoid, if $\\alpha|_{\\mathcal{K}} : \\mathcal{K} \\rightarrow M$ and\n$\\beta|_{\\mathcal{K}} : \\mathcal{K} \\rightarrow M$ are diffeomorphisms.\n\\end{defn}\n The existence of local bisections through any point $g \\in G$ is always guaranted. The space of bisections $\\mathcal{B}(G)$ form an infinite dimensional (Fr\\'{e}chet) Lie group under the multiplication of subsets induced from the partial multiplication of $G$. Note that the left (right) multiplication is defined only on $\\alpha$-fibers ($\\beta$-fibers), therefore, we can not define a diffeomorphism of $G$ using left (right) multiplication by an element, like a Lie group. However we can do so by using bisection instead of an element.\nGiven a bisection $\\mathcal{K} \\in \\mathcal{B}(G)$, let\n$l_{\\mathcal{K}}$ and $r_{\\mathcal{K}}$ be the diffeomorphisms on $G$ defined by\n$$ l_{\\mathcal{K}} (h) = gh, \\hspace{0.15cm} \\text{where} \\hspace{0.15cm} g \\in \\mathcal{K} \\hspace{0.15cm} \\text{is the unique element such that} \\hspace{0.15cm} \\beta(g) = \\alpha(h)$$\nand\n$$ r_{\\mathcal{K}} (h) = hg', \\hspace{0.15cm} \\text{where} \\hspace{0.15cm} g' \\in \\mathcal{K} \\hspace{0.15cm} \\text{is the unique element such that} \\hspace{0.15cm} \\alpha(g') = \\beta(h).$$\n\n\\begin{remark}\n Suppose $\\mathcal{K}$ is any (local) bisection of $G$ through $g \\in G$. Then the restriction of the map\n$l_{\\mathcal{K}}$ to $\\alpha^{-1} (\\beta(g))$ is the left translation $l_g$ by $g$:\n$$ l_g : \\alpha^{-1}(\\beta(g)) \\rightarrow \\alpha^{-1}(\\alpha(g)), ~ h \\mapsto gh .$$\n\\end{remark}\n\nThen we have the following result \\cite{xu}.\n\n\\begin{prop}\\label{left-invariant}\n Let $G \\rightrightarrows M$ be a Lie groupoid and $P$ be an $n$-vector field on $G$. Suppose for any $g \\in G$ with $\\beta(g) = u$, $P$ satisfies $P(g) = (l_{\\mathcal{G}})_* P(\\epsilon_u)$, where $\\mathcal{G} \\in \\mathcal{B}(G)$ is any arbitrary bisection through the point $g$. Then $P$ is left invariant.\n\\end{prop}\n\n\n\n\\begin{defn}\n A {\\it Lie algebroid} $(A, [~, ~], a)$ over a smooth manifold M is a smooth vector bundle $A$ over\nM together with a Lie algebra structure $[~, ~]$ on the space $\\Gamma{A}$ of the smooth sections of $A$ and a bundle map $a : A \\rightarrow T M $ , called the {\\it anchor}, such that\n\\begin{enumerate}\n \\item the induced map $ a : \\Gamma{A} \\rightarrow \\mathcal{X}^1(M) $ is a Lie algebra homomorphism, where $\\mathcal{X}^1(M)$ is the usual Lie algebra of vector fields on $M$.\n \\item For any $ X, Y \\in \\Gamma{A} $ and $f \\in C^\\infty (M)$, we have\n$$ [X, f Y ] = f [X, Y ] + (a(X)f )Y. $$\n\\end{enumerate}\n\\end{defn}\nWe may denote a Lie algebroid simply by $A$, when all the structures are understood. Any Lie algebra is a Lie algebroid over a point with zero anchor. The tangent bundle of any smooth manifold is a Lie algebroid with usual Lie bracket of vector fields and identity as anchor.\\\\\n{\\bf Lie algebroid of a Lie groupoid.}\nGiven a Lie groupoid $G \\rightrightarrows M$, its Lie algebroid consists of the vector bundle $AG \\rightarrow M$ whose fiber at $x \\in M$ coincides with the tangent space\nat the unit element $\\epsilon_x$ of the $\\alpha$-fiber at $x$. Then the space of sections of $AG$ can be identified with the left invariant vector fields\n\\begin{align*}\n \\mathcal{X}^1_{\\text{inv}}(G) = \\{ X \\in \\Gamma (T^\\alpha G) = \\Gamma (\\text{ker} (d\\alpha))| X_{gh} = (l_g)_* X_h, \\forall (g, h) \\in G_{(2)} \\}\n\\end{align*}\non $G$. Since the space of left invariant vector fields on $G$\nis closed under the Lie bracket, therefore it defines a Lie bracket on $\\Gamma AG$. The anchor $a$ of $AG$ is defined to be the differential of the target map $\\beta$ restricted to $AG$. \n\n\n\n\n\n\n\n\n\nLet $AG $ be the Lie algebroid of the Lie groupoid $G \\rightrightarrows M$. Given any $X \\in \\Gamma AG$, let $\\overleftarrow{X}$ be the corresponding left invariant vector field on $G$. Then there exists an $\\epsilon > 0$ and a $1$-parameter family of transformations $\\phi_t$ $(|t| < \\epsilon)$, generated by $\\overleftarrow{X}$ (\\cite{mac-book}). Suppose each\n$\\phi_t$ is defined on all of $M$, where $M$ is identified with a closed embedded submanifold of $G$ via the unit map. We denote the image of $M$ via $\\phi_t$\nby exp $tX$. Then exp $tX$ is a bisection of the groupoid (for all $|t| < \\epsilon$) and satisfies $1$-parameter group like conditions, namely\n\\begin{align*}\n \\text{exp} (t+s) X = \\text{exp} \\hspace*{0.1cm} tX \\cdot \\text{exp} \\hspace*{0.1cm}sX, \\hspace*{1cm} \\text {whenever} \\hspace*{0.5cm} |t| , |s| , |t+s| < \\epsilon,\n\\end{align*}\nwhere on the right hand side, we used the multiplication of bisections.\n\n\n\\section{Coisotropic submanifolds}\nLet $M$ be a manifold and $\\Pi \\in \\mathcal{X}^n(M)$ be a $n$-vector field on $M$. Let\n\\begin{align*}\n \\Pi^{\\sharp} : {\\bigwedge}^{n-1}T^*M \\rightarrow TM\n\\end{align*}\nbe the induced bundle map given by\n\\begin{align*}\n \\langle \\beta, \\Pi^{\\sharp} (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}) \\rangle = \\Pi (\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta)\n\\end{align*}\nfor all $\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta \\in T_x^* M,$ $x \\in M$.\n\nWe recall the following definition from \\cite{pont-geng-xu}.\n\\begin{defn}\nA submanifold $C \\hookrightarrow M$ is said to be {\\it coisotropic} with respect to $\\Pi$,\nif\n\\begin{align*}\n \\Pi^{\\sharp} ({\\bigwedge}^{n-1} (TC)^0) \\subset TC\n\\end{align*}\nwhere $$(TC)^0_x = \\{ \\alpha \\in T_x^* M | \\hspace*{0.1cm} \\alpha (v) = 0, \\forall v \\in T_xC\\},~~x \\in C,$$\nor equivalently,\n\\begin{align*}\n \\Pi_x (\\alpha_1, \\ldots, \\alpha_n) = 0, \\forall \\alpha_i \\in (TC)^0_x, x \\in C.\n\\end{align*}\n\\end{defn}\n\nWe have the following easy observation for coisotropic submanifolds of a Nambu-Poisson manifold.\n\\begin{prop}\n Let $(M, \\Pi)$ be a Nambu-Poisson manifold of order $n$ and $C$ be a closed embedded submanifold of $M$. Let $\\mathcal{I} (C) = \\{ f \\in C^\\infty(M)\\big| f|_C \\equiv 0 \\}$ denote the vanishing ideal of $C$. Then the followings are equivalent:\n\\begin{enumerate}\n\\item $C$ is a coisotropic submanifold;\n\\item $\\mathcal{I} (C)$ is a Nambu-Poisson subalgebra;\n\\item for every $f_1, \\ldots, f_{n-1} \\in \\mathcal{I} (C)$, the Hamiltonian vector field $X_{f_1...f_{n-1}}$ is tangent to $C$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n (1) $\\Rightarrow$ (2) Let $f_1, \\ldots, f_n \\in \\mathcal{I} (C) $.\nThen for any $x \\in C$, $d_xf_i \\in (TC)^0_x$, for all $i=1, \\ldots, n.$ Now since $C$ is a coisotropic submanifold, we have\n\\begin{align*}\n \\{f_1, \\ldots, f_n\\}(x) = \\Pi_x (d_xf_1, \\ldots, d_xf_n) = 0,~ \\forall x \\in C.\n\\end{align*}\nHence $\\{f_1, \\ldots, f_n\\} \\in \\mathcal{I} (C)$. Therefore $\\mathcal{I} (C)$ is a Nambu-Poisson subalgebra.\n\n(2) $\\Rightarrow$ (3) Let $f_1, \\ldots, f_{n-1} \\in \\mathcal{I} (C)$ and $x \\in C$.\nLet $\\alpha \\in (TC)^0_x.$ Then there exists a function $g$ vanishing on $C$ such that $d_xg = \\alpha$. Since $\\mathcal{I} (C)$ is a Nambu-Poisson subalgebra, we have\n\\begin{align*}\n \\{f_1, \\ldots, f_{n-1}, g \\}(x) = 0.\n\\end{align*}\nThus,\n\\begin{align*}\n X_{f_1\\ldots f_{n-1}}\\big|_x (\\alpha) = X_{f_1\\ldots f_{n-1}}\\big|_x (d_xg) = \\{f_1, \\ldots, f_{n-1}, g \\}(x) = 0\n\\end{align*}\nand consequently, $X_{f_1\\ldots f_{n-1}}$ is tangent to $C$.\n\n(3) $\\Rightarrow$ (1) Let $x \\in C$ and $\\alpha_1, \\ldots, \\alpha_n \\in (TC)^0_x.$ Then there exist functions $f_1, \\ldots, f_n \\in \\mathcal{I}(C)$ such that\n$d_xf_i = \\alpha_i,$ $\\forall i = 1, \\ldots, n.$ Therefore,\n\\begin{align*}\n \\Pi_x (\\alpha_1, \\ldots, \\alpha_n) = \\Pi_x (d_xf_1, \\ldots, d_xf_n) = X_{f_1...f_{n-1}} \\big|_x (d_xf_n) = 0.\n\\end{align*}\nHence $C$ is a coisotropic submanifold of $M$.\n\\end{proof}\n\n\n\n\\begin{prop}\n Let $(M, \\Pi_M)$ and $(N, \\Pi_N)$ be two Nambu-Poisson manifolds of same order $n$ and $C \\hookrightarrow N$ be a coisotropic submanifold of $N$ with respect to $\\Pi_N$. If $\\phi: M \\rightarrow N$\nis a Nambu-Poisson map transverse to $C$, then $\\phi^{-1}(C)$ is a coisotropic submanifold of $M$ with respect to $\\Pi_M$ (the result holds true for manifolds with $n$-vector fields such that\n$\\phi_* \\Pi_M = \\Pi_N$).\n\\end{prop}\n\n\n\\begin{proof}\nSince $\\phi$ is transverse to $C$, therefore $\\phi^{-1}(C)$ is a submanifold of $M$. Moreover\n\\begin{align*}\n T(\\phi^{-1}(C)) = (T \\phi)^{-1} TC.\n\\end{align*}\nTherefore $T(\\phi^{-1}(C))^0 = (T \\phi)^{*} (TC)^0$. Observe that\n\\begin{align*}\n T\\phi (\\Pi_M^{\\sharp} \\big( {\\bigwedge}^{n-1} T(\\phi^{-1}(C))^0 \\big)) =& T\\phi ( \\Pi_M^{\\sharp} \\big((T \\phi)^{*} {\\bigwedge}^{n-1} (TC)^0 \\big))\\\\\n=& \\Pi_N^{\\sharp} ({\\bigwedge}^{n-1} (TC)^0)\\\\\n\\subseteq & TC.\n\\end{align*}\nThus,\n\\begin{align*}\n \\Pi_M^{\\sharp} \\big( {\\bigwedge}^{n-1} T(\\phi^{-1}(C))^0 \\big) \\subseteq (T\\phi)^{-1} TC = T(\\phi^{-1}(C))\n\\end{align*}\nand hence $\\phi^{-1}(C)$ is coisotropic with respect to $\\Pi_M.$\n\\end{proof}\n\n\\begin{prop}\\label{image-coiso}\n Let $\\phi: (M, \\Pi_M) \\rightarrow (N, \\Pi_N)$ be a Nambu Poisson map between two Nambu-Poisson manifolds $(M, \\Pi_M)$ and $(N, \\Pi_N)$ and $C\\hookrightarrow M$ be a coisotropic submanifold\nof $M$. Assume that $\\phi(C)$ is a submanifold of $N$. Then $\\phi(C)$ is a coisotropic submanifold of $N$ (the result holds true for manifolds with $n$-vector fields such that $\\phi_{*}\\Pi_M = \\Pi_N$).\n\\end{prop}\n\n\\begin{proof}\nWe have $T(\\phi(C)) \\supseteq T\\phi (TC)$ and $(T\\phi)^* (T(\\phi(C)))^0 \\subseteq (TC)^0$. Therefore,\n\\begin{align*}\n \\Pi_N^{\\sharp} ({\\bigwedge}^{n-1} T(\\phi(C))^0) =& T\\phi (\\Pi_M^{\\sharp} ((T\\phi)^* {\\bigwedge}^{n-1} T(\\phi(C))^0)) \\hspace*{1cm}(\\text{since ~} \\phi \\text{~ is a N-P map})\\\\\n\\subseteq & T\\phi (\\Pi_M^{\\sharp} ({\\bigwedge}^{n-1} (TC)^0))\\\\\n\\subseteq & T\\phi (TC) \\hspace*{1cm}(\\text{since~} C \\hookrightarrow M \\text{~ is coisotropic})\\\\\n\\subseteq & T(\\phi(C))\n\\end{align*}\nwhich shows that $\\phi(C)$ is a coisotropic submanifold of $N$.\n\\end{proof}\n\nUsing the terminology of coisotropic submanifold with respect to any multivector field allows us to extend the results of Weinstein \\cite{wein} from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field.\n\n\\begin{prop}\\label{nambu map-coiso}\n Let $(M, \\Pi_M)$ and $(N, \\Pi_N)$ be two manifolds with $n$-vector fields and $\\phi: M \\rightarrow N$ be a smooth map. Then $\\phi$ is a $(\\Pi_M, \\Pi_N)$-map, that is $\\phi_* \\Pi_M = \\Pi_N$ if and only if its graph\n\\begin{align*}\n \\text{Gr}(\\phi) = \\{(m, \\phi(m))| m \\in M \\}\n\\end{align*}\nis a coisotropic submanifold of $M \\times N$ with respect to $\\Pi_M \\oplus (-1)^{n-1} {\\Pi}_N$.\n\\end{prop}\n\n\\begin{proof}\n Let $C = \\text{Gr}(\\phi) \\subset M \\times N$. Then $C$ is a closed embedded submanifold of $M \\times N.$\nNote that, a tangent vector to the graph consist of a pair $(v_m, (T\\phi)(v_m))$, where $m \\in M$, $v_m \\in T_mM$. Therefore,\n$(TC)^0$ consists of a pair of covectors $(-(T\\phi)^*\\psi, \\psi)$, where $\\psi \\in T_{\\phi(m)}^*N.$\nTherefore, Gr($\\phi)$ is a coisotropic submanifold of $M \\times N$ with respect to $\\Pi_M \\oplus (-1)^{n-1} {\\Pi}_N$ if and only if\n$(\\Pi_M^{\\sharp} \\times (-1)^{n-1} \\Pi_N^{\\sharp})$ maps $(-(T\\phi)^*\\psi_1, \\psi_1) \\wedge \\cdots \\wedge (-(T\\phi)^*\\psi_{n-1}, \\psi_{n-1}) $ into $TC$,\nfor all $\\psi_1, \\ldots, \\psi_{n-1} \\in T^*_{\\phi(m)}N$ and $m \\in M$. In other words,\n\\begin{align*}\n(T\\phi) \\bigg(\\Pi_M^{\\sharp} (- (T\\phi)^*\\psi_1, \\ldots, - (T\\phi)^*\\psi_{n-1})\\bigg) = (-1)^{n-1} {\\Pi}_N^{\\sharp} (\\psi_1, \\ldots, \\psi_{n-1})\n\\end{align*}\nthat is,\n\\begin{align*}\n(T\\phi) \\bigg(\\Pi_M^{\\sharp} ( (T\\phi)^*\\psi_1, \\ldots, (T\\phi)^*\\psi_{n-1})\\bigg) = {\\Pi}_N^{\\sharp} (\\psi_1, \\ldots, \\psi_{n-1}).\n\\end{align*}\nThis is equivalent to the condition that $\\phi$ is a $(\\Pi_M, \\Pi_N)$-map.\n\\end{proof}\n\n\\begin{defn}\n Let $(M, \\Pi_M)$ be a Nambu-Poisson manifold of order $n$ and $\\phi: M \\rightarrow N$ be a smooth surjective map. If there exist a Nambu-Poisson structure $\\Pi_N$ (of order $n$) on $N$\nwhich makes $\\phi$ into a Nambu-Poisson map, then $\\Pi_N$ is called the Nambu-Poisson structure {\\it coinduced} by the mapping $\\phi$.\n\\end{defn}\n\nThe following is a characterization of coinduced Nambu-Poisson structure.\n \n\\begin{prop}\\label{coinduced}\n Let $(M, \\Pi_M)$ be a Nambu-Poisson manifold of order $n$ and $\\phi: M \\rightarrow N$ be a smooth surjective map from $M$ to some manifold $N$. Then $N$ has a Nambu-Poisson structure coinduced by $\\phi$ if and only if for all $f_1, \\ldots, f_{n} \\in C^\\infty(N)$, the function\n$\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M$ is constant along the fibers of $\\phi$.\n\\end{prop}\n\n\n\\begin{proof}\n Let $f_1, \\ldots, f_n \\in C^\\infty(N)$. If the function $\\{ \\phi^*f_1, \\ldots, \\phi^*f_n \\}_M$ is constant along the $\\phi$-fibers, then there exists a function on $N$, which we denote by $\\{f_1, \\ldots, f_n\\}_N$ such that\n$\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M = \\phi^* \\{f_1, \\ldots, f_n\\}_N $. Clearly this bracket defines a coinduced Nambu-Poisson structure on $N$.\n\nConversely, suppose that there is a Nambu-Poisson bracket $\\{~, \\ldots, ~\\}_N$ on $N$ coinduced by $\\phi$. Then for any $y \\in N,$\n\\begin{align*}\n\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M(\\phi^{-1}\\{y\\}) =& (\\phi^* \\{f_1, \\ldots, f_n\\}_N )(\\phi^{-1}y)\\\\\n=& \\{f_1, \\ldots, f_n\\}_N (y)\n\\end{align*}\nproving $\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M$ is constant along the $\\phi$-fibers.\n\\end{proof}\n\n\\begin{remark}\\label{rem-coinduced}\n Let $(M, \\Pi_M)$ be a manifold with an $n$-vector field and $\\phi: M \\rightarrow N$ be a smooth map. Then there exists an $n$-vector field\n$\\Pi_N$ on $N$ such that $\\phi$ is a $(\\Pi_M, \\Pi_N)$-map if and only if for all $f_1, \\ldots, f_n \\in C^\\infty(N)$, the function\n$\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}_M$ is constant along the fibers of $\\phi$.\n\\end{remark}\n\n\n\\begin{prop}\nLet $(M, \\Pi_M)$ be a Nambu-Poisson manifold and $\\phi : M \\rightarrow N$ be a surjective submersion with connected fibers. Let $\\text{ker} \\hspace*{0.1cm} \\phi_*(m)$ is spanned by local Hamiltonian vector fields (that is, $\\text{ker} \\hspace*{0.1cm} \\phi_*(m) \\subset \\mathcal{D}_mM $), for all $m \\in M$. Then \n$N$ has a Nambu-Poisson structure coinduced by $\\phi.$\n\\end{prop}\n\n\n\\begin{proof}\n Since $\\phi$ is a submersion, the fibers of $\\phi$ are submanifolds of $M$. Then for $y \\in N,$ $\\phi^{-1}(\\{y\\}) = C$ is a submanifold of $M$. Let $g_1, \\ldots, g_{n-1}$ be locally defined functions on $M$ such that $X_{g_1...g_{n-1}} \\in \\text{ker} \\hspace*{0.1cm} \\phi_{*}$. Let $f_1, \\ldots, f_n \\in C^\\infty(N).$ To prove that $\\{\\phi^*f_1, \\ldots, \\phi^*f_n\\}$\nis constant on the fibers, it is enough to prove that\n\\begin{align*}\n X_{g_1...g_{n-1}} \\{\\phi^*f_1, \\ldots, \\phi^*f_n\\} = 0.\n\\end{align*}\nNote that\n\\begin{align*}\n X_{g_1...g_{n-1}} \\{\\phi^*f_1, \\ldots, \\phi^*f_n\\} = \\sum_{k=1}^n \\{\\phi^*f_1, \\ldots, X_{g_1...g_{n-1}} (\\phi^*f_k), \\ldots, \\phi^*f_n\\}\n\\end{align*}\nand the functions $\\phi^*f_i$ are constant along the fibers. Hence by the Proposition \\ref{coinduced}, there exists a coinduced Nambu-Poisson structure on $N.$\n\\end{proof}\n\n\\begin{prop}\\label{coinduced-coiso}\nLet $(M, \\Pi_M)$ be a manifold with an $n$-vector field and $\\phi: M \\rightarrow N$ be a surjective submersion. Then $N$ has an (unique) $n$-vector field $\\Pi_N$ such that\n$\\phi$ is a $(\\Pi_M, \\Pi_N)$-map if and only if\n $R(\\phi) = \\{(x,y) \\in M \\times M|\\phi(x)=\\phi(y)\\}$ is a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M \\oplus (-1)^{n-1}{\\Pi}_M$.\n\\end{prop}\n\n\\begin{proof}\nNote that $R(\\phi) = (\\phi \\times \\phi)^{-1} (\\Delta_N)$, where\n$\\Delta_N$ is the diagonal of $N \\times N$. Since $\\phi$ is surjective submersion $R(\\phi)$ is a submanifold of $M \\times M$.\nMoreover, for $(x,y) \\in R(\\phi)$\n\\begin{align*}\n T_{(x,y)} (R(\\phi)) = \\{(X, Y)\\in T_xM \\times T_yM | (T\\phi)_x (X) = (T\\phi)_y (Y)\\}.\n\\end{align*}\nTherefore, $T(R(\\phi))^0$ consists of covectors $(-(T\\phi)_x^*\\psi, (T\\phi)_y^*\\psi)$, where $\\psi \\in T_{\\phi(x)}^*N$.\n\nThus, $R(\\phi)$ be a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M \\oplus (-1)^{n-1}{\\Pi}_M$\nif and only if for all $\\psi_1, \\ldots, \\psi_{n-1} \\in T_{\\phi(x)}^*N$ and $(x, y) \\in R(\\phi),$ $\\Pi_M^{\\sharp} \\oplus (-1)^{n-1}{\\Pi}_M^{\\sharp}$ maps \n$$(-(T\\phi)_x^*\\psi_1, (T\\phi)_y^*\\psi_1) \\wedge \\cdots \\wedge (-(T\\phi)_x^*\\psi_{n-1}, (T\\phi)_y^*\\psi_{n-1})$$ into $T(R(\\phi)).$ \nThat is\n$$(T\\phi)_x \\Pi_M^{\\sharp} (-(T\\phi)_x^*\\psi_1, \\ldots, -(T\\phi)_x^*\\psi_{n-1} )= (-1)^{n-1} (T\\phi)_y \\Pi_M^{\\sharp} ((T\\phi)_y^*\\psi_1, \\ldots, (T\\phi)_y^*\\psi_{n-1}),$$\nor equivalently,\n\\begin{align} \\label{firsteqn}\n (T\\phi)_x \\Pi_M^{\\sharp} ((T\\phi)_x^*\\psi_1, \\ldots, (T\\phi)_x^*\\psi_{n-1} ) = (T\\phi)_y \\Pi_M^{\\sharp} ((T\\phi)_y^*\\psi_1, \\ldots, (T\\phi)_y^*\\psi_{n-1})\n\\end{align}\nholds. Let $f_1, \\ldots ,f_n \\in C^\\infty(N)$ and $x \\in M.$ Then\n\\begin{align*}\n\\{\\phi^*f_1, \\ldots , \\phi^*f_n\\}_M (x) =& \\langle \\Pi_M^{\\sharp} (d_x (\\phi^*f_1) \\wedge \\cdots \\wedge d_x (\\phi^*f_{n-1})), d_x (\\phi^*f_n) \\rangle\\\\\n =& \\langle \\Pi_M^{\\sharp} \\big((T\\phi)_x^* \\psi_1 \\wedge \\cdots \\wedge (T\\phi)_x^* \\psi_{n-1}\\big), (T\\phi)_x^* \\psi_n \\rangle\\\\\n =& \\langle (T\\phi)_x \\Pi_M^{\\sharp} \\big((T\\phi)_x^* \\psi_1 \\wedge \\cdots \\wedge (T\\phi)_x^* \\psi_{n-1}\\big), \\psi_n \\rangle\n\\end{align*}\nwhere $\\psi_i = d_{\\phi(x)}f_i = d_{\\phi(y)}f_i\\in T_{\\phi(x)}^*N$, for all $1 \\leqslant i \\leqslant n.$ It follows from the Equation (\\ref{firsteqn}) that the function $\\{\\phi^*f_1, \\ldots , \\phi^*f_n\\}_M$ is constant along the $\\phi$-fibers if and only if $R(\\phi)$ is a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M^{\\sharp} \\oplus (-1)^{n-1}{\\Pi}_M^{\\sharp}$. Hence the result follows by the Remark \\ref{rem-coinduced}. The uniqueness follows from the surjectivity of $\\phi.$\n\\end{proof}\n\n\\section{Nambu-Lie groupoids}\nIn this section, we recall the definition of multiplicative multivector fields on Lie groupoid (\\cite{pont-geng-xu}) and define Nambu-Lie groupoid (of order $n$) as a Lie groupoid with a multiplicative\n$n$-vector field which is also a Nambu-Poisson tensor.\n\n\\begin{defn}\nLet $G \\rightrightarrows M$ be a Lie groupoid and $\\Pi \\in \\mathcal{X}^n(G)$ be an $n$-vector field on $G$. Then $\\Pi$ is called {\\it multiplicative} if the graph of the groupoid multiplication\n\\begin{align*}\n \\{ (g, h, gh) \\in G \\times G \\times G | ~ \\beta(g) = \\alpha(h)\\}\n\\end{align*}\nis a coisotropic submanifold of $G \\times G \\times G$ with respect to $\\Pi \\oplus \\Pi \\oplus (-1)^{n-1} \\Pi$.\n\\end{defn}\n \nThen we have the following characterization of multiplicative multivector fields \\cite{pont-geng-xu}:\n\\begin{thm} \\label{multiplicative}\n Let $G \\rightrightarrows M$ be a Lie groupoid and $\\Pi \\in \\mathcal{X}^n (G)$ be an $n$-vector field on $G$. Then $\\Pi$ is multiplicative if and only if the following conditions are satisfied.\n\\begin{enumerate}\n \\item $\\Pi$ is an affine tensor. In other words\n\\begin{align*}\n \\Pi(gh) = (r_{\\mathcal{H}})_* \\Pi(g) + (l_{\\mathcal{G}})_* \\Pi(h) - (r_{\\mathcal{H}})_* (l_{\\mathcal{G}})_* \\Pi(u)\n\\end{align*}\nwhere $u = \\beta(g) = \\alpha(h)$ and $\\mathcal{G}, \\mathcal{H}$ are (local) bisections through the points $g, h$ respectively.\n \\item $M$ is a coisotropic submanifold of $G$ with respect to $\\Pi$.\n \\item For all $g \\in G$, $\\alpha_* \\Pi(g)$ and $\\beta_* \\Pi(g)$ depend only on the base points $\\alpha(g)$ and $\\beta(g)$ respectively.\n \\item For all $f, f' \\in C^\\infty(M)$, the $(n-2)$-vector field $\\iota_{d(\\alpha^*f) \\wedge d(\\beta^*f')} \\Pi$ is zero. In other words,\n\\begin{align*}\n \\{~, \\ldots, \\alpha^*f, \\beta^*f' \\} = 0.\n\\end{align*}\n \\item For all $f_1, \\ldots, f_k \\in C^\\infty(M)$, $\\iota_{d(\\beta^*f_1) \\wedge \\cdots \\wedge d(\\beta^*f_k)} \\Pi$ is a left invariant $(n-k)$-vector field on $G$, $1 \\leqslant k < n.$\n\\end{enumerate}\n\\end{thm}\n\n\\begin{remark}\\label{lie-gp-multiplicative}\n Suppose $G$ be a Lie group considered as a Lie groupoid over a point. Then the conditions {\\it (3)} - {\\it (5)} of the Theorem \\ref{multiplicative} are satisfied automatically. The condition {\\it (2)} implies that $\\Pi (e) = 0$ (where $e$ is the identity element of the group), which together with condition {\\it (1)} implies\nthat $\\Pi$ satisfies the usual multiplicativity condition\n$$ \\Pi (gh) = (r_h)_* \\Pi(g) + (l_g)_* \\Pi(h) .$$\n\\end{remark}\n\n\n\n\\begin{defn}\\label{nambu-lie groupoid} \n A {\\it Nambu-Lie groupoid of order $n$} is a Lie groupoid $G \\rightrightarrows M$ with a multiplicative Nambu tensor $\\Pi \\in \\mathcal{X}^n(G)$ of order $n.$ \n\\end{defn}\nA Nambu Lie groupoid (of order $n$) will be denoted\nby $(G \\rightrightarrows M, \\Pi) $.\n\n\\begin{exam}\\label{exam-nlg}\n\\begin{enumerate}\n\\item Poisson groupoids \\cite{wein} are examples of Nambu-Lie groupoids with $n=2.$\n\\item Any Lie groupoid with zero Nambu structure is a Nambu-Lie groupoid.\n\\item Let $(G, \\Pi)$ be a Nambu-Lie group (of order $n$) \\cite{vais}. Thus $G$ is a Lie group equipped with a Nambu structure $\\Pi$ of order $n$ on $G$\nsuch that\n\\begin{align*}\n \\Pi(gh) = (r_h)_* \\Pi(g) + (l_g)_* \\Pi(h)\n\\end{align*}\nfor all $g, h \\in G$. Note that the right hand side of the above equality is equal to $m_* (\\Pi(g), \\Pi(h)),$ where $m_* : \\bigwedge^n T_{(g,h)} (G \\times G) \\rightarrow \\bigwedge^n T_{gh}G$\nis the map induced by the multiplication map $m : G \\times G \\rightarrow G$. Therefore,\n\\begin{align*}\n \\Pi(gh) = m_* (\\Pi(g), \\Pi(h)).\n\\end{align*}\nThus, the group multiplication map $m : G \\times G \\rightarrow G$ is a $(\\Pi \\oplus \\Pi, \\Pi)$-map. Therefore, by the Proposition \\ref{nambu map-coiso}, the graph of the group multiplication map is a coisotropic submanifold of $G \\times G \\times G$ with respect to $\\Pi \\oplus \\Pi \\oplus (-1)^{n-1} \\Pi$. Hence $(G, \\Pi)$ is a Nambu-Lie groupoid over a point.\nConversely, if $(G, \\Pi)$ is a Nambu-Lie groupoid over a point, then the group multiplication map $m : G \\times G \\rightarrow G$ is a \n$(\\Pi \\oplus \\Pi, \\Pi)$-map. Hence $(G, \\Pi)$ is a Nambu-Lie group in the sense of \\cite{vais}. One can also see the equivalence between Nambu-Lie groupoid\nover a point and Nambu-Lie group by using Remark \\ref{lie-gp-multiplicative}.\n\\end{enumerate}\n\\end{exam}\n\nFor a Poisson groupoid the following facts are well known \\cite{wein}.\n\\begin{itemize}\n\\item The groupoid inversion map is a anti-Poisson map. \n\\item The Poisson structure on the total space induces a Poisson structure on the base such that the source map is a Poisson map and the target map is a anti-Poisson map. \n\\end{itemize}\n\nIn the next proposition we generalize the above facts to the Nambu-Poisson setting.\n\n\\begin{prop}\\label{inverse-basenambu}\n Let $(G \\rightrightarrows M , \\Pi)$ be a Nambu-Lie groupoid. Then\n\\begin{enumerate} \n\\item The inverse map $i : G \\rightarrow G$, $g \\mapsto g^{-1}$ is an anti-Nambu Poisson map.\n\\item There is a unique Nambu-Poisson structure on $M$ which we denote by $\\Pi_M$ for which $\\alpha$ is a Nambu-Poisson map and $\\beta$ is an anti Nambu-Poisson map.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\n(1) It is proved in \\cite{pont-geng-xu} that given a Lie groupoid $G \\rightrightarrows M$ with multiplicative $n$-vector field $\\Pi \\in \\mathcal{X}^n(G)$,\nthe groupoid inversion map $i : G \\rightarrow G$ satisfies\n\\begin{align*} \n i_* \\Pi = (-1)^{n-1} \\Pi.\n\\end{align*}\nHence the result follows as $\\Pi$ is a Nambu tensor.\n\n(2) Let $f_1, \\ldots, f_n \\in C^\\infty(M)$ be any functions on $M$. Then for any $g \\in G$, we have\n\\begin{align*}\n\\{\\alpha^* f_1, \\ldots, \\alpha^* f_n \\} (g) =& \\Pi (g) (d_g (\\alpha^* f_1), \\ldots, d_g (\\alpha^* f_n))\\\\\n =& \\Pi(g) (\\alpha^* (d_{\\alpha(g)} f_1), \\ldots, \\alpha^* (d_{\\alpha(g)} f_n))\\\\\n =& \\alpha_* \\Pi (g) (d_{\\alpha(g)} f_1, \\ldots, d_{\\alpha(g)} f_n).\n\\end{align*}\nSince $\\alpha_* \\Pi(g)$ depends only on the value of $\\alpha(g)$, it follows that the function $\\{ \\alpha^* f_1, \\ldots, \\alpha^* f_n \\}$ is constant on the $\\alpha$-fibers. Therefore, by the Proposition \\ref{coinduced}, there exists a Nambu-Poisson structure $\\Pi_M$ with the induced bracket denoted by $\\{ ~, \\ldots, ~\\}_M,$ on $M$\nfor which $\\alpha$ is a Nambu-Poisson map. Since $\\beta = \\alpha \\circ i$ and $i$ is anti Nambu-Poisson, therefore $\\beta$\nis an anti Nambu-Poisson map.\n\\end{proof}\n\n\\begin{remark}\nConsider the map $(\\alpha,\\beta): G \\rightarrow M \\times M.$\nSince we have $\\alpha_{*} \\Pi = \\Pi_M$ and $\\beta_{*}\\Pi = (-1)^{n-1} \\Pi_M$, using property {\\it (4)} of the Theorem \\ref{multiplicative} we obtain \n\\begin{align*}\n(\\alpha,\\beta)_{*} \\Pi = \\alpha_* \\Pi \\oplus \\beta_* \\Pi = \\Pi_M \\oplus (-1)^{n-1} \\Pi_M.\n\\end{align*}\n\\end{remark}\n\n\n\\begin{prop}\n Let $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid. If the orbit space $M\/\\sim$ is a smooth manifold, then $M\/\\sim$ carries a Nambu-Poisson structure such that the projection $q: M \\rightarrow M\/\\sim $ is a Nambu-Poisson map.\n\\end{prop}\n\n\\begin{proof}\nLet $\\Pi_M$ be the induced Nambu structure on the base $M$. For the projection map $q: M \\rightarrow M\/\\sim $, we have\n\\begin{align*}\n R(q) =& \\{(x,y) \\in M \\times M| q(x) = q(y)\\}\\\\\n=& \\{(\\alpha(g), \\beta(g))| g \\in G\\}\\\\\n=& (\\alpha,\\beta)(G).\n\\end{align*}\nConsider $G$ as a coisotropic submanifold of $G$ with respect to $\\Pi$ and also consider the map $(\\alpha, \\beta) : G \\rightarrow M \\times M$. By the above Remark we have $(\\alpha, \\beta)_* \\Pi = \\Pi_M \\oplus (-1)^{n-1} \\Pi_M.$ Therefore, by the Proposition \\ref{image-coiso}, $R(q) = (\\alpha,\\beta)(G)$ is a coisotropic submanifold of $M \\times M$ with respect to $\\Pi_M \\oplus (-1)^{n-1}\\Pi_M.$ Hence the result follows from the Proposition \\ref{coinduced-coiso}.\n\\end{proof}\n\n\n\n\\section{Infinitesimal form of Nambu-Lie groupoid}\n\nThe aim of this section, is to study the infinitesimal form of a\nNambu-Lie groupoid. We show that if $(G \\rightrightarrows M, \\Pi)$ is a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$, then\n$(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ introduced in \\cite{bas-bas-das-muk}. Before proceeding further, let us briefly recall from \\cite{bas-bas-das-muk} the notion of a weak Lie-Filippov bialgebroid.\n\nLie bialgebroids are generalization of both Poisson manifolds and Lie bialgebras. Recall that a Lie bialgebroid, introduced by Mackenzie and Xu \\cite{mac-xu} is also the infinitesimal form of a Poisson groupoid. It is defined as a pair $(A, A^\\ast)$ of Lie algebroids in duality, where the Lie bracket of $A$ satisfies the following compatibility condition expressed in terms of the differential $d_\\ast$ on $\\Gamma (\\bigwedge^\\bullet A)$ \n$$d_\\ast[X, Y] = [d_\\ast X, Y] + [X , d_\\ast Y],$$ \nfor all $X,~ Y \\in \\Gamma A.$ \n\nWe note that if $M$ is a Poisson manifold then the Lie algebroid structures on $TM$ and $T^\\ast M$ form a Lie bialgebroid. On the other hand, it is well known \\cite{kosmann,mac-xu} that if $(A, A^\\ast)$ is a Lie bialgebroid over a smooth manifold $M$ then there is a canonical Poisson structure on the base manifold $M$.\n\nThus it is natural to ask the following question which was posed in \\cite{bas-bas-das-muk}:\\\\ \n{\\it Does there exist some notion of bialgebroid associated to a Nambu-Poisson manifold of order $n > 2$ }? \n\nTo answer this question, the authors \\cite{bas-bas-das-muk} introduced the notion of weak Lie-Filippov bialgebroid.\n\nIt is well known \\cite{gra-mar, vais} that for a Nambu-Poisson manifold $M$ of order $n \\geqslant 2$, the space $\\Omega^1(M)$ of $1$-forms admits an $n$-ary bracket, called {\\it Nambu-form bracket}, such that the bracket satisfies almost all the properties of an $n$-Lie algebra (also known as Filippov algebra of order $n$) bracket \\cite{fil} except that the Fundamental identity is satisfied only in a restricted sense as described below.\n\nLet $(M, \\{~, \\ldots, ~\\})$ be a Nambu-Poisson manifold of order $n$ with associated Nambu-Poisson tensor $\\Pi$. Then one can define the Nambu form-bracket on the space of $1$-forms\n\\begin{align*}\n [~, \\ldots, ~]: \\Omega^1(M) \\times \\cdots \\times \\Omega^1(M) \\rightarrow \\Omega^1(M)\n\\end{align*}\n by the following\n\\begin{align*}\n [\\alpha_1, \\ldots, \\alpha_n] = \\sum_{k=1}^n (-1)^{n-k} \\mathcal{L}_{\\Pi^{\\sharp}(\\alpha_1 \\wedge\\cdots \\wedge \\hat{\\alpha}_k \\wedge\\cdots \\wedge \\alpha_n)} \\alpha_k - (n-1) d (\\Pi(\\alpha_1, \\ldots, \\alpha_n))\n\\end{align*}\n\\begin{align}\\label{nambu-bracket}\n\\hspace*{1cm} = d (\\Pi(\\alpha_1, \\ldots, \\alpha_n)) + \\sum_{k=1}^n (-1)^{n-k} \\iota_{\\Pi^{\\sharp}(\\alpha_1 \\wedge\\cdots \\wedge \\hat{\\alpha}_k \\wedge\\cdots \\wedge \\alpha_n)} d\\alpha_k\n\\end{align}\nfor $\\alpha_i \\in \\Omega^1(M), i=1,\\ldots ,n.$ Here $\\hat{\\alpha}_k$ in a monomial $\\alpha_1 \\wedge \\cdots \\wedge \\hat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n$ means that the symbol $\\alpha_k$ is missing in the monomial. The above bracket satisfies the following properties (\\cite{vais}).\n\\begin{enumerate} \n\\item The bracket is skew-symmetric.\n\\item $[df_1, \\ldots, df_n] = d \\{f_1, \\ldots, f_n \\}$.\n\\item $[\\alpha_1,\\ldots, \\alpha_{n-1}, f \\alpha_n] = f [\\alpha_1,\\ldots, \\alpha_{n-1}, \\alpha_n] + \\Pi ^{\\sharp} (\\alpha_1 \\wedge\\cdots \\wedge \\alpha_{n-1})(f) \\alpha_n$.\n\\item The bracket satisfies the Fundamental identity\n\\begin{align*}\n [\\alpha_1, \\ldots, \\alpha_{n-1}, [\\beta_1, \\ldots, \\beta_n]] = \\sum_{k=1}^n [\\beta_1, \\ldots , \\beta_{k-1},[\\alpha_1, \\ldots, \\alpha_{n-1}, \\beta_k], \\ldots, \\beta_n]\n\\end{align*}\nwhenever the $1$-forms $\\alpha_i \\in \\Omega^1(M)$ are closed, $1 \\leqslant i \\leqslant n-1$ and for any $\\beta_j$.\n\\item $[\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}), \\Pi^{\\sharp}(\\beta_1 \\wedge \\cdots \\wedge \\beta_{n-1})]\\\\ \n= \\sum_{k=1}^{n-1} \\Pi^{\\sharp} (\\beta_1 \\wedge\\cdots \\wedge [\\alpha_1,\\ldots , \\alpha_{n-1}, \\beta_k] \\wedge \\cdots \\wedge \\beta_{n-1})$\\\\\nfor closed $1$-forms $\\alpha_i \\in \\Omega^1 (M)$ and for any $1$-forms $\\beta_j$. \n\\end{enumerate}\n\n\nThe Nambu-form bracket on $\\Omega^1(M)$, together with the usual Lie algebroid structure on $TM$ yields an example of a notion called a {\\it weak Lie-Filippov algebroid pair of order $n$}, $n>2,$ on a smooth vector bundle (cf. Definition $5.5$, \\cite{bas-bas-das-muk}). \n\nIn order to classify such structures, the authors formulate a notion of {\\it Nambu-Gerstenhaber algebra of order $n$}. It turns out, weak-Lie-Filippov algebroid pair structures of order $n$, $n>2$, on a smooth vector bundle $A$ over $M$, are in bijective correspondence with Nambu-Gerstenhaber brackets of order $n$ on the graded commutative, associative algebra $\\Gamma \\bigwedge^\\bullet A^*$ of multisections of $A^*$, where $A^*$ is the dual bundle (cf. Definition $5.7$, Theorem $5.8$, \\cite{bas-bas-das-muk}).\n\nMoreover, for a Nambu-Poisson manifold $M$ of order $n > 2$, the Nambu-Gerstenhaber bracket on $\\Omega^\\bullet(M)$, extending the Nambu-form bracket on $\\Omega^1(M)$ satisfies certain suitable compatibility condition similar to the compatibility condition of a Lie bialgebroid. This motivates the authors to introduce the notion of a {\\it weak Lie-Filippov bialgebroid structure} of order $n$ on a smooth vector bundle. \n\n\n\\begin{defn}\\label{lie-fill-defn}\nA {\\it weak Lie-Filippov bialgebroid of order $n>2$} over a smooth manifold $M$ consists of a pair $(A, A^*)$, where $A$ is a smooth vector bundle over $M$ with dual bundle $A^*$ satisfying the following properties:\n\\begin{enumerate}\n\\item $A$ is a Lie algebroid with $d_A$ being the differential of the Lie algebroid cohomology of $A$ with trivial representation;\n\\item the space of smooth sections $\\Gamma A^*$ admits a skew-symmetric $n$-ary bracket\n$$[~, \\ldots ,~]: {\\Gamma A^* \\times \\cdots \\times \\Gamma A^*} \\longrightarrow \\Gamma A^*$$\nsatisfying \n$$[\\alpha_1, \\ldots , \\alpha_{n-1}, [\\beta_1, \\ldots , \\beta_n]] = \\sum_{k=1}^n [\\beta_1, \\ldots , \\beta_{k-1}, [\\alpha_1, \\ldots , \\alpha_{n-1}, \\beta_k], \\ldots ,\\beta_n]$$\nfor all $d_A$-closed sections $\\alpha_i \\in \\Gamma A^*,~ 1 \\leqslant i \\leqslant n-1$ and for any sections $\\beta_j \\in \\Gamma A^*,~ 1\\leqslant j\\leqslant n;$\n\\item there exists a vector bundle map $\\rho : \\bigwedge^{n-1}A^* \\longrightarrow TM$, called the {\\it anchor} of the pair $(A, A^*)$, such that the identity\n$$ [\\rho (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}), \\rho (\\beta_1 \\wedge \\cdots \\wedge \\beta_{n-1})] = \\sum_{k=1}^{n-1}\\rho (\\beta_1 \\wedge \\cdots \\wedge [\\alpha_1, \\ldots , \\alpha_{n-1}, \\beta_k] \\wedge \\cdots \\wedge \\beta_{n-1})$$\nholds for all $d_A$-closed sections $\\alpha_i \\in \\Gamma A^*,~ 1\\leqslant i \\leqslant n-1$ and for any sections $\\beta_j \\in \\Gamma A^*,~ 1\\leqslant j\\leqslant n-1;$\n\\item for all sections $\\alpha_i \\in \\Gamma A^*,~ 1\\leqslant i \\leqslant n$ and any $f \\in C^\\infty(M)$,\n$$[\\alpha_1, \\ldots , \\alpha_{n-1}, f\\alpha_n] = f [\\alpha_1, \\ldots , \\alpha_{n-1}, \\alpha_n] + \\rho (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1})(f)\\alpha_n$$ holds;\n\\item the following compatibility condition holds:\n$$ d_A[\\alpha_1, \\ldots , \\alpha_n] = \\sum_{k=1}^n [\\alpha_1, \\ldots , d_A\\alpha_k, \\ldots , \\alpha_n],$$ for any $\\alpha_i \\in \\Gamma A^\\ast,$ $1 \\leqslant i \\leqslant n$, where the bracket $[~, \\ldots , ~]$ on the right hand side is the graded extension of the bracket on $\\Gamma A^\\ast$.\n\\end{enumerate}\n\\end{defn}\n\nA weak Lie-Filippov bialgebroid (of order $n$) over $M$ is denoted by $(A, A^*)$ when all the structures are understood. A Lie bialgebroid is a Lie-Filippov\nbialgebroid of order $2$ such that the conditions (2) and (3) of the above definition has no restriction on $\\alpha$.\n\nIn \\cite{bas-bas-das-muk}, the authors have shown that for a Nambu-Poisson manifold $M$ of order $n > 2$, the pair $(TM, T^*M)$ is a weak Lie-Filippov bialgebroid of order $n$ (cf. Corollary $6.3$, \\cite{bas-bas-das-muk}). It is also proved that if $(G, \\Pi)$ is a Nambu-Lie group \\cite{vais} of order $n$ with its Lie algebra $\\mathfrak{g},$ then $(\\mathfrak{g}, \\mathfrak{g}^*)$ forms a (weak) Lie-Filippov bialgebroid of order $n$ over a Point.\n\nIt is known that the base of a Lie bialgebroid carries a natural Poisson structure. In \\cite{bas-bas-das-muk} it has been extended to the Nambu-Poisson set up.\n\n\\begin{prop} \\label{wlfb-base-nambu}(\\cite{bas-bas-das-muk})\nLet $(A, A^*)$ be a weak Lie-Filippov bialgebroid (of order n) over $M$. Then the bracket\n\\begin{align*}\n \\{f_1, \\ldots, f_n\\}_{(A, A^*)} := \\rho (d_A f_1 \\wedge \\cdots \\wedge d_A f_{n-1}) f_n\n\\end{align*}\ndefines a Nambu-Poisson structure of order n on $M$.\n\\end{prop}\n\n\nIt is known that, given a coisotropic submanifold $C$ of a Poisson manifold $M$, the conormal bundle $(TC)^0 \\rightarrow C$ is a Lie subalgebroid\nof the cotangent Lie algebroid $T^*M$ \\cite{wein}. If $M$ is a Nambu-Poisson manifold of order $n$ ($n \\geqslant 3$), the cotangent bundle $T^*M$ is not a Filippov\nalgebroid. However we have the following useful result.\n\n\\begin{prop}\\label{coiso-n-bracket}\n Let $C$ be a closed embedded coisotropic submanifold of a Nambu-Poisson manifold $(M, \\Pi)$ of order $n$. Then\n \\begin{enumerate}\n\\item the bundle map $\\Pi^{\\sharp} : \\bigwedge^{n-1}T^*M \\rightarrow TM$ maps $\\bigwedge^{n-1} (TC)^0$ to $TC$;\n\\item the Nambu-form bracket on the space of $1$-forms $\\Omega^1(M)$ can be restricted to the sections of the conormal bundle $ (TC)^0 \\rightarrow C$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nThe assertion $(1)$ follows from the definition of coisotropic submanifold. To prove $(2)$, let $\\alpha_1, \\ldots ,\\alpha_n \\in \\Gamma (TC)^0.$ We extend them to $1$-forms on $M$, which we denote by the same notation. Let $X \\in \\mathcal{X}^1(M)$ be such that $X\\big|_C$ is tangent to $C$. From the definition\nof Nambu-form bracket on $1$-forms, we have\n\\begin{align*}\n\\langle [\\alpha_1, \\ldots, \\alpha_n], X \\rangle = \\sum_{k=1}^n (-1)^{n-k} \\langle \\mathcal{L}_{\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n)} \\alpha_k, X \\rangle\n- (n-1) \\langle d (\\Pi(\\alpha_1,\\ldots, \\alpha_n)), X \\rangle. \n\\end{align*}\nObserve that\n\\begin{align*}\n \\langle \\mathcal{L}_{\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n)} \\alpha_k, X \\rangle \n=& \\mathcal{L}_{ \\Pi^\\sharp (\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n )} \\langle \\alpha_k, X \\rangle \\\\\n&- \\langle \\alpha_k, [\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n), X]\\rangle .\n\\end{align*}\nThis is zero on $C$, because, \n\\begin{itemize}\n\\item $\\langle \\alpha_k, X\\rangle$ is zero on $C$; \n\\item $\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\widehat{\\alpha_k} \\wedge \\cdots \\wedge \\alpha_n)$ and $X$ are both tangent to $C$ and hence their Lie bracket is also tangent to C. Thus its pairing with $\\alpha_k$ vanish on $C$.\n\\end{itemize}\nNote that $\\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1})\\big|_C$ is tangent to $C$ and $\\alpha_n \\big|_C \\in (TC)^0.$ As a consequence, the function\n\\begin{align*}\n\\Pi (\\alpha_1,\\ldots, \\alpha_n) = \\langle \\alpha_n, \\Pi^{\\sharp}(\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1})\\rangle\n\\end{align*}\nis zero on $C.$ Therefore, the differential\n$d (\\Pi(\\alpha_1, \\ldots, \\alpha_n))$ restricted to $C$ is in $(TC)^0$, which in turn implies that the second term of the right hand side also vanish on $C$. Hence\n\\begin{align*}\n [\\alpha_1, \\ldots, \\alpha_n]\\big|_C \\in (TC)^0.\n\\end{align*}\nOne can check that the restriction to $C$ does not depend on the chosen extension. Hence it defines a bracket on the sections of the conormal bundle $(TC)^0 \\rightarrow C$.\n\\end{proof}\n\n\\begin{remark}\n\\begin{enumerate}\n\\item Let $m_0 \\in M$ such that $\\Pi (m_0) = 0$. Then $\\{m_0\\}$ is a coisotropic submanifold of $M$. In this case, the conormal structure becomes $T_{m_0}^*M$, which is a Filippov algebra.\n\\item The Nambu structure of a Nambu-Lie group G vanishes at the identity element and therefore the dual $\\mathfrak{g}^*$ of the Lie algebra $\\mathfrak{g}$ of G has a Filippov algebra structure \\cite{vais}.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{remark}\\label{nlg-bracket-anchor}\nLet $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$. By the Proposition \\ref{coiso-n-bracket}, we see that the space of sections of the conormal bundle $ A^*G = (TM)^0 \\rightarrow M$ admits a skew-symmetric $n$-bracket $[~,\\ldots,~]$ and there exists a bundle map\n$$\\rho := \\Pi^{\\sharp}\\big|_{{\\bigwedge}^{n-1} (TM)^0} : {\\bigwedge}^{n-1} A^*G = {\\bigwedge}^{n-1} (TM)^0 \\rightarrow TM ,$$ as $M$ is a coisotropic submanifold of $G$.\n\\end{remark}\n\n\n\n\nLet $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$.\n Let $f \\in C^\\infty(M).$ Then by part ($5$) of the Theorem \\ref{multiplicative}, $\\iota_{d(\\beta^*f)} \\Pi$ is a left invariant $(n-1)$-vector field\non $G$. Therefore, there exists an $(n-1)$-multisection $\\delta^0_\\Pi (f)\\in \\Gamma{\\bigwedge^{n-1}AG}$ of the Lie algebroid $AG$ such that\n\\begin{align*}\n \\iota_{d(\\beta^*f)} \\Pi = \\overleftarrow{\\delta^0_\\Pi (f)} .\n\\end{align*}\nThen we have the following result.\n\n\\begin{prop}\n Let $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid of order $n$ and $AG \\rightarrow M$ be its Lie algebroid. Then for any $ X \\in \\Gamma AG$,\n\\begin{align*}\n \\mathcal{L}_{\\overleftarrow{X}} \\Pi := [\\overleftarrow{X}, \\Pi]\n\\end{align*}\nis a left invariant $n$ vector field on $G$, where $\\overleftarrow{X}$ is the left invariant vector field on $G$ corresponding to $X$. Moreover $\\mathcal{L}_{\\overleftarrow{X}} \\Pi$ corresponds to the $n$-multisection $- \\delta^1_\\Pi (X) \\in \\Gamma{\\bigwedge^n AG}$,\nthat is,\n\\begin{align*}\n \\mathcal{L}_{\\overleftarrow{X}} \\Pi = -\\overleftarrow{\\delta^1_\\Pi (X)}\n\\end{align*}\nwhere $\\delta^1_\\Pi (X) \\in \\Gamma{\\bigwedge^n AG}$ is given by\n\\begin{align*}\n\\delta^1_\\Pi (X) (\\alpha_1, \\ldots, \\alpha_n) = \\sum_{k=1}^{n} (-1)^{n-k} \\Pi^{\\sharp} (\\alpha_1 \\wedge \\cdots \\wedge \\hat{\\alpha}_k \\wedge \\cdots \\wedge \\alpha_n) (X(\\alpha_k))\n- X([\\alpha_1,\\ldots, \\alpha_n])\n\\end{align*}\nfor $\\alpha_1, \\ldots, \\alpha_n \\in \\Gamma A^*G = \\Gamma (TM)^0.$\n\\end{prop}\n\n\\begin{proof}\nLet $\\mathcal{X}_t = \\text{exp} \\hspace*{0.05cm} tX$ be the one-parameter family of bisections generated by $X \\in \\Gamma AG.$ Let $g \\in G$ with $\\beta(g) = u$. Let\n$u_t = (\\text{exp} \\hspace*{0.05cm} tX) (u)$ be the integral curve of $\\overleftarrow{X}$ starting from $u$. If $\\mathcal{G}$ is any (local) bisection through $g$, then from the multiplicativity condition of $\\Pi$ (cf. Theorem \\ref{multiplicative}), we have\n\\begin{align*}\n \\Pi ( g u_t) = (r_{\\mathcal{X}_t})_* \\Pi(g) + (l_{\\mathcal{G}})_* \\Pi(u_t)- (r_{\\mathcal{X}_t})_* (l_{\\mathcal{G}})_* \\Pi(u).\n\\end{align*}\nTherefore,\n\\begin{align*}\n(r_{\\mathcal{X}^{-1}_t})_* \\Pi ( g u_t) - \\Pi(g) = (r_{\\mathcal{X}^{-1}_t})_* (l_{ \\mathcal{G}})_* \\Pi(u_t) - (l_{\\mathcal{G}})_* \\Pi (u).\n\\end{align*}\nTaking derivative at $t = 0$, one obtains\n\\begin{align*}\n (\\mathcal{L}_{\\overleftarrow{X}} \\Pi)(g) = (l_{\\mathcal{G}})_* ((\\mathcal{L}_{\\overleftarrow{X}} \\Pi)(u)).\n\\end{align*}\nTherefore, $\\mathcal{L}_{\\overleftarrow{X}} \\Pi $ is left invariant by the Proposition \\ref{left-invariant} and hence it corresponds to some $n$-multisection of $AG$. To show that\n$\\mathcal{L}_{\\overleftarrow{X}} \\Pi$ corresponds to $- \\delta^1_\\Pi (X) \\in \\Gamma {\\bigwedge^n AG}$, we have to check that $\\mathcal{L}_{\\overleftarrow{X}} \\Pi$ and $- \\overleftarrow{\\delta^1_\\Pi (X)}$ coincide on the unit space $M$ (both being left invariant). Since both of them are tangent to $\\alpha$-fibers, it is enough to show that they coincide on the conormal bundle $(TM)^0$.\nLet $\\alpha_1, \\ldots, \\alpha_n$ be any sections of $(TM)^0$ and $\\tilde{\\alpha}_1, \\ldots, \\tilde{\\alpha}_n$ be their respective extensions to one forms on $G$. Observe that\n\\begin{align*}\n& (\\mathcal{L}_{\\overleftarrow{X}} \\Pi)\\big|_M (\\alpha_1, \\ldots, \\alpha_n)\\\\\n =& \\big[\\langle \\overleftarrow{X}, d (\\Pi(\\tilde{\\alpha}_1,\\ldots,\\tilde{\\alpha}_n)) \\rangle - \\sum_{k=1}^{n} \\Pi(\\tilde{\\alpha}_1 ,\\ldots, \\mathcal{L}_{\\overleftarrow{X}} \\tilde{\\alpha}_k,\\ldots, \\tilde{\\alpha}_n)\\big]\\big|_M\\\\\n =& \\big[\\langle \\overleftarrow{X}, [\\tilde{\\alpha}_1,\\ldots,\\tilde{\\alpha}_n] \\rangle - \\sum_{k=1}^n (-1)^{n-k} \\langle \\overleftarrow{X}, \\iota_{\\Pi^{\\sharp} (\\tilde{\\alpha}_1 \\wedge\\cdots \\wedge \\hat{\\tilde{\\alpha}}_k \\wedge\\cdots \\wedge \\tilde{\\alpha}_n)} d\\tilde{\\alpha}_k \\rangle\\\\\n & - \\sum_{k=1}^{n} (-1)^{n-k} \\langle \\Pi^{\\sharp} (\\tilde{\\alpha}_1 \\wedge\\cdots \\wedge \\hat{\\tilde{\\alpha}}_k \\wedge\\cdots \\wedge \\tilde{\\alpha}_n), \\mathcal{L}_{\\overleftarrow{X}} \\tilde{\\alpha}_k \\rangle \\big]\\big|_M\\\\\n & \\hspace*{5cm} (\\text{from the Equation}~ (\\ref{nambu-bracket}))\\\\\n =& \\big[\\langle \\overleftarrow{X}, [\\tilde{\\alpha}_1,\\ldots,\\tilde{\\alpha}_n] \\rangle - \\sum_{k=1}^n (-1)^{n-k} \\langle \\Pi^{\\sharp} (\\tilde{\\alpha}_1 \\wedge\\cdots \\wedge \\hat{\\tilde{\\alpha}}_k \\wedge\\cdots \\wedge \\tilde{\\alpha}_n), d \\iota_{\\overleftarrow{X}} \\tilde {\\alpha}_k \\rangle \\big]\\big|_M\\\\\n & \\hspace*{5cm} (\\text{using Cartan formula})\\\\\n =& \\langle X, [\\alpha_1,\\ldots,\\alpha_n] \\rangle - \\sum_{k=1}^n (-1)^{n-k} \\Pi^{\\sharp} ({\\alpha}_1 \\wedge\\cdots \\wedge \\hat{\\alpha}_k \\wedge\\cdots \\wedge {\\alpha_n}) (X (\\alpha_k))\\\\\n = & -\\delta^1_\\Pi (X) (\\alpha_1,\\ldots, \\alpha_n).\n\\end{align*}\n\\end{proof}\n\nTo make our notation simple, let us denote $\\delta^0_\\Pi, \\delta^1_\\Pi$ by the same symbol $\\delta_\\Pi$. We extend $\\delta_\\Pi$ to the graded algebra $\\Gamma{\\bigwedge^{\\bullet}A}$ of multisections of $AG$ by the following rule\n\\begin{align*}\n \\delta_\\Pi (P \\wedge Q) = \\delta_\\Pi (P) \\wedge Q + (-1)^{|P| (n-1)} P \\wedge \\delta_\\Pi (Q)\n\\end{align*}\nfor $P \\in \\Gamma{\\bigwedge^{|P|}A}, Q \\in \\Gamma{\\bigwedge^{|Q|}A}$. Then the operator\n\\begin{align*}\n \\delta_\\Pi : \\Gamma{{\\bigwedge}^kAG} \\rightarrow \\Gamma{{\\bigwedge}^{k+n -1} AG}\n\\end{align*}\nsatisfies\n\\begin{align*}\n \\delta_\\Pi ([P,Q]) = [ \\delta_\\Pi (P), Q] + (-1)^{(|P|-1)(n-1)} [P, \\delta_\\Pi (Q)].\n\\end{align*}\nNote that the operator $\\delta_\\Pi$ need not satisfy condition $\\delta_\\Pi \\circ \\delta_\\Pi = 0 .$\n\n\n\n\n\\medskip\n\nWe known that, Lie bialgebroids are infinitesimal form of Poisson groupoids. More precisely, given a Poisson groupoid $G \\rightrightarrows M$ with Lie algebroid $AG$, it is known that its dual bundle $A^*G$\nalso carries a Lie algebroid structure and $(AG, A^*G)$ forms a Lie bialgebroid. In the next theorem we show that weak Lie-Filippov bialgebroids are infinitesimal form of Nambu-Lie groupoids.\n\n\n\n\n\n\n\\begin{thm}\\label{nambu-grpd-bialgbd}\nLet $(G \\rightrightarrows M , \\Pi) $ be a Nambu-Lie groupoid of order $n$ with Lie algebroid $AG \\rightarrow M$. Then $(AG, A^*G)$ forms a weak Lie-Filippov bialgebroid of order $n$ over $M$.\n\\end{thm}\n\n\n\\begin{proof}\nFrom the Remark \\ref{nlg-bracket-anchor}, we have the space of sections of the bundle $A^*G = (TM)^0 \\rightarrow M$ admits a skew-symmetric $n$-bracket\n$[~, \\ldots, ~]$ and there exists a bundle map $$\\rho : {\\bigwedge}^{n-1}A^*G \\rightarrow TM.$$\n\nLet $\\alpha_1, \\ldots, \\alpha_{n-1} \\in \\Gamma (TM)^0=\\Gamma{(A^*G)} $ with $d_A \\alpha_i = 0$, for all $i= 1, \\ldots ,n-1.$ Let $\\tilde{\\alpha}_1, \\ldots, \\tilde{\\alpha}_{n-1}$ be, respectively, their extensions to left invariant $1$-forms on $G$ such that $d \\tilde{\\alpha}_i = 0$, for all\n$i= 1, \\ldots ,n-1.$ Then the conditions $(2)$ and $(3)$ of the Definition \\ref{lie-fill-defn} of a weak Lie-Filippov algebroid pair follows from the weak Lie-Filippov bialgebroid structure $(TG, T^*G)$ (Note that $G$ is a Nambu-Poisson manifold). \n\nLet $f \\in C^\\infty(M).$ Then observe that\n\\begin{align*}\n [\\tilde {\\alpha}_1, \\ldots, \\tilde{\\alpha}_{n-1}, (\\beta^*f) \\tilde{\\alpha}_n ] = (\\beta^*f) [\\tilde{\\alpha}_1, \\ldots, \\tilde{\\alpha}_{n-1}, \\tilde{\\alpha}_n ] + \\Pi^{\\sharp}(\\tilde{\\alpha}_1 \\wedge \\cdots \\wedge \\tilde{\\alpha}_{n-1})(\\beta^*f)\\tilde{\\alpha}_n.\n\\end{align*}\nSince $(\\beta^*f) \\tilde{\\alpha}_n = \\widetilde{f \\alpha}_n$, by assertion $(2)$ of the Propostion \\ref{coiso-n-bracket}, we get\n\\begin{align*}\n[\\alpha_1,\\ldots, \\alpha_{n-1}, f \\alpha_n] = f [\\alpha_1, \\ldots, \\alpha_n] + \\rho (\\alpha_1 \\wedge \\cdots \\wedge \\alpha_{n-1}) (f) \\alpha_n\n\\end{align*}\nproving condition (4) of the Definition \\ref{lie-fill-defn}. Moreover the compatibility condition of the weak Lie-Filippov bialgebroid (condition $(5)$ of the Definition \\ref{lie-fill-defn}) follows from the observation that for any $\\alpha \\in \\Gamma{(A^*G)} = \\Gamma (TM)^0$ and any left invariant extension $\\tilde{\\alpha} \\in \\Omega^1(G)$, we have\n\\begin{align*}\n d_A \\alpha = (d \\tilde \\alpha)|_M.\n\\end{align*}\nThus, $(AG, A^*G)$ is a weak Lie-Filippov bialgebroid of order $n$.\n\\end{proof}\n\n\n\\begin{remark}If $(G, \\Pi)$ is a Nambu-Lie group with Lie algebra $\\mathfrak{g}$, the dual vector space $\\mathfrak{g}^*$\ncarries a Filippov algebra structure \\cite{vais}. Moreover the pair $(\\mathfrak{g} , \\mathfrak{g}^*)$ forms a (weak) Lie-Filippov bialgebra (\\cite{bas-bas-das-muk, vais}).\nThe Lie-Filippov bialgebra $(\\mathfrak{g}, \\mathfrak{g}^*)$ is the infinitesimal form of the Nambu-Lie group $(G, \\Pi).$ \nA Lie-Filippov bialgebra $(\\mathfrak{g}, \\mathfrak{g}^*)$ can also be seen a Lie \nalgebra $\\mathfrak{g}$ together with a Filippov algebra structure on the dual vector space $\\mathfrak{g}^*$ such that the map $\\delta : \\mathfrak{g} \\rightarrow \\bigwedge^n \\mathfrak{g}$ dual\nto the Filippov bracket on $\\mathfrak{g}^*$, defines a $1$-cocycle of $\\mathfrak{g}$ with respect to the adjoint representation on $\\bigwedge^n \\mathfrak{g}$.\n\\end{remark}\n\nWe have seen that given a Nambu-Lie groupoid of order $n$, there is an induced Nambu-Poisson structure on the base manifold (cf. Proposition \\ref{inverse-basenambu}). On the other hand, \ngiven a weak Lie-Filippov bialgebroid, there is an induced Nambu-Poisson structure on the base (cf. Theorem \\ref{wlfb-base-nambu}).\nThe next proposition compares these Nambu-Poisson structures on the base induced from the Nambu Lie groupoid and its infinitesimal. \n\n\n\\begin{prop}\\label{compare-two-NP-structure}\nLet $(G \\rightrightarrows M , \\Pi) $ be a Nambu-Lie groupoid (of order $n$) with associated weak Lie-Filippov bialgebroid $(AG, A^*G).$ Then the induced Nambu structures on $M$\ncoming from the Nambu-Lie groupoid and the weak Lie-Filippov bialgebroid are related by\n\\begin{align*}\n \\{ ~, \\ldots, ~\\}_M = (-1)^{n-1}\\{ ~, \\ldots, ~\\}_{(AG, A^*G)} .\n\\end{align*}\n\\end{prop}\n\n\n\\begin{proof}\nFor any functions $f_1, \\ldots, f_n \\in C^\\infty(M)$, we have\n\\begin{align*}\n \\{f_1, \\ldots, f_n\\}_{(AG, A^*G)} =& \\Pi^{\\sharp}\\big|_M (d_A f_1 \\wedge \\cdots \\wedge d_A f_{n-1}) f_n\\\\\n=& \\Pi^{\\sharp} ( d (\\beta^* f_1) \\wedge \\cdots \\wedge d (\\beta^* f_{n-1}))\\big|_M f_n\\\\\n=& \\Pi (\\beta^* f_1 , \\ldots, \\beta^* f_{n-1}, \\beta^* f_n)\\big|_M\\\\\n=& (-1)^{n-1} \\big( \\beta^*\\{f_1, \\ldots, f_n\\} \\big) \\big|_M\\\\\n=& (-1)^{n-1} \\{f_1, \\ldots, f_n\\}_M.\n\\end{align*}\n\\end{proof}\n\\begin{remark}\n It is known that under some connectedness and simply connectedness assumption, any Lie bialgebra integrates to a Poisson-Lie group \\cite{lu-wein}, and any Lie bialgebroid\nintegrates to a Poisson groupoid \\cite{mac-xu2}. These results does not hold in the context of Nambu structures of order $\\geqslant 3$. Let $G$ be a connected and simply-connected Lie group with Lie algebra $\\mathfrak{g}.$ Given a Lie-Filippov bialgebra structure $(\\mathfrak{g}, \\mathfrak{g}^*)$ on $\\mathfrak{g}$, the $1$-cocycle $\\delta : \\mathfrak{g} \\rightarrow \\bigwedge^n \\mathfrak{g}$\ndual to the Filippov algebra bracket on $\\mathfrak{g}^*$ integrates a multiplicative $n$-vector field $\\Pi$ on the Lie group. However this $n$-vector field (for $n \\geqslant 3$) need not be a Nambu tensor \\cite{vais}, that is, need not be locally decomposable. Thus (weak) Lie-Filippov bialgebra\ndoes not integrate to a Nambu-Lie group in general. \n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Coisotropic subgroupoids of a Nambu-Lie groupoid}\n\nIn this final section, we introduce the notion of coisotropic subgroupoid of a Nambu-Lie groupoid \n and study the infinitesimal object corresponding to it.\n\n\\begin{defn}\n Let $(G \\rightrightarrows M , \\Pi)$ be a Nambu-Lie groupoid of order $n$. Then a subgroupoid $H \\rightrightarrows N$ is called a {\\it coisotropic subgroupoid} if $H$ is a coisotropic submanifold of $G$ with respect to $\\Pi$.\n\\end{defn}\n\n\\begin{exam}\n\\begin{enumerate}\n\\item For $n=2$, that is, when $G \\rightrightarrows M$ is a Poisson groupoid, this notion is same as the coisotropic subgroupoid of a Poisson groupoid introduced in \\cite{xu}.\n\\item Let $(G, \\Pi)$ be a Nambu-Lie group. Then a subgroup of $G$ is called coisotropic if it is also a coisotropic submanifold of $G$. Any coisotropic subgroup\nof $G$ is a coisotropic subgroupoid over a point.\n\\item Let $(G \\rightrightarrows M , \\Pi)$ be a Nambu-Lie groupoid. Then by the Proposition \\ref{inverse-basenambu}, there exist an induced Nambu-structure\non $M$ for which the source map $\\alpha$ is a Nambu-Poisson map. Let $N \\hookrightarrow M$ be a coisotropic submanifold of $M$ with respect to this induced Nambu\nstructure. Consider the restriction $G\\big|_N := \\alpha^{-1}(N) \\cap \\beta^{-1}(N)$, then $G\\big|_N \\rightrightarrows N$ is a coisotropic subgroupoid.\n\\item Let $G \\rightrightarrows M$ be a Nambu-Lie groupoid. If the set of all elements of $G$ which has same source and target, is a submanifold of $G$, then it is a coisotropic subgroupoid.\n\\end{enumerate}\n\\end{exam}\n\nNote that, the infinitesimal object corresponding to a Nambu-Lie groupoid $(G\\rightrightarrows M , \\Pi)$ is the weak Lie-Filippov bialgebroid\n$(AG, A^*G)$. Therefore it is natural to ask how the Lie algebroid of a coisotropic subgroupoid $H \\rightrightarrows N$ is related to the weak Lie-Filippov bialgebroid $(AG, A^*G)$. To answer this question, we introduce a notion of {\\it coisotropic subalgebroid} of a weak Lie-Filippov bialgebroid and show that infinitesimal forms of coisotropic subgroupoids of a Nambu-Lie groupoid $(G \\rightrightarrows M, \\Pi)$ appear as coisotropic subalgebroids of the corresponding weak Lie-Filippov bialgebroid $(AG, A^*G)$.\n\n\\begin{defn}\\label{coiso-subalg}\n Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid of order $n$ over $M$. Then a Lie subalgebroid $B \\rightarrow N$ of $A\\rightarrow M$ is\ncalled a {\\it coisotropic subalgebroid} if the anchor $\\rho: \\bigwedge^{n-1}A^* \\rightarrow TM$ and the $n$-bracket $[~, \\ldots, ~]$ on $\\Gamma{A^*}$\nsatisfy the following properties.\n\\begin{enumerate}\n\\item The anchor $\\rho$ maps $\\bigwedge^{n-1}B^{0} \\rightarrow TN$.\n\\item If $\\alpha_1, \\ldots, \\alpha_n \\in \\Gamma{A^*}$ with ${\\alpha_i}\\big|_N \\in B^{0}$ for all $i$, then $[\\alpha_1, \\ldots, \\alpha_n]\\big|_N \\in B^{0}$.\n\\item If $\\alpha_1, \\ldots, \\alpha_n \\in \\Gamma{A^*}$ with ${\\alpha_i}\\big|_N \\in B^{0}$ for all $i$ and $\\alpha_n\\big|_N = 0$, then $[\\alpha_1, \\ldots, \\alpha_n]\\big|_N = 0,$\nwhere $B_x^{0} = \\{\\gamma \\in A_x^* | \\gamma (v) = 0, \\forall v \\in B_x\\}$, is the annihilator of $B_x$, $x \\in N$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{exam}\n Let $M$ be a Nambu-Poisson manifold, then $(TM, T^*M)$ is a weak Lie-Filippov bialgebroid over $M$. Let $N \\hookrightarrow M$ be a coisotropic submanifold. Then from the Proposition \\ref{coiso-n-bracket}, it follows that the tangent bundle $TN \\rightarrow N$ is a coisotropic subalgebroid. \n\\end{exam}\n\nIt is known that (Proposition \\ref{wlfb-base-nambu}, see also \\cite{bas-bas-das-muk}), the base of a weak Lie-Filippov bialgebroid carries a Nambu structure.\nThe next Proposition shows that the base of a coisotropic subalgebroid is a coisotropic submanifold with respect to this induced Nambu structure.\n\n\\begin{prop}\n Let $(A, A^*)$ be a weak Lie-Filippov bialgebroid over $M$ and $B \\rightarrow N$ be a coisotropic subalgebroid. Then $N$ is a coisotropic submanifold of $M$.\n\\end{prop}\n\n\\begin{proof}\nLet $a : A \\rightarrow TM$ denote the anchor of the Lie algebroid $A$ and $\\rho : \\bigwedge^{n-1}A^* \\rightarrow TM$ be the anchor of pair $(A, A^*).$\nWe first show that, $a^* (TN)^0 \\subseteq B^{0}$. This is true because, $ \\langle a^*\\xi_x, v \\rangle = \\langle \\xi_x, a(v)\\rangle = 0$\nfor $\\xi_x \\in (TN)_x^0$ and $v \\in B_x$.\n\nLet $\\Pi_{(A, A^*)}$ be the induced Nambu structure on $M$ coming from the weak Lie-Filippov bialgebroid $(A, A^*).$ Then the induced map\n$\\Pi^{\\sharp}_{(A, A^*)} : \\bigwedge^{n-1}T^*M \\rightarrow TM$ is given by $\\Pi^{\\sharp}_{(A, A^*)} = \\rho \\circ \\bigwedge^{n-1}a^*$. Therefore, for any\n$\\xi_1, \\ldots, \\xi_{n-1} \\in (TN)^0$, we have\n$$ \\Pi^{\\sharp}_{(A, A^*)} (\\xi_1, \\ldots, \\xi_{n-1}) = \\rho (a^*\\xi_1, \\ldots, a^*\\xi_{n-1}) \\in TN $$\nas $a^*\\xi_i \\in B^{0}$ and $B$ is a coisotropic subalgebroid. Therefore $N$ is a coisotropic submanifold of $M$.\n\\end{proof}\n\n\nThe next proposition shows that the infinitesimal object corresponding to coisotropic subgroupoids are coisotropic subalgebroids.\n\n\\begin{prop}\\label{coiso-subgrpd-coiso-subalgbd}\nLet $(G \\rightrightarrows M, \\Pi)$ be a Nambu-Lie groupoid with weak Lie-Filippov bialgebroid $(AG, A^*G)$. Let $H \\rightrightarrows N$\nbe a coisotropic subgroupoid of $G \\rightrightarrows M$ with Lie algebroid $AH \\rightarrow N$. Then $AH \\rightarrow N$ is a coisotropic subalgebroid.\n\\end{prop}\n\n\\begin{proof}\n Since $H \\rightrightarrows N$ is a Lie subgroupoid of $G \\rightrightarrows M$, therefore $AH \\rightarrow N$ is a Lie subalgebroid of\n$AG \\rightarrow M$. We claim that the anchor $\\rho = \\Pi^{\\sharp}\\big|_{\\bigwedge^{n-1}(TM)^0} = \\Pi^{\\sharp}\\big|_{\\bigwedge^{n-1} (A^*G)}$ of the weak\nLie-Filippov bialgebroid $(AG, A^*G)$ maps $\\bigwedge^{n-1}(AH)^0$ to $TN$. First observe that, for any $x \\in N$, $(AH)_x^0 = (TM)_x^0 \\cap (TH)_x^0$\nand $T_xN = T_xM \\cap T_xH$. Therefore, \n$\\rho$ maps $\\bigwedge^{n-1} (AH)^0$ to \n$$\\Pi^{\\sharp}({\\bigwedge}^{n-1} (TM)^0) \\cap \\Pi^{\\sharp}({\\bigwedge}^{n-1} (TH)^0) \\subseteq TM \\cap TH \\cong TN,$$ \nhere we have used the fact that $M$ and $H$ are both coisotropic submanifolds of $G$.\n\nLet $\\alpha_1, \\ldots, \\alpha_n \\in \\Gamma{A^*G} = \\Gamma{(TM)^0}$ such that $\\alpha_i\\big|_N \\in (AH)^0$, for all $i=1, \\ldots, n$. Let\n$\\tilde{\\alpha}_1, \\ldots, \\tilde{\\alpha}_n$ be one forms on $G$ extending $\\alpha_1, \\ldots, \\alpha_n$ and are conormal to $H$. Then\nby the Proposition \\ref{coiso-n-bracket}, the $1$-form $[\\tilde{\\alpha_1}, \\ldots, \\tilde{\\alpha_n}]$ is conormal to both $M$ and $H$, as $M$\nand $H$ are both coisotropic submanifolds of $G$. Therefore, \n$$[\\tilde{\\alpha_1}, \\ldots, \\tilde{\\alpha_n}]\\big|_N \\in (TM)^0 \\cap (TH)^0 \\cong (AH)^0.$$ \nVerification of the last condition of the Definition \\ref{coiso-subalg} is similar. Hence $AH \\rightarrow N$ is a coisotropic subalgebroid of $(AG, A^*G).$\n\\end{proof}\n\n\\begin{corollary}\n Let $(G \\rightrightarrows M , \\Pi)$ be a Nambu-Lie groupoid and $H \\rightrightarrows N$ be a coisotropic subgroupoid. Then $N$ is a coisotropic submanifold of $M$.\n\\end{corollary}\n\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1} \\markboth{#1}{#1}\n \\leavevmode\\par}\n\\def\\paragraph#1{{\\bf #1\\ }}\n\n\\newtheorem{lemma}{Lemma}[section]\n\n\\newtheorem{theorem}[lemma]{Theorem}\n\n\\newtheorem{corollary}[lemma]{Corollary}\n\n\\newtheorem{definition}[lemma]{Definition}\n\n\\newtheorem{proposition}[lemma]{Proposition}\n\n\\newtheorem{remark}{Remark}[section]\n\n\\newtheorem{example}{Example}[section]\n\n\\newtheorem{conject}{Conjecture}[section]\n\n\\newtheorem{hypothesis}{Hypothesis}[section]\n\n\n\n\n\\def\\varepsilon{\\varepsilon}\n\\def\\<{\\langle}\n\\def\\>{\\rangle}\n\\def\\subset\\subset{\\subset\\subset}\n\\def {\\mathbb R} {{\\mathbb R}}\n\\def {\\mathbb Q} {{\\mathbb Q}}\n\\def {\\mathbb Z} {{\\mathbb Z}}\n\\def {\\mathbb C} {{\\mathbb C}}\n\\def {\\mathbb N} {{\\mathbb N}}\n\n\n\\def {\\mathbb S} {{\\mathbb S}}\n\\def {\\frac{1}{2}} {{\\frac{1}{2}}}\n\\def \\varepsilon {\\varepsilon}\n\\def \\sigma {\\sigma}\n\\def \\alpha {\\alpha}\n\\def \\beta {\\beta}\n\\def \\omega {\\omega}\n\\def \\Omega{\\Omega}\n\\def \\Theta {\\Theta}\n\\def \\delta {\\delta}\n\\def \\rho {\\rho}\n\\def \\phi {\\phi}\n\n\n\n\n\n\\title{Macroscopic models of collective motion with repulsion}\n\\author{Pierre Degond$^{1}$, Giacomo Dimarco$^{2}$, Thi Bich Ngoc Mac$^{3,4}$, Nan Wang $^{5} $ }\n\\date{}\n\\begin{document}\n\n\\maketitle\n\n\\begin{center}\n1- Department of Mathematics, Imperial college London, \\\\\nLondon SW7 2AZ, United Kingdom.\\\\\nemail:pdegond@imperial.ac.uk\\\\\n$\\mbox{}$ \\\\\n2-Department of Mathematics, University of Ferrara, 44100 Ferrara, Italy\\\\\nemail: giacomo.dimarco@unife.it \\\\\n$\\mbox{}$ \\\\\n3-Universit\\'e de Toulouse; UPS, INSA, UT1, UTM ;\\\\\nInstitut de Math\\'ematiques de Toulouse ; \\\\\nF-31062 Toulouse, France. \\\\\n$\\mbox{}$ \\\\\n4-CNRS; Institut de Math\\'ematiques de Toulouse UMR 5219 ;\\\\\nF-31062 Toulouse, France.\\\\\nemail: thi-bich-ngoc.mac@math.univ-toulouse.fr \\\\\n$\\mbox{}$ \\\\\n5-National University of Singapore, Department of Mathematics,\\\\\nLower Kent Ridge Road, Singapore 119076.\\\\\nemail: g0901252@nus.edu.sg \\\\\n\\end{center}\n\n\n\n\\vspace{0.5 cm}\n\\begin{abstract}\n\nWe study a system of self-propelled particles which interact with their neighbors via alignment and repulsion. The particle velocities result from self-propulsion and repulsion by close neighbors. The direction of self-propulsion is continuously aligned to that of the neighbors, up to some noise. A continuum model is derived starting from a mean-field kinetic description of the particle system. It leads to a set of non conservative hydrodynamic equations. We provide a numerical validation of the continuum model by comparison with the particle model. We also provide comparisons with other self-propelled particle models with alignment and repulsion. \n\\end{abstract}\n\n\\medskip\n\\noindent\n{\\bf Acknowledgements:} PD is on leave from CNRS, Institut de Math\\'ematiques de Toulouse, France, and GD is on leave from Universit\\'e Paul Sabatier, Institut de Math\\'ematiques de Toulouse, France, where part of this research was conducted. TBNM wishes to thank the hospitality of the Department of Mathematics, Imperial College London for its hospitality. NW wishes to thank the Universit\\'e Paul Sabatier, Institut de Math\\'ematiques de Toulouse, France, for its hospitality. This work has been supported by the French 'Agence Nationale pour la Recherche (ANR)' in the frame of the contract 'MOTIMO' (ANR-11-MONU-009-01). \n\n\n\\medskip\n\\noindent\n{\\bf Key words: } Fokker-Planck equation, macroscopic limit, Von Mises-Fisher distribution, Generalized Collision Invariants, Non-conservative equations, Self-Organized Hydrodynamics, self-propelled particles, alignment, repulsion.\n\n\\medskip\n\\noindent\n{\\bf AMS Subject classification: } 35Q80, 35L60, 82C22, 82C70, 92D50\n\\vskip 0.4cm\n\n\n\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Introduction}\n\\label{sec:intro}\n\nThe study of collective motion in systems consisting of a large number of agents, such as bird flocks, fish schools, suspensions of active swimmers (bacteria, sperm cells ), etc has triggered an intense literature in the recent years. We refer to \\cite{Vicsek_Zafeiris_PhysRep12, Koch_FluidMech2011} for recent reviews on the subject. Many of such studies rely on a particle model or Individual Based Model (IBM) that describes the motion of each individual separately (see e.g in \\cite{Aoki_BullJapSocSciFish92, Chate_etal_PRE08, Chuang_etal_PhysicaD07, Couzin_etal_JTB02, Cucker_Smale_IEEETransAutCont07, Henkes_etal_PRE11, Mogilner_etal_JMB03, Motsch_Tadmor_JSP11, Szabo_PRL06}).\n\n\nIn this work, we aim to describe dense suspensions of elongated self-propelled particles in a fluid, such as sperm. In such dense suspensions, steric repulsion is an essential ingredient of the dynamics. A large part of the literature is concerned with dilute suspensions \\cite{Hernandez-Ortiz_etal_JPhysCondMat09, Koch_FluidMech2011, Pedley_etal_JFM88, Saintillan_Fluids08, Woodhouse_Goldstein_PRL12}. In these approaches, the Stokes equation for the fluid is coupled to the orientational distribution function of the self-propelled particles. However, these approaches are of ``mean-field type'' i.e. assume that particle interactions are mediated by the fluid through some kinds of averages. These approaches do not deal easily with short-range interactions such as steric repulsion or interactions mediated by lubrication forces. Additionally, these models assume a rather simple geometry of the swimmers, which are reduced to a force dipole, while the true geometry and motion of an actual swimmer, like a sperm cell, is considerably more complex. \n\nIn a recent work \\cite{Peruani_etal_PRE06}, Peruani et al showed that, for dense systems of elongated self-propelled particles, steric interaction results in alignment. Relying on this work, and owing to the fact that the description of swimmer interactions from first physical principles is by far too complex, we choose to replace the fluid-mediated interaction by a simple alignment interaction of Vicsek type \\cite{Vicsek_etal_PRL95}. In the Vicsek model, the agents move with constant speed and attempt to align with their neighbors up to some noise. Many aspects of the Vicsek model have been studied, such as phase transitions \\cite{Aldana_etal_PRL07, Chate_etal_PRE08, Degond_etal_JNonlinearSci13, Degond_etal_preprint13, Gretoire_Chate_PRL04, Vicsek_etal_PRL95}, numerical simulations \\cite{Motsch_Navoret_MMS11}, derivation of macroscopic models \\cite{Bertin_etal_JPA09, Degond_Motsch_M3AS08}.\n\nThe alignment interaction acting alone may trigger the formation of high particle concentrations. However, in dense suspensions, volume exclusion prevent such high densities to occur. When distances between particles become too small, repulsive forces are generated by the fluid or by the direct reaction of the bodies one to each other. These forces contribute to repel the particles and to prevent further contacts. To model this behavior, we must add a repulsive force to the Vicsek alignment model. Inspired by \\cite{Baskaran_Marchetti_PRL10, Henkes_etal_PRE11, Szabo_PRL06} we consider the possibility that the particle orientations (i.e the directions of the self-propulsion force) and the particle velocities may be different. Indeed, steric interaction may push the particles in a direction different from that of their self-propulsion force. \n\nWe consider an overdamped regime in which the velocity is proportional to the force through a mobility coefficient. The overdamped limit is justified by the fact that the background fluid is viscous and thus the forces due to friction are very large compared to those due to motion. Indeed, for micro size particles, the Reynolds number is very small ($\\sim 10^{-4}$) and thus the effect of inertia can be neglected. Finally, differently from \\cite{Baskaran_Marchetti_PRL10, Henkes_etal_PRE11, Szabo_PRL06} we consider an additional term describing the relaxation of the particle orientation towards the direction of the particle velocity. We also take into account a Brownian noise in the orientation dynamics of the particles. This noise may take into account the fluid turbulence for instance. Therefore, the particle dynamics \nresults from an interplay between relaxation towards the mean orientation of the surrounding particles, relaxation towards the direction of the velocity vector and Brownian noise. From now on we refer to the above described model as the Vicsek model with repulsion.\n\nStarting from the above described microscopic dynamical system we successively derive mean-field equations and hydrodynamic equations. Mean field equations are valid when the number of particles is large and describe the evolution of the one-particle distribution, i.e. the probability for a particle to have a given orientation and position at a given instant of time. Expressing that the spatio-temporal scales of interest are large compared to the agents' scales leads to a singular perturbation problem in the kinetic equation. Taking the hydrodynamic limit, (i.e. the limit of the singular perturbation parameter to zero) leads to the hydrodynamic model. Hydrodynamic models are particularly well-suited to systems consisting of a large number of agents and to the observation of the system's large scale structures. Indeed, the computational cost of IBM increases dramatically with the number of agents, while that of hydrodynamic models is independent of it. With IBM, it is also sometimes quite cumbersome to access observables such as order parameters, while these quantities are usually directly encoded into the hydrodynamic equations. \n\nThe derivation of hydrodynamic models has been intensely studied by many authors. Many of these models are based on phenomenological considerations \\cite{Toner_etal_AnnPhys05} or derived from moment approaches and ad-hoc closure relations \\cite{Baskaran_Marchetti_PRL10, Bertin_etal_JPA09, Ratushnaya_etal_PhysicaA07}. The first mathematical derivation of a hydrodynamic system for the Vicsek model has been proposed in \\cite{Degond_Motsch_M3AS08}. We refer to this model to as the Self-Organized Hydrodynamic (SOH) model. One of the main contributions of \\cite{Degond_Motsch_M3AS08} is the concept of ``Generalized Collision Invariants'' (GCI) which permits the derivation of macroscopic equations for a particle system in spite of its lack of momentum conservation. The SOH model has been further refined in \\cite{Degond_etal_MAA13, Frouvelle_M3AS12}. \n\nPerforming the hydrodynamic limit in the kinetic equations associated to the Vicsek model with repulsion leads to the so-called ``Self-Organized Hydrodynamics with Repulsion'' (SOHR) system. The SOHR model consists of a continuity equation for the density $\\rho$ and an evolution equation for the average orientation $\\Omega \\in {\\mathbb S}^{n-1}$ where $n$ indicates the spatial dimension. More precisely, the model reads\n\\begin{eqnarray}\n& \\partial_t\\rho + \\nabla_x \\cdot(\\rho U ) = 0, \\label{macro1_intro} \\\\\n& \\rho \\partial_t \\Omega + \\rho (V \\cdot \\nabla_x)\\Omega + P_{\\Omega^\\perp}\\nabla_x p(\\rho) = \\gamma P_{\\Omega^\\perp} \\Delta_x(\\rho \\Omega), \\label{macro2_intro}\t\\\\\n& \\vert \\Omega \\vert = 1,\t\\label{macro3_intro}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n& U = c_1v_0 \\Omega - \\mu \\Phi_0 \\nabla_x \\rho , \\quad V = c_2 v_0 \\Omega - \\mu \\Phi_0 \\nabla_x \\rho , \\label{macro4_intro}\\\\\n& p(\\rho ) = v_0 d \\rho + \\alpha \\mu \\Phi_0 \\big( (n-1)d + c_2 \\big) \\dfrac{\\rho^2}{2}, \\quad \\gamma = k_0 \\big( (n-1)d + c_2\\big). \\label{macro5_intro}\n\\end{eqnarray}\nThe coefficients $c_1$, $c_2$, $v_0$, $\\mu$, $\\Phi_0$, $d$, $\\alpha$, $k_0$ are associated to the microscopic dynamics and will be defined later on. The symbol $P_{\\Omega^\\bot}$ stands for the projection matrix $ P_{\\Omega^\\bot} = \\mbox{Id} - \\Omega \\otimes \\Omega$ of ${\\mathbb R}^n$ on the hyperplane $\\Omega^\\bot$. The SOHR model is similar to the SOH model obtained in \\cite{Degond_etal_MAA13}, but with several additional terms which are consequences of the repulsive force at the particle level. The repulsive force intensity is characterized by the parameter $\\mu \\Phi_0$. In the case $\\mu \\Phi_0 = 0$, the SOHR system is reduced to the SOH one. \n\nWe first briefly describe the original SOH model. Inserting (\\ref{macro4_intro}), (\\ref{macro5_intro}) with $\\mu \\Phi_0 = 0$ into (\\ref{macro1_intro}), (\\ref{macro2_intro}) leads to \n\\begin{eqnarray}\n& \\partial_t\\rho + c_1 v_0 \\, \\nabla_x \\cdot(\\rho \\Omega ) = 0, \\label{macro6_intro} \\\\\n& \\rho \\partial_t \\Omega + c_2 v_0 \\rho (\\Omega \\cdot \\nabla_x)\\Omega + v_0 d \\, P_{\\Omega^\\perp}\\nabla_x \\rho = \\gamma P_{\\Omega^\\perp} \\Delta_x(\\rho \\Omega), \\label{macro7_intro}\n\\end{eqnarray}\ntogether with (\\ref{macro3_intro}). This model shares similarities with the isothermal compressible Navier-Stokes (NS) equations. Both models consist of a non linear hyperbolic part supplemented by a diffusion term. Eq. (\\ref{macro6_intro}) expresses conservation of mass, while Eq. (\\ref{macro7_intro}) is an equation for the mean orientation of the particles. It is not conservative, contrary to the corresponding momentum conservation equation in NS. The two equations are supplemented by the geometric constraint (\\ref{macro3_intro}). This constraint is satisfied at all times, as soon as it is satisfied initially. The preservation of this property is guaranteed by the projection operator $P_{\\Omega^\\perp}$. A second important difference between the SOH model and NS equations is that the convection velocities for the density and the orientation, $v_0 c_1$ and $v_0 c_2$ respectively are different while for NS they are equal. That $c_1 \\not = c_2$ is a consequence of the lack of Galilean invariance of the model (there is a preferred frame, which is that of the fluid). The main consequence is that the propagation of sound waves is anisotropic for this type of fluids \\cite{Toner_etal_AnnPhys05}. \n\nThe first main difference between the SOH and the SOHR system is the presence of the terms $\\mu \\Phi_0 \\nabla_x \\rho$ in the expressions of the velocities $U$ and $V$. Inserting this term in the density Eq. (\\ref{macro1_intro}) results in a diffusion-like term $- \\mu \\Phi_0 \\, \\nabla_x \\cdot(\\rho ( \\nabla_x \\rho))$ which avoids the formation of high particle concentrations. This term shows similarities with the non-linear diffusion term in porous media models. Similarly, inserting the term $\\mu \\Phi_0 \\nabla_x \\rho$ in the orientation Eq. (\\ref{macro2_intro}) results in a convection term in the direction of the gradient of the density. Its effect is to force particles to change direction and move towards regions of lower concentration. The second main difference is the replacement of the linear (with respect to $\\rho$) pressure term $v_0 d \\, P_{\\Omega^\\perp}\\nabla_x \\rho$ by a nonlinear pressure $p(\\rho)$ in the orientation Eq. (\\ref{macro2_intro}). The nonlinear part of the pressure enhances the effects of the repulsion forces when concentrations become high. \n\nTo further establish the validity of the SOHR model (\\ref{macro1_intro})-(\\ref{macro5_intro}), we perform numerical simulations and compare them to those of the underlying IBM. To numerically solve the SOHR model, we adapt the relaxation method of \\cite{Motsch_Navoret_MMS11}. In this method, the unit norm constraint (\\ref{macro3_intro}) is abandonned and replaced by a fully conservative hyperbolic model in which $\\Omega$ is supposed to be in ${\\mathbb R}^n$. However, at the end of each time step of this conservative model, the vector $\\Omega$ is normalized. Motsch and Navoret showed that the relaxation method provides numerical solutions of the SOH model which are consistent with those of the particle model. The resolution of the conservative model can take advantage of the huge literature on the numerical resolution of hyperbolic conservation laws (here specifically, we use \\cite{Degond_Peyrard_99}). We adapt the technique of \\cite{Motsch_Navoret_MMS11} to include the diffusion fluxes. Using these approximations, we numerically demonstrate the good convergence of the scheme for smooth initial data and the consistency of the solutions with those of the particle Vicsek model with repulsion. \n\nThe outline of the paper is as follows. In section 2, we introduce the particle model, its mean field limit, the scaling and the hydrodynamic limit. In Section 3 we present the numerical discretization of the SOHR model, while in Section 4 we present several numerical tests for the macroscopic model and a comparison between the microscopic and microscopic models. Section 5 is devoted to draw a conclusion. Some technical proofs will be given in the Appendices.\n\n\n\n\\setcounter{equation}{0}\n\\section{ Model hierarchy and main results }\n\\label{sec:IBM}\n\n\\subsection{The individual based model and the mean field limit}\n\nWe consider a system of $N$-particles each of which is described by its position $ X_k(t) \\in {\\mathbb R}^n$, its velocity $v_k(t) \\in {\\mathbb R}^n,$ and its direction $ \\omega_k(t) \\in {\\mathbb S}^{n-1} ,$ where $ k \\in \\{ 1, \\cdots , N \\}$, $n$ is the spatial dimension and ${\\mathbb S}^{n-1}$ denotes the unit sphere. The particle ensemble satisfies the following stochastic differential equations \n\\begin{eqnarray} \n\\dfrac{dX_k}{dt} & = & v_k,\n\\label{overdamped1}\\\\\nv_k\t& = & v_0 \\, \\omega_k - \\mu \\, \\nabla_x \\Phi(X_k(t),t),\n\\label{overdamped2}\\\\\nd\\omega_k\t & = & P_{\\omega_k^\\perp}\\circ (\\nu \\, \\bar{\\omega}(X_k(t),t) \\, dt + \\alpha \\, v_k \\, dt + \\sqrt{2D} \\, dB^k_t ).\\label{overdamped3}\n\\end{eqnarray}\t\nEq. (\\ref{overdamped1}) simply expresses the spatial motion of a particle of velocity $v_k$. Eq (\\ref{overdamped2}) shows that the velocity $v_k$ is composed of two components: a self-propulsion velocity of constant magnitude $v_0$ in direction $\\omega_k$ and a repulsive force proportional to the gradient of a potential $\\Phi(x,t)$ with mobility coefficient $\\mu$. Equation (\\ref{overdamped3}) describes the time evolution of the orientation. The first term models the relaxation of the particle orientation towards the average orientation $\\bar{\\omega}(X_k(t),t)$ of its neighbors with rate $\\nu$. The second term models the relaxation of the particle orientation towards the direction of the particle velocity $v_k$ with rate $\\alpha$. Finally, the last term describes standard independent white noises $dB^k_t$ of intensity $\\sqrt{2D}$. The symbol $\\circ$ reminds that the equation has to be understood in the Stratonovich sense. Under this condition and thanks to the presence of $P_{\\omega^\\perp}$, the orthogonal projection onto the plane orthogonal to $\\omega$ (i.e $P_{\\omega^\\perp} = (\\mbox{Id} - \\omega \\otimes \\omega),$ where $\\otimes $ denotes the tensor product of two vectors and $\\mbox{Id}$ is the identity matrix), the orientation $\\omega_k$ remains on the unit sphere. We assume that $v_0$, $\\mu$, $\\nu$ $\\alpha$, $D$ are strictly positive constants. \n\nThe repulsive potential $\\Phi(x,t)$ is the resultant of binary interactions mediated by the binary interaction potential $ \\phi $. It is given by: \n\\begin{equation}\n\\Phi(x,t) = \\dfrac{1}{N} \\sum_{i=1}^{N} \\nabla \\phi \\Big( \\frac{|x - X_i|}{r} \\Big)\n\\label{eq:pot}\n\\end{equation}\nwhere the binary repulsion potential $\\phi(|x|)$ only depends on the distance. We suppose that $x \\mapsto \\phi(|x|)$ is smooth (in particular implying that $\\phi'(0) =0$ where the prime denotes the derivative with respect to $|x|$). We also suppose that \n$$ \\phi \\geq 0, \\qquad \\int_{{\\mathbb R}^n} \\phi(|x|) \\, dx < \\infty, $$\nin particular implying that $\\phi(|x|) \\to 0$ as $ |x| \\rightarrow \\infty$. The quantity $r$ denotes the typical range of $\\phi$ and the fact that $\\phi \\geq 0$ ensures that $\\phi$ is repulsive. In the numerical test Section, we will propose precise expressions for this potential force. \n\nThe mean orientation $\\bar{\\omega}(x,t)$ is defined by\n\\begin{equation}\n\\bar{\\omega}(x,t) = \\dfrac{{\\mathcal J}(x,t)}{\\vert {\\mathcal J}(x,t) \\vert}, \\qquad {\\mathcal J} (x,t) = \\frac{1}{N} \\sum\\limits_{i=1}^{N} K\\Big(\\frac{|x-X_i|}{R}\\Big) \\omega_i.\n\\label{eq:meanorient}\n\\end{equation}\t\nIt is constructed as the normalization of the vector $ {\\mathcal J} (x,t)$ which sums up all orientation vectors $\\omega_i$ of all the particles which belong to the range of the ``influence kernel'' $K(|x|)$. The quantity $R>0$ is the typical range of the influence kernel $K (|x|\/R)$, which is supposed to depend only on the distance. It measures how the mean orientation at the origin is influenced by particles at position $x$. Here, we assume that $x \\to K(|x|)$ is smooth at the origin and compactly supported. For instance, if $K$ is the indicator function of the ball of radius $1$, the quantity $\\bar \\omega (x,t)$ computes the mean direction of the particles which lie in the sphere of radius $R$ centered at $x$ at time $t$. \n\n\n\n\n\\begin{remark}\n(i) In the absence of repulsive force (i.e. $\\mu = 0$), the system reduces to the time continuous version of the Vicsek model proposed in \\cite{Degond_Motsch_M3AS08}. \\\\\n(ii) The model presented is the so called overdamped limit of the model consisting of (\\ref{overdamped1}) and (\\ref{overdamped3}) and where (\\ref{overdamped2}) is replaced by: \n\\begin{eqnarray} \\label{nonoverdamped4}\n\\epsilon \\dfrac{dv_k}{dt}\t & = & \\lambda_1 (v_0 \\omega_k - v_k) - \\lambda_2 \\nabla_x \\Phi(X_k(t),t).\n\\end{eqnarray}\t\nwith $\\mu = \\lambda_2\/\\lambda_1.$ Taking the limit $ \\epsilon \\rightarrow 0 $ in (\\ref{nonoverdamped4}), we obtain (\\ref{overdamped2}). As already mentioned in the Introduction, for microscopic swimmers, this limit is justified by the very small Reynolds number and the very small inertia of the particles. \n\\end{remark}\n\nWe now introduce the mean field kinetic equation which describes the time evolution of the particle system in the large $N$ limit. The unknown here is the one particle distribution function $f(x, \\omega, t)$ which depends on the position $x \\in {\\mathbb R}^n$, orientation $\\omega\\in {\\mathbb S}^{n-1}$ and time $t$. The evolution of $f$ is governed by the following system\n\\begin{eqnarray}\n& & \\hspace{-1cm} \\partial_t f + \\nabla_x \\cdot (v_f f) + \\nu \\, \\nabla_\\omega \\cdot( P_{\\omega^\\perp} \\bar{\\omega}_f f) + \\alpha \\, \\nabla_\\omega \\cdot (P_{\\omega^\\perp} v_f f ) - D \\Delta_\\omega f = 0, \n\\label{eqmeanfield1}\\\\\n& & \\hspace{-1cm} v_f(x, t) = \tv_0 \\omega - \\mu \\nabla_x \\Phi_f(x,t), \n\\label{eqmeanfield2}\n\\end{eqnarray}\nwhere the repulsive potential and the average orientation are given by \t\n\\begin{eqnarray}\n& & \\Phi_f(x,t) = \\int_{{\\mathbb S}^{n-1} \\times {\\mathbb R}^n} \\phi\\left(\\dfrac{|x-y|}{r}\\right) \\,f(y, w, t) \\, dw \\, dy, \n\\label{mfeq1} \\\\\n & &\\bar{\\omega}_f (x,\\omega, t) = \\dfrac{\\mathcal J_f(x,t)}{|\\mathcal J_f(x,t)|}, \n\\label{mfeq2}\\\\\t \t\n& & \\mathcal J_f (x,t) = \t\\int_{{\\mathbb S}^{n-1} \\times {\\mathbb R}^n} K\\left(\\dfrac{|x-y|}{R}\\right) \\, f(y, w, t) \\, w \\, dw \\, dy \n\\label{mfeq3}.\t\t \n\\end{eqnarray}\t\nEquation (\\ref{eqmeanfield1}) is a Fokker-Planck type equation. The second term at the left-hand side of (\\ref{eqmeanfield1}) describes particle transport in physical space with velocity $ v_f$ and is the kinetic counterpart of Eq. (\\ref{overdamped1}). The third, fourth and fifth terms describe transport in orientation space and are the kinetic counterpart of Eq. (\\ref{overdamped3}). The alignment interaction is expressed by the third term, while the relaxation force towards the velocity $v_f$ is expressed by the fourth term. The fifth term represents the diffusion due to the Brownian noise in orientation space. The projection $P_{\\omega^\\perp}$ insures that the force terms are normal to $\\omega$. The symbols $\\nabla_\\omega \\cdot$ and $\\Delta_\\omega$ respectively stand for the divergence of tangent vector fields to ${\\mathbb S}^{n-1}$ and the Laplace-Beltrami operator on ${\\mathbb S}^{n-1}$. Eq. (\\ref{eqmeanfield2}) is the direct counterpart of (\\ref{overdamped2}). \n\nEq. (\\ref{mfeq1}) is the continuous counterpart of Eq. (\\ref{eq:pot}). Indeed, letting $f$ be the empirical measure\n$$f = \\frac{1}{N} \\sum_{i=1}^N \\delta_{(x_i(t), \\omega_i(t))} (x,\\omega), $$\nin (\\ref{mfeq1}) (where $\\delta_{(x_i(t), \\omega_i(t))} (x,\\omega)$ is the Dirac delta at $(x_i(t), \\omega_i(t))$) leads to (\\ref{eq:pot}). Similarly, Eqs. (\\ref{mfeq2}), (\\ref{mfeq3}) are the continuous counterparts of (\\ref{eq:meanorient}) (by the same kind of argument). The rigorous convergence of the particle system to the above Fokker-Planck equation (\\ref{eqmeanfield1}) is an open problem. We recall however that, the derivation of the kinetic equation for the Vicsek model without repulsion has been done in \\cite{Bolley_etal_AML11} in a slightly modified context.\n\n\n\\subsection{Scaling}\n\t\nIn order to highlight the role of the various terms, we first write the system in dimensionless form. We chose $t_0$ as unit of time and choose \n$$ x_0 = v_0 t_0, \\qquad f_0 = \\frac{1}{x_0^n}, \\qquad \\phi_0 = \\frac{v_0^2 \\, t_0}{\\mu}, $$\nas units of space, distribution function and potential. We introduce the dimensionless variables:\n$$\n\\tilde{x} = \\dfrac{x}{x_0}, \\qquad \\tilde{t} = \\dfrac{t}{t_0}, \\qquad \\tilde{f} = \\dfrac{f}{f_0}, \\qquad \\tilde{\\phi} = \\dfrac{\\phi}{\\phi_0}, \n$$\nand the dimensionless parameters\n$$\n\\breve R = \\dfrac{R}{x_0}, \\qquad \\breve r = \\dfrac{r}{x_0}, \\qquad \\breve D = t_0 D, \\qquad \\breve \\nu = t_0 \\nu, \\qquad \\breve \\alpha = \\alpha x_0.\n$$\nIn the new set of variables $(\\tilde x,\\tilde t)$, Eq. (\\ref{eqmeanfield2}) becomes\t(dropping the tildes and the $\\breve{}$ for simplicity): \n$$\nv_f =\t \\omega - \\nabla_x \\Phi_f(x,t), \n$$\n\\noindent\t\nwhile $f$, $\\Phi_f$, $\\bar \\omega_f$, ${\\mathcal J}_f$are still given by (\\ref{eqmeanfield2}), (\\ref{mfeq1}), (\\ref{mfeq2}), (\\ref{mfeq3}) (now written in the new variables). \n\nWe now define the regime we are interested in. We assume that the ranges $R$ and $r$ of the interaction kernels $K$ and $\\phi $ are both small but with $R$ much larger than $r$. More specifically, we assume the existence of a small parameter $\\varepsilon \\ll 1$ such that:\n$$ R = \\sqrt{\\varepsilon} \\hat{R}, \\qquad r = \\varepsilon \\hat{r} \\qquad \\mbox{ with } \\qquad \\hat{R}, \\, \\hat{r} = \\mathcal O(1).$$ \nWe also assume that the diffusion coefficient $D$ and the relaxation rate to the mean orientation $\\nu$ are large and of the same orders of magnitude (i.e. $d = D\/\\nu = {\\mathcal O}(1)$), while the relaxation to the velocity $\\alpha$ stays of order $1$, i.e.\n$$ \\nu = \\dfrac{1}{\\varepsilon}, \\qquad d = \\dfrac{D}{\\nu} = {\\mathcal O}(1), \\qquad \\alpha = {\\mathcal O}(1). $$\nWith these new notations, dropping all 'hats', the distribution function $f^\\varepsilon(x,\\omega,t)$ (where the superscript $\\varepsilon$ now higlights the dependence of $f$ upon the small parameter $\\varepsilon$) satisfies the following Fokker-Plank equation \t\n\\begin{eqnarray}\n&&\\hspace{-1.2cm} \\varepsilon \\Big( \\partial_t f^\\varepsilon + \\nabla_x \\cdot (v_{f^\\varepsilon}^\\varepsilon f^\\varepsilon) \\Big)+ \\nabla_\\omega \\cdot( P_{\\omega^\\perp} \\bar \\omega^\\varepsilon_{f^\\varepsilon} f^\\varepsilon) + \\varepsilon \\alpha \\nabla_\\omega \\cdot (P_{\\omega^\\perp} v^\\varepsilon_{f^\\varepsilon} f^\\varepsilon) - d \\Delta_\\omega f^\\varepsilon = 0, \n\\label{eqmeanfield5} \\\\\n&&\\hspace{-1cm} v^\\varepsilon_f =\t \\omega - \\nabla_x \\Phi_f^\\varepsilon(x,t),\t\n\\label{eqmeanfield6}\n\\end{eqnarray}\nwhere the repulsive potential and the average orientation are now given by\n\\begin{eqnarray*}\n& & \\hspace{-1cm} \\Phi_f^\\varepsilon(x,t) = \\int_{{\\mathbb S}^{n-1} \\times {\\mathbb R}^n} \\phi\\Big(\\dfrac{|x-y|}{\\varepsilon r } \\Big) \\, f^\\varepsilon(y, w, t) \\, dw \\, dy, \n\\\\\n& & \\hspace{-1cm} \t\\bar{\\omega}^\\varepsilon_f = \\dfrac{\\mathcal J^\\varepsilon_{f}(x,t)}{|\\mathcal J^\\varepsilon_f(x,t)|} , \\quad \\mathcal J^\\varepsilon_f (x,t) = \t \\int_{{\\mathbb S}^{n-1} \\times {\\mathbb R}^n} K\\Big(\\dfrac{|x-y|}{\\sqrt{\\varepsilon}R } \\Big) \\, f^\\varepsilon (y, w, t) \\, w \\, dw \\, dy \n\\end{eqnarray*}\t\n\n\nNow, by Taylor expansion and the fact that the kernels $K$, $\\phi$ only depend on $|x|$, we obtain (provided that $K$ is normalized to $1$ i.e. $\\int_{\\mathbb R} K(|x|) \\, dx = 1$) :\n\\begin{eqnarray}\n& & \\hspace{-1cm} v_f^\\varepsilon(x,t) =\\omega - \\Phi_0 \\nabla_x \\rho_f^\\varepsilon + \\mathcal O(\\varepsilon^2), \\label{expansion1}\\\\\n& & \\hspace{-1cm} \\bar{\\omega}^\\varepsilon_f (x,t) = G_f^0(x,t) + \\varepsilon G^1_f(x,t) + \\mathcal O(\\varepsilon^2), \\label{expansion2}\\\\\n& & \\hspace{-1cm} G_f^0(x,t) = \\Omega_f(x,t) , \\quad G^1_f(x,t) = \\dfrac{k_0}{ | J_f|} P_{\\Omega_f^\\perp} \\Delta_x J_f, \\nonumber\n\\end{eqnarray}\nwhere the coefficients $k_0, \\Phi_0$ are given by\n\\begin{equation}\n\\label{k0}\nk_0 = \\dfrac{R^2}{2n} \\int_{x\\in {\\mathbb R}^n} K (|x|)|x|^2 dx , \\quad \\Phi_0 = \\int_{x\\in {\\mathbb R}^n} \\phi(x) dx .\n\\end{equation}\nFor example, if $K$ is the indicator function of the ball of radius $1,$ then $ k_0 = \\vert {\\mathbb S}^{n-1} \\vert \/ 2n(n+2) ,$ where $\\vert {\\mathbb S}^{n-1} \\vert $ is the volume of the sphere ${\\mathbb S}^{n-1}$. In the cases $d = 2$ and $d = 3$, we respectively get $ k _0= \\pi\/8 $ and $ k_0 =2\\pi\/15 .$ The local density $\\rho_f,$ the local current density $J_f$ and local average orientation $\\Omega_f$ are defined by\n\\begin{eqnarray}\n& & \\hspace{-1cm} \\rho_f(x,t) = \\int_{{\\mathbb S}^{n-1} } f(x, w, t) \\, dw, \\quad\n\\label{f1}\\\\\t\n& & \\hspace{-1cm} \nJ_f(x,t) = \\int_{\\omega \\in {\\mathbb S}^{n-1}} f(x, w,t) \\, w \\, d w, \\quad\n\\Omega_f(x,t) = \\dfrac{J_f(x,t)}{|J_f(x,t)|}. \n\\label{f2} \n\\end{eqnarray}\nMore details about this Taylor expansion are given in Appendix \\ref{sec:taylorexpasion} .\tLet us observe that this scaling, first proposed in \\cite{Degond_etal_MAA13} is different from the one used in \\cite{Degond_Motsch_M3AS08} and results in the appearance of the viscosity term at the right-hand side of Eq. (\\ref{macro2_intro}). \n\nFinally, if we neglect the terms of order $\\varepsilon^2$ and we define the so-called collision operator $Q(f)$ by\n$$\n Q(f) = - \\nabla_\\omega \\cdot (P_{\\omega^\\perp} \\Omega_f f) + d \\Delta_\\omega f,\n$$\nthe rescaled system (\\ref{eqmeanfield5}), (\\ref{eqmeanfield6}) can be rewritten as follows\n\\begin{eqnarray}\n&&\\hspace{-1cm} \\varepsilon \\Big ( \\partial_t f^\\varepsilon + \\nabla_x \\cdot (v^\\varepsilon_f f^\\varepsilon) + \\alpha \\, \\nabla_\\omega \\cdot (P_{\\omega^\\perp} v^\\varepsilon_f f^\\varepsilon) + \\nabla_\\omega \\cdot (P_{\\omega^\\perp} G^1_{f^\\varepsilon} f^\\varepsilon) \\Big ) = Q(f^\\varepsilon), \n\\label{lastscaling1}\\\\\n&&\\hspace{-1cm} v_{f^\\varepsilon}(x, \\omega,t) = \\omega - \\Phi_0 \\nabla_x \\rho_{f^\\varepsilon} , \\quad G^1_{f^\\varepsilon}(x,t) = \\dfrac{k_0}{ | J_{f^\\varepsilon}|} P_{\\Omega_f^\\perp} \\Delta_x J_{f^\\varepsilon} \n\\label{lastscaling2}\n\t\\end{eqnarray}\t\n\n\n\\subsection{Hydrodynamic limit}\n\nThe aim is now to derive a hydrodynamic model by taking the limit $\\varepsilon \\rightarrow 0$ of system (\\ref{lastscaling1}), (\\ref{lastscaling2}) where the local density $\\rho_f,$ the local current $J_f$ and the local average orientation $\\Omega_f$ are defined by (\\ref{f1}), (\\ref{f2}). \n\nWe first introduce the von Mises-Fisher (VMF) probability distribution $M_{\\Omega}(\\omega)$ of orientation $\\Omega \\in {\\mathbb S}^{n-1}$ defined for $\\omega \\in {\\mathbb S}^{n-1}$ by: \n$$\nM_\\Omega(\\omega)=Z^{-1} \\, \\exp\\left(\\dfrac{\\omega \\cdot \\Omega}{d}\\right), \\qquad Z = \\int_{\\omega \\in {\\mathbb S}^{n-1}} \\exp\\left(\\dfrac{\\omega \\cdot \\Omega}{d}\\right) \\, d\\omega\n$$\nAn important parameter will be the flux of the VMF distribution, i.e. $\\int_{\\omega \\in {\\mathbb S}^{n-1}} M_\\Omega(\\omega) \\omega d\\omega$. By obvious symmetry consideration, we have \n$$\n\\int_{\\omega \\in {\\mathbb S}^{n-1}} M_\\Omega(\\omega) \\, \\omega \\, d\\omega = c_1 \\Omega, \n$$\nwhere the quantity $c_1 = c_1(d)$ does not depend on $\\Omega$, is such that $0 \\leq c_1(d) \\leq 1$ and is given by \n\\begin{equation}\n\\label{c1}\nc_1(d) = \\int_{\\omega \\in {\\mathbb S}^{n-1}} M_\\Omega(\\omega) \\, (\\omega \\cdot \\Omega) \\, d\\omega.\n\\end{equation}\nWhen $d$ is small, $M_\\Omega$ is close to a Dirac delta $\\delta_\\Omega$ and represents a distribution of perfectly aligned particles in the direction of $\\Omega$. When $d$ is large, $M_\\Omega$ is close to a uniform distribution on the sphere and represents a distribution of almost totally disordered orientations. \nThe function $d \\in {\\mathbb R}_+ \\mapsto c_1(d) \\in [0,1]$ is strictly decreasing with $\\lim_{d \\to 0} c_1(d) = 1$, $\\lim_{d \\to \\infty} c_1(d) = 0$. Therefore, $c_1(d)$ represents an order parameter, which corresponds to perfect disorder when it is close to $0$ and perfect alignment order when it is close to $1$. \n\n\\medskip\n\\noindent\nWe have following theorem:\n\n\\begin{theorem}\n\\label{thm:sm}\nLet $f^\\varepsilon$ be the solution of (\\ref{lastscaling1}), (\\ref{lastscaling2}). Assume that there exists $f$ such that \n\\begin{equation}\nf^\\varepsilon \\rightarrow f \\quad \\mbox{ as } \\quad \\varepsilon \\rightarrow 0, \n\\label{eq:converg}\n\\end{equation} \npointwise as well as all its derivatives. Then, there exist $ \\rho(x,t)$ and $\\Omega(x,t)$ such that\n\t\\begin{equation}\\label{f0}\n\t\tf(x,\\omega,t) = \\rho(x,t) M_{\\Omega(x,t)}(\\omega),\n\t\\end{equation}\t\t\t\nMoreover, the functions $\\rho(x,t), \\Omega(x,t)$ satisfy the following equations\\begin{eqnarray}\n&&\\hspace{-1cm} \\partial_t\\rho + \\nabla_x \\cdot(\\rho U ) = 0, \n\\label{macro1} \\\\\n&&\\hspace{-1cm} \\rho \\big(\\partial_t \\Omega + (V \\cdot \\nabla_x)\\Omega \\big) + P_{\\Omega^\\perp}\\nabla_x p(\\rho) = \\gamma P_{\\Omega^\\perp} \\Delta_x(\\rho \\Omega), \n\\label{macro2}\t\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&&\\hspace{-1cm} U = c_1 \\Omega - \\Phi_0 \\nabla_x \\rho , \\quad V = c_2 \\Omega - \\Phi_0 \\nabla_x \\rho ,\n\\label{macro21}\\\\\n&&\\hspace{-1cm} p(\\rho ) = d \\rho + \\alpha \\Phi_0 \\big( (n-1)d + c_2 \\big) \\dfrac{\\rho^2}{2}, \\quad \\gamma = k_0 \\big( (n-1)d + c_2 \\big).\n\\label{macro22}\n\\end{eqnarray}\nand the coefficients $c_1, c_2$ will be defined in formulas (\\ref{c1}), (\\ref{c2}) below.\t\t\n\\end{theorem}\n\n\n\\noindent\nGoing back to unscaled variables, we find the model (\\ref{macro1_intro})-(\\ref{macro5_intro}) presented in the Introduction. \n\n\n\\noindent\n{\\bf{Proof:} }\n The proof of this theorem is divided into three steps: (i) determination of the equilibrium states ; (ii) determination of the Generalized Collision Invariants ; (iii) hydrodynamic limit. We give a sketch of the proof for each step.\n\n\\medskip\n\\noindent\n{\\bf Step (i): determination of the equilibrium states} We define the equilibria as the elements of the null space of $Q$, considered as an operator acting on functions of $\\omega$ only. \n\n\\begin{definition} \\label{def:small_equi}\nThe set ${\\mathcal E}$ of equilibria of $Q$ is defined by\n$$\n{\\mathcal E} = \\big\\{ f \\in H^1({\\mathbb S}^{n-1}) \\, \\, | \\, \\, f \\geq 0 \\mbox{ and } Q(f) = 0 \\big\\} .\n$$\n\\end{definition}\n\\noindent\n\nWe have the following:\n\n\\begin{lemma}\nThe set ${\\mathcal E}$ is given by\n$$\n\\mathcal E = \\Big\\{ \\rho M_\\Omega(\\omega) \\, \\, | \\, \\, \\rho \\in {\\mathbb R}_+, \\, \\, \\Omega \\in {\\mathbb S}^{n-1} \\Big\\}\n$$\n\\label{lem:smallkin_equi}\n\\end{lemma}\n\n\\noindent\n\\vspace{-0.2cm}\nFor a proof of this lemma, see \\cite{Degond_Motsch_M3AS08}. The proof relies on writing the collision operator as\n$$\n Q(f) = \\nabla_\\omega \\cdot \\Big( M_{\\Omega_f} \\nabla_\\omega \\big( \\frac{f}{M_{\\Omega_f}} \\big) \\Big). \n$$\n\n\n\\medskip\n\\noindent\n{\\bf Step (ii): Generalized Collision Invariants (GCI).} We begin with the definition of a collision invariant. \n\n\\begin{definition}\nA collision invariant (CI) is a function $\\psi(\\omega)$ such that for all functions $f(\\omega)$ with sufficient regularity we have\n$$\n\\int_{\\omega \\in {\\mathbb S}^{n-1}} Q (f) \\, \\psi \\, d\\omega = 0.\n$$\nWe denote by ${\\mathcal C}$ the set of CI. The set ${\\mathcal C}$ is a vector space.\n\\label{def:CI}\n\\end{definition}\n\n\n\\noindent\nAs seen in \\cite{Degond_Motsch_M3AS08}, the space of CI is one dimensional and spanned by the constants. Physically, this corresponds to conservation of mass during particle interactions. Since energy and momentum are not conserved, we cannot hope for more physical conservations. Thus the set of CI is not large enough to allow us to derive the evolution of the macroscopic quantities $\\rho$ and $\\Omega.$ To overcome this difficulty, a weaker concept of collision invariant, the so-called ``Generalized collisional invariant'' (GCI) has been introduced in \\cite{Degond_Motsch_M3AS08}. To introduce this concept, we first define the operator ${\\mathcal Q}(\\Omega, f)$, which, for a given $\\Omega \\in {\\mathbb S}^{n-1}$, is given by\n$$\n {\\mathcal Q}(\\Omega, f) = \\nabla_\\omega \\cdot \\Big( M_\\Omega \\nabla_\\omega \\big( \\frac{f}{M_\\Omega} \\big) \\Big). \n$$\nWe note that \n\\begin{equation}\nQ(f) = {\\mathcal Q}(\\Omega_f, f), \n\\label{eq:calQ_def}\n\\end{equation}\nand that for a given $\\Omega \\in {\\mathbb S}^{n-1}$, the operator $f \\mapsto \n {\\mathcal Q}(\\Omega, f)$ is a linear operator. Then we have the\n\n\n\\begin{definition}\nLet $\\Omega \\in {\\mathbb S}^{n-1}$ be given. A Generalized Collision Invariant (GCI) associated to $\\Omega$ is a function $\\psi \\in H^1({\\mathbb S}^{n-1})$ which satisfies:\n\\begin{equation}\n\\int_{\\omega \\in {\\mathbb S}^{n-1}} {\\mathcal Q} (\\Omega, f) \\, \\psi(\\omega) \\, d\\omega = 0, \\quad \\forall f \\in H^1({\\mathbb S}^{n-1}) \\quad \\mbox{ such that } \\quad P_{\\Omega^\\bot} \\Omega_f = 0. \n\\label{eq:sm_def_GCI}\n\\end{equation}\nWe denote by ${\\mathcal G}_\\Omega$ the set of GCI associated to $\\Omega$.\n\\label{def:GCI}\n\\end{definition}\n\n\\noindent\nThe following Lemma characterizes the set of generalized collision invariants.\n\n\\begin{lemma}\nThere exists a positive function $h$: $[-1,1] \\to {\\mathbb R}$ such that \n\\begin{equation*}\n{\\mathcal G}_\\Omega = \\{ C + h(\\omega \\cdot \\Omega) \\beta \\cdot \\omega \\, \\, \\mbox{with arbitrary} \\, \\, C\\in {\\mathbb R} \\mbox{ and } \\beta \\in {\\mathbb R}^n \\mbox{ such that } \\beta \\cdot \\Omega = 0 \\}.\n \\end{equation*}\nThe function $h$ is such that $h(\\cos \\theta) = \\frac{g(\\theta)}{ \\sin \\theta}$ and $ g(\\theta)$ is the unique solution in the space $V$ defined by\n\\begin{eqnarray*}\n&&\\hspace{-1cm}\nV = \\{ g \\, \\, \\vert \\, \\, (n-2)(\\sin \\theta)^{\\frac{n}{2} - 2} g \\in L^2(0, \\pi), \\quad (\\sin \\theta)^{\\frac{n}{2} - 1} g \\in H^1_0(0, \\pi) \\},\n \\end{eqnarray*}\n(denoting by $H^1_0(0, \\pi) $ the Sobolev space of functions which are square integrable as well as their derivative and vanish at the boundary) of the problem\n\\begin{eqnarray*}\n&&\\hspace{-1cm}\n-\\sin^{2-n} \\theta \\, e^{ -\\frac{\\cos\\theta}{d}} \\, \\frac{d}{d \\theta} \\big( \\sin^{n-2} \\theta \\, e^{ \\frac{\\cos\\theta}{d}} \\, \\,\\frac{dg}{d \\theta}(\\theta) \\big) + \\dfrac{n-2}{\\sin^2 \\theta } \\, g(\\theta)= \\sin \\theta.\n \\end{eqnarray*}\nThe set ${\\mathcal G}_\\Omega$ is a $n$-dimensional vector space.\n\\end{lemma}\n\\noindent\nFor a proof we refer to \\cite{Degond_Motsch_M3AS08} for $n=3$ and to \\cite{Frouvelle_M3AS12} for general $n \\geq 2$. We denote by $\\psi_\\Omega$ the vector GCI \n\\begin{equation}\n\\psi_\\Omega = h(\\omega \\cdot \\Omega) \\, P_{\\Omega^\\bot} \\omega, \n\\label{eq:psi}\n\\end{equation}\n\n\\noindent\nWe note that, thanks to (\\ref{eq:calQ_def}) and (\\ref{eq:sm_def_GCI}), we have\n\\begin{equation}\n\\int_{\\omega \\in {\\mathbb S}^{n-1}} Q(f) \\, \\psi_{\\Omega_f}(\\omega) \\, d\\omega = 0, \\quad \\forall f \\in H^1({\\mathbb S}^{n-1}). \n\\label{eq:intGCI}\n\\end{equation}\n\n\\medskip\n\\noindent\n{\\bf Step (iii): Hydrodynamic limit $\\varepsilon \\rightarrow 0.$} In the limit $\\varepsilon \\rightarrow 0$, we assume that (\\ref{eq:converg}) holds. Then , thanks to (\\ref{lastscaling1}), we have $Q(f) = 0 $. In view of Lemma \\ref{lem:smallkin_equi}, this implies that $f$ has the form (\\ref{f0}). We now need to determine the equations satisfied by $\\rho$ and $\\Omega$. \n\nFor this purpose, we divide Eq. (\\ref{lastscaling1}) by $\\varepsilon$ and integrate it with respect to $\\omega$. Writing (\\ref{lastscaling1}) as \n\\begin{equation}\n({\\mathcal T}_1+{\\mathcal T}_2 + {\\mathcal T}_3)f^{\\varepsilon}=\\dfrac{1}{\\varepsilon} \\, Q(f^\\varepsilon), \n\\label{eq:kin_reduced}\n\\end{equation}\nwhere \n\\begin{equation}\n{\\mathcal T}_1 f =\\partial_t f + \\nabla_x \\cdot (v_f f), \\quad {\\mathcal T}_2 f = \\alpha \\, \\nabla_\\omega \\cdot (P_{\\omega^\\perp} \\, v_f \\, f), \\quad {\\mathcal T}_3 f = \\nabla_\\omega \\cdot (P_{\\omega^\\perp} \\, G^1_{f} \\, f), \n\\label{eq:T123}\n\\end{equation}\nwe observe that the integral of ${\\mathcal T}_2 f^\\varepsilon$ and ${\\mathcal T}_3 f^\\varepsilon$over $\\omega$ is zero since it is in divergence form and the integral of the right-hand side of (\\ref{eq:kin_reduced}) is zero since $1$ is a CI. The integral of ${\\mathcal T}_1 f^\\varepsilon$ gives: \n$$ \\partial_t \\rho_{f^\\varepsilon} + \\nabla_x \\cdot \\Big( \\int_{{\\mathbb S}^{n-1}} f^\\varepsilon(x,\\omega,t) \\, v_{f^\\varepsilon} (x,\\omega,t) \\, d\\omega \\Big) = 0. $$\nWe take the limit $\\varepsilon \\rightarrow 0$ and use (\\ref{eq:converg}) to get Eq. (\\ref{macro1}) with \n$$ U = \\int_{{\\mathbb S}^{n-1}} \\rho(x,t) \\, M_{\\Omega(x,t)} (\\omega) \\, v_{\\rho M_\\Omega} (x,\\omega,t) \\, d\\omega. $$\nUsing (\\ref{lastscaling2}), we get $v_{\\rho M_\\Omega} (x,\\omega,t) = \\omega - \\Phi_0 \\nabla_x \\rho(x,t)$. With (\\ref{c1}), this leads to the first equation (\\ref{macro21}). \n\n\nMultiplying (\\ref{eq:kin_reduced}) by $\\psi_{\\Omega_{f^\\varepsilon}}$, integrating with respect to $\\omega$ and using (\\ref{eq:intGCI}), we get\n$$ \\int_{{\\mathbb S}^{n-1}} ({\\mathcal T}_1+{\\mathcal T}_2 + {\\mathcal T}_3)f^{\\varepsilon}(x,\\omega,t) \\, \\psi_{\\Omega_{f^\\varepsilon}} (x,\\omega,t) \\, d \\omega = 0. \n$$\n and taking the limit $\\varepsilon \\rightarrow 0,$ we get\n\\begin{equation}\n\\label{vitesse}\n\\int_{{\\mathbb S}^{n-1}} ( ({\\mathcal T}_1+{\\mathcal T}_2 + {\\mathcal T}_3) (\\rho M_{\\Omega}))(x,\\omega,t) \\, \\psi_{\\Omega(x,t)} (\\omega) \\, d \\omega = 0. \n\\end{equation}\nThis equation describes the evolution of the mean direction $\\Omega.$ The computations which lead to (\\ref{macro2}) are proved in the Appendix \\ref{subsec:small_equi}.\nThe coefficient $c_2$ in (\\ref{macro2}) is defined by\n\\begin{eqnarray}\n\\label{c2}\n& & c_2(d) = \\dfrac{\\langle \\sin^2 \\theta \\cos \\theta \\, h \\rangle_{M_\\Omega} }{ \\langle \\sin^2 \\theta \\, h \\rangle_{M_\\Omega}}= \\dfrac{\\int_{0}^{\\pi} \\sin^n \\theta \\cos \\theta \\, M_\\Omega \\,h \\,d\\theta}{\\int_{0}^{\\pi} \\sin^n \\theta \\,M_\\Omega \\,h \\,d\\theta},\n\\end{eqnarray}\nwhere for any function $g(\\cos \\theta),$ we denote $\\langle g \\rangle$ by\n$$\n\\langle g \\rangle_{M_\\Omega} = \\int_{\\omega \\in {\\mathbb S}^{n-1}} M_\\Omega(\\omega)\\,g(\\omega \\cdot \\Omega) \\,d\\omega = \\dfrac{\\int_{0}^{\\pi} g(\\cos \\theta) \\,e^{\\frac{\\cos \\theta}{d}}\\, \\sin^{n-2} \\theta \\,d\\theta}{\\int_{0}^{\\pi} e^{\\frac{\\cos \\theta}{d}} \\,\\sin^{n-2} \\,\\theta \\,d\\theta}.\n$$\n\n\\medskip\n\\begin{remark}\nThe SOHR model (\\ref{macro1}), (\\ref{macro2}) can be rewritten as follows\n\\begin{eqnarray*}\n&&\\hspace{-1cm} \\partial_t \\rho + c_1 \\nabla_x \\cdot(\\rho \\Omega) = \\Phi_0 \\Delta_x \\left(\\dfrac{\\rho^2}{2}\\right), \n\\\\\n&&\\hspace{-1cm} \\partial_t \\Omega + (\\bar V \\cdot \\nabla_x)\\Omega + P_{\\Omega^\\perp}\\nabla_x h(\\rho) = \\gamma P_{\\Omega^\\perp} \\Delta_x \\Omega, \n\\end{eqnarray*}\nwhere the vectors $ \\bar V$ and the function $h(\\rho)$ are defined by\n\\begin{eqnarray*}\n\\bar V = c_2 \\Omega - (\\Phi_0 + 2 \\gamma) \\nabla_x \\rho, \\quad h'(\\rho) = \\frac{1}{\\rho} \\, p'(\\rho), \n\\end{eqnarray*}\nand where the primes denote derivatives with respect to $\\rho$. This writing displays this system in the form of coupled nonlinear advection-diffusion equations. \n \\end{remark}\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Numerical discretization of the SOHR model}\n\\label{sec:num_solution}\nIn this section, we develop the numerical approximation of the system (\\ref{macro1})-(\\ref{macro22}) in the two dimensional case. As mentioned above, this system is not conservative because of the geometric constraint $|\\Omega| = 1.$ Weak solutions of non-conservative systems are not unique because jump relations across discontinuities are not uniquely defined. This indeterminacy cannot be waived by means of an entropy inequality, by contrast to the case of conservative systems. In \\cite{Motsch_Navoret_MMS11} the authors address this problem for the SOH model. They show that the model is a zero-relaxation limit of a conservative system where the velocity $\\Omega$ is non-constrained (i.e. belongs to ${\\mathbb R}^n$). Additionally, they show that the numerical solutions build from the relaxation system are consistent with those of the underlying particle model, while other numerical solutions built directly from the SOH model are not. Here we extend this idea to the SOHR model. More precisely, we introduce the following relaxation model (in dimension $n=2$):\n\\begin{eqnarray}\n&&\\hspace{-1cm} \\partial_t \\rho^\\eta + \\nabla_x \\cdot (\\rho^\\eta U^\\eta) = 0, \\label{relax1}\\\\\n&&\\hspace{-1cm} \\partial_t(\\rho^\\eta \\Omega^\\eta) + \\nabla_x \\cdot ( \\rho^\\eta V^\\eta \\otimes \\Omega^\\eta) + \\nabla_x p( \\rho^\\eta ) - \\gamma \\Delta_x(\\rho^\\eta \\Omega^\\eta) = \\dfrac{\\rho^\\eta}{\\eta} (1 - |\\Omega^\\eta|^2) \\Omega^\\eta, \\label{relax2} \\\\\n&&\\hspace{-1cm} U^\\eta = c_1 \\Omega^\\eta - \\Phi_0 \\nabla_x \\rho^\\eta , \\quad V^\\eta = c_2 \\Omega^\\eta - \\Phi_0 \\nabla_x \\rho^\\eta,\\label{relax22}\\\\\n&&\\hspace{-1cm} p(\\rho^\\eta) = d \\rho^\\eta + \\alpha \\Phi_0 \\big( d + c_2 \\big) \\dfrac{(\\rho^\\eta)^2}{2}, \\quad \\gamma = k_0 \\big( d + c_2 \\big).\n\\label{relax23}\n\\end{eqnarray}\nThe left-hand sides form a conservative system. We get the following proposition:\n\n\\begin{proposition}\n\\label{prop1}\nThe relaxation model (\\ref{relax1})-(\\ref{relax23}) converges to the SOHR model (\\ref{macro1})-(\\ref{macro22}) as $\\eta $ goes to zero.\n\\end{proposition}\n\n\\noindent\nThe proof of proposition \\ref{prop1} is given in Appendix \\ref{numerical:convergence}. This allows us to use well-established numerical techniques for solving the conservative system (i.e. the left-hand side of (\\ref{relax1}), (\\ref{relax2})). The scheme we propose relies on a time splitting of step $\\Delta t$ between the conservative part\n\\begin{eqnarray}\n&&\\hspace{-1cm} \\partial_t \\rho^{\\eta} + \\nabla_x \\cdot (\\rho^{\\eta} U^{\\eta}) = 0, \n\\label{split1}\\\\\n&&\\hspace{-1cm} \\partial_t(\\rho^{\\eta} \\Omega^{\\eta}) + \\nabla_x \\cdot ( \\rho^{\\eta} V^{\\eta} \\otimes \\Omega) + \\nabla_x p(\\rho^{\\eta}) - \\gamma \\Delta_x (\\rho^{\\eta} \\Omega^{\\eta}) = 0, \n\\label{split2}\n\\end{eqnarray}\nand the relaxation part\n\\begin{eqnarray}\n&&\\hspace{-1cm}\\partial_t \\rho^\\eta = 0, \n\\label{relax3} \\\\\n&&\\hspace{-1cm}\\partial_t(\\rho^\\eta \\Omega^\\eta) = \\dfrac{\\rho^\\eta}{\\eta} (1 - |\\Omega^\\eta|^2) \\Omega^\\eta. \n\\label{relax4}\n\\end{eqnarray}\n\n\\medskip\n\\noindent\nSystem (\\ref{split1}-\\ref{split2}) can be rewritten in the following form (we omit the superscript $\\eta$ for simplicity)\n$$\n\tQ_t + (F(Q, Q_x))_x + (G(Q, Q_y))_y = 0,\n$$\nwhere\n$$\n\tQ = \\begin{pmatrix}\n\t\t\t\t\\rho \\\\ \\rho \\Omega_1 \\\\ \\rho \\Omega_2\n\t\t\\end{pmatrix}, \\quad\n\tF(Q, Q_x) = \\begin{pmatrix}\n\t\t\t\t \\rho U_1 \\\\\n\t\t\t\t\\rho \\Omega_1 V_1 + p(\\rho) - \\gamma \\partial_x(\\rho \\Omega_1), \\\\\n\t\t\t\t\\rho \\Omega_1 V_2 - \\gamma \\partial_x(\\rho \\Omega_2)\n\t\t\\end{pmatrix}, \n$$\n$$\n\tG(Q, Q_y) = \\begin{pmatrix}\n\t\t\t\t \\rho U_2 \\\\\n\t\t\t\t \\rho \\Omega_2 V_1 - \\gamma \\partial_y(\\rho \\Omega_1) \\\\\n\t\t\t\t\\rho \\Omega_2 V_2 + p(\\rho) - \\gamma \\partial_y(\\rho \\Omega_2)\n\t\t\t\t\\end{pmatrix}.\n$$\n\n\n\\medskip\n\\noindent\nWe consider now the following numerical scheme where we denoted $Q^*_{i,j}$ the approximation of $Q$ at time $t^{n+1} = (n+1) \\Delta t $ and position $x_i = i \\Delta x, y_j = j \\Delta y$:\n\\begin{eqnarray*}\n Q^{*}_{i,j} & = & Q^{n}_{i, j} - \\dfrac{\\Delta t}{\\Delta x} \\{F^n_{i +1\/2,j} - F^n_{i-1\/2,j}\\} \n- \\dfrac{\\Delta t}{\\Delta y} \\{ G^n_{i,j+1\/2} - G^n_{i,j-1\/2} \\}, \n\\end{eqnarray*}\nwhere the numerical flux $F^n_{i +1\/2,j}$ is given by\n\\begin{equation*}\n\t F^n_{i +1\/2,j} = \\dfrac{F^n(Q^n_{i,j}, Q^n_{xi, j}) + F^n(Q^n_{i+1,j}, Q^n_{x(i+1),j})}{2} - P^{i+\\frac{1}{2}}_2 \\Big( \\dfrac{\\partial F}{\\partial Q}(\\bar Q^n_{i,j}, \\bar{Q}^n_{xi,j} ) \\Big) (Q^n_{i+1,j} - Q^n_{i,j}),\n\\end{equation*}\nwith\n\\begin{equation*}\nQ^n_{xi,j} = \\frac{(Q^n_{i+1,j} - Q^n_{i,j} )}{\\Delta x} , \\quad \\bar{Q}^n_{i,j} = \\dfrac{Q^n_{i,j} + Q^n_{i+1, j}}{2},\t \\quad \\bar{Q}^n_{xi,j} = \\dfrac{Q^n_{xi,j} + Q^n_{x(i+1), j}}{2},\n\\end{equation*}\nand the analogous discretization holds for $G^n_{i ,j+ \\frac{1}{2}}$.\n\n\\noindent\nIn the above formula, $P^{i+\\frac{1}{2}}_2$ is a polynomial of matrices of degree $2$ calculated with the eigenvalues of the Jacobian matrices $\\dfrac{\\partial F}{\\partial Q}$ at an intermediate state depending on $(Q^n_{i,j}, Q^n_{xi, j})$ and $(Q^n_{i+1,j}, Q^n_{x(i+1), j})$ as detailed in \\cite{Degond_Peyrard_99}.\nTo ensure stability of the scheme, the time step $\\Delta t$ satisfies a Courant-Friedrichs-Lewy (CFL) condition computed as the minimum of the CFL conditions required for the hyperbolic and diffusive parts of the system. \n\n\nOnce the approximate solution of the conservative system is computed, equations (\\ref{relax3}) and (\\ref{relax4}) can be solved explicitly. In the limit $\\eta\\rightarrow 0$ they give\n$$\n \\rho^{n+1}=\\rho^{*}, \\qquad \\Omega^{n+1}=\\frac{\\Omega^{*}}{|\\Omega^{*}|}\n$$\nwhere $(\\rho^*,\\Omega^{*})$ is the numerical solution of system (\\ref{split1}-\\ref{split2}). This ends one step of the numerical scheme for the system (\\ref{relax1}-\\ref{relax2}).\n\n\n\\section{Numerical tests}\nThe goal of this section is to present some numerical solutions of the system (\\ref{macro1})-(\\ref{macro22}) which validate the numerical scheme proposed in the previous section. We will first perform a convergence test. We then successively compare the solutions obtained with the SOHR model with those computed by numerically solving the individual based model (\\ref{overdamped1}) in regimes in which the two models should provide similar results. We will finally perform some comparisons between the SOH and the SOHR system to highlight the difference between the two models. We will compare the SOHR model with another way to incorporate repulsion in the SOH Model, the so-called DLMP model of \\cite{Degond_etal_MAA13}.\n\nFor all the tests, we use the model in uscaled variables as described in the Introduction (see (\\ref{macro1_intro})-(\\ref{macro5_intro}). The potential kernel $\\phi$ is chosen as\n\\begin{equation}\n\\label{phi}\n\\phi(x) = \\begin{cases}\n (|x| -1)^2 \\quad \\mbox{if} \\quad |x| \\leq1,\\\\\n0 \\quad \\mbox{if} \\quad |x| > 1,\n\\end{cases}\n\\end{equation}\nwhich gives $\\Phi_0 = \\dfrac{\\pi}{6}$, while for $K$, by assumption normalized to $1,$ we choose the following form\n$$\nK(\\vert z \\vert ) = \\begin{cases}\n \\dfrac{1}{\\pi} \\quad \\mbox{if} \\quad |z| \\leq 1,\\\\\n0 \\quad \\mbox{if} \\quad |z| > 1.\n\\end{cases}\n$$\nThis leads to $k_0 = \\dfrac{1}{8}.$ The other parameters, which are fixed for all simulations if not differently stated, are :\n$$ v_0 = 1, \\,\\, \\mu = \\dfrac{1}{2}, \\,\\, \\alpha = 1, \\,\\, d = 0.1, \\,\\, L_x = 10, \\,\\, L_y = 10, $$\nwhich, in dimension $n=2$, lead to (after numerically computing the GCI and the associated integrals):\n$$ c_1 = 0.9486, \\,\\, c_2 =0.8486. $$\nIn the visualization of the results, we will use the angle $\\theta$ of the vector $\\Omega$ relative to the $x$-axis, i.e. $\\Omega = (\\cos \\theta, \\sin \\theta)$. \n\n\\subsection{Convergence test}\nThe first test is targeted at the validation of the proposed numerical scheme. For this purpose, we investigate the convergence when the space step $(\\Delta x, \\Delta y )$ tends to $(0,0)$, refining the grid and checking how the error behaves asymptotically. The initial mesh size is $\\Delta x = \\Delta y = 0.25$ while the time step is $\\Delta t = 0.001.$ We repeat the computation for $(\\dfrac{\\Delta x}{2}, \\dfrac{\\Delta y}{2}),$ $(\\dfrac{\\Delta x}{4}, \\dfrac{\\Delta y}{4}),$ $(\\dfrac{\\Delta x}{8}, \\dfrac{\\Delta y}{8})$. The convergence rate is estimated through the measure of the $L^1$ norm of the error for the vectors $(\\rho, \\cos\\theta)$ by using for each grid the next finer grid as reference solution. The initial data is\n\\begin{equation}\n\\label{Vortex}\n\\rho_0 = 1, \\quad \\theta_0(x,y) = \\begin{cases}\n\t\t\t\t\t\t\t\t\t\t\\arctan(\\dfrac{y_1}{x_1} ) + \\dfrac{\\pi}{2} sign(x_1) \\,\\, \\mbox{if} \\,\\, x_1 \\neq 0, \\\\\n\t\t\t\t\t\t\t\t\t\t\\pi \\,\\,\\mbox{if} \\,\\, x_1 = 0 \\,\\, \\mbox{and} \\,\\, y_1 > 0, \\\\\n 0 \\,\\, \\mbox{if} \\,\\, x_1 = 0 \\,\\, \\mbox{and} \\,\\, y_1 < 0,\n \\end{cases}\n\\end{equation}\nwhere\n$$ x_1 = x - \\dfrac{L_x}{2}, \\quad y_1 = y - \\dfrac{L_y}{2}.$$\nThe boundary conditions are fixed in time on the four sides of the square : $( \\rho^n, \\theta^n ) = (\\rho_0, \\theta_0 )$.\nThe error curves for the density and for $ \\cos \\theta $ are plotted in figure \\ref{fig:scheme} as a function of the space step in log-log scale at time $T = 1s.$ The slope of the error curves are compared to a straight line of slope 1.\nFrom the figure, we observe the convergence of the scheme with accuracy close to $1.$\n\n\\noindent\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[height=8cm,width=11cm]{errorMacro3.png}\n\\caption{$L^1$-error for the density $\\rho$ and the flux direction $ \\cos\\theta $ as a function of $\\Delta x$ in log-log scale. A straight line of slope $1$ is plotted for reference. This figure shows that the scheme is numerically of order $1$. \n}\n\\label{fig:scheme}\n\\end{figure}\n\n\n\\subsection{Comparison between the SOHR and the Vicsek model with repulsion}\n\nIn this subsection, we validate the SOHR model by comparing it to the Vicsek model with repulsion on two different test cases. We investigate the convergence of the microscopic IBM to the macroscopic SOHR model when the scaling parameter $\\varepsilon$ tend to zero. The scaled IBM is written: \n\\begin{eqnarray*} \n&&\\hspace{-1cm} \\dfrac{dX_k}{dt} = v_k, \\qquad \nv_k\t= \\omega_k - \\, \\nabla_x \\Phi(X_k(t),t),\\\\\n&&\\hspace{-1cm} d\\omega_k\t= P_{\\omega_k^\\perp} \\circ \\Big( \\frac{1}{\\varepsilon} \\, \\bar{\\omega}(X_k(t),t) \\, dt + \\alpha \\, v_k \\, dt + \\sqrt{\\frac{2d}{\\varepsilon}} \\, dB^k_t \\Big).\\\\\n&&\\hspace{-1cm} \\Phi(x,t) = \\dfrac{1}{\\varepsilon^2 \\, N} \\sum_{i=1}^{N} \\nabla \\phi \\Big( \\frac{|x - X_i|}{\\varepsilon \\, r} \\Big), \\\\\n&&\\hspace{-1cm} \\bar{\\omega}(x,t) = \\dfrac{{\\mathcal J}(x,t)}{\\vert {\\mathcal J}(x,t) \\vert}, \\qquad {\\mathcal J} (x,t) = \\frac{1}{N} \\sum\\limits_{i=1}^{N} K\\Big(\\frac{|x-X_i|}{\\sqrt{\\varepsilon} \\, R}\\Big) \\omega_i. \n\\end{eqnarray*}\t\nThe solution of the individual based model (\\ref{overdamped1}-\\ref{overdamped3}) is computed by averaging different realizations in order to reduce the statistical errors. The coefficient of the IBM are fixed to $r = 0.0625$ for the repulsive range, $R = 0.25$ for the alignment interaction range, while $N = 10^5$ particles are used for each simulation. The details of the particles simulation can be found in \\cite{Fehske_2007, Hockney_1988} for classical particle approaches or in \\cite{Motsch_Navoret_MMS11} for a direct application to the SOH model.\n\n\\medskip\n\\noindent\n\\paragraph{Riemann problem:}\nThe convergence of the two models is measured on a Riemann problem with the following initial data\n\\begin{equation}\n\\label{RM}\n(\\rho_l, \\theta_l ) = (0.0067, 0.7 ) ,\\quad (\\rho_r, \\theta_r ) = (0.0133, 2.3).\n\\end{equation}\nand with periodic boundary condition in $x$ and $y$. The parameters of the SOHR model are: $\\Delta t = 0.01, \\,\\, \\Delta x = \\Delta y = 0.25$. In figure \\ref{fig:macro_micro} we report the relative $L^1$ norm of the error for the macroscopic quantities $(\\rho, \\theta)$ between the SOHR model and the particle model with respect to the number of averages for different values of $\\varepsilon$ : $ \\varepsilon = 1$ (x-mark), $\\varepsilon = 0.5$ (plus), $\\varepsilon = 0.1$ (circle), $\\varepsilon = 0.05$ (square) at time $T = 1s.$ This figure shows, as expected, that the distance between the two solutions goes to zero when $\\varepsilon$ goes to zero.\nIn figure \\ref{fig:solution_RM} we report the density $\\rho$ and the flux direction $\\theta$ for the same Riemann problem along the $x$-axis for $\\varepsilon = 0.05$ at time $T = 1s$, the solution being constant in the $y$-direction. Again we clearly observe that the two models provide very close solutions, the small differences being due to the different numerical schemes employed for their discretizations.\n\n\n\\begin{figure}[!ht]\n\\subfloat[For density $\\rho$ \\label{dd}]{%\n\\includegraphics[width=0.51\\textwidth]{dens_macro_micro_RM.png}\n }\n\\hfill\n\\subfloat[For $\\theta$ \\label{velo}]{%\n\\includegraphics[width=0.51\\textwidth]{theta_macro_micro_RM.png}\n }\n\\caption{Relative error between the macroscopic and the microscopic model for density (a) and $ \\theta $ (b) as a function of the number of averages for different values of $\\varepsilon$. The error decreases with both decreasing $\\varepsilon$ and increasing number of averages, showing that the SOHR model provides a valid approximation of the IBM for $\\rho$ and $\\theta$. }\n\\label{fig:macro_micro}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\subfloat[density $\\rho$ \\label{density}]{%\n\\includegraphics[width=0.5\\textwidth]{density_RM.png}\n }\n\\hfill\n\\subfloat[ $\\theta$ \\label{velocity}]{%\n\\includegraphics[width=0.5\\textwidth]{theta_RM.png}\n }\n\\caption{Solution of the Riemann problem (\\ref{RM}) along the $x$-axis for the SOHR model (blue line) and for the IBM with $\\varepsilon = 0.05$ (red line) at $T = 1s.$ The agreement between the two models is excellent. For the SOHR model, the mesh size is $\\Delta x = \\Delta y = 0.0625$. }\n\\label{fig:solution_RM}\n\\end{figure}\n\n\\medskip\n\\noindent\n\\paragraph{Taylor-Green vortex problem:}\nIn this third test case, we compare the numerical solutions provided by the two models in a more complex case. The initial data are\n\\begin{equation}\n\\label{Taylor_Green}\n\\rho_0 = 0.01, \\quad \\Omega_0(x,y) = \\dfrac{\\tilde{\\Omega}_0(x,y)}{\\vert \\tilde{\\Omega}_0(x,y) \\vert},\n\\end{equation}\n where the vector $ \\tilde{\\Omega}_0 = (\\tilde{\\Omega}_{01}, \\tilde{\\Omega}_{02}) $ is given by\n\\begin{eqnarray*}\t\t\n\t&& \\tilde{\\Omega}_{01}(x,y) = \\dfrac{1}{3}\\sin( \\dfrac{\\pi}{5} x ) \\cos( \\dfrac{\\pi}{5} y ) + \\dfrac{1}{3} \\sin( \\dfrac{3\\pi}{10} x) \\cos( \\dfrac{3\\pi}{10} y ) + \\dfrac{1}{3} \\sin( \\dfrac{\\pi}{2} x) \\cos( \\dfrac{\\pi}{2} y),\\\\\n\t && \\tilde{\\Omega}_{02}(x,y) = - \\dfrac{1}{3} \\cos( \\dfrac{\\pi}{5} x) \\sin( \\dfrac{\\pi}{5} y )- \\dfrac{1}{3} \\cos( \\dfrac{3\\pi}{10} x ) \\sin( \\dfrac{3 \\pi}{10} y) - \\dfrac{1}{3} \\cos( \\dfrac{\\pi}{2} x) \\sin( \\dfrac{\\pi}{2} y ).\t\n\\end{eqnarray*}\nwith periodic boundary conditions in both directions. The numerical parameters for the SORH model are : $\\Delta x = \\Delta y = 0.2,$ $\\Delta t = 0.01$, while for the particle simulations we choose : $N = 10^5$ particles, $\\varepsilon = 0.05,$ $r = 0.04,$ $R = 0.2.$ \nIn figure \\ref{fig:dens_macro_micro_t06_GT} and \\ref{fig:velo_macro_micro_t06_GT}, we report the density $\\rho$ and the flux direction $\\Omega$ at time $t = 0.6$s. In both figures, the left picture is for the IBM and the right one for the SOHR model. Again, we find a very good agreement between the two models in spite of the quite complex structure of the solution.\n\n\n\n\\begin{figure}[!ht]\n\\subfloat[ density $\\rho$ for the IBM \\label{density_micro_t06}]{%\n\\includegraphics[width=0.5\\textwidth]{density_micro_t06_TG.png}\n }\n\\hfill\n\\subfloat[density $\\rho$ for the SOHR model \\label{density_macro_t06}]{%\n\\includegraphics[width=0.5\\textwidth]{density_macro_t06_TG.png}\n }\n\\caption{Density $\\rho$ for Taylor-Green vortex problem \\ref{Taylor_Green} at time $t = 0.6s$. Left: IBM. Right: SOHR model. }\n\\label{fig:dens_macro_micro_t06_GT}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!ht]\n\\subfloat[ $\\Omega$ for the IBM \\label{velo_micro_t06}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_micro_t06_TG.png}\n }\n\\hfill\n\\subfloat[$\\Omega$ for the SOHR model \\label{velo_macro_t06}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_macro_t06_TG.png}\n }\n\\caption{Mean direction $\\Omega$ for Taylor-Green vortex problem \\ref{Taylor_Green} at time $t = 0.6s$. Left: IBM. Right: SOHR model. }\n\\label{fig:velo_macro_micro_t06_GT}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Comparison between the SOH and the SOHR model}\nIn this part, we show the difference between the SOH system (\\ref{macro6_intro}), (\\ref{macro7_intro}) and the SOHR one for different values of the repulsive force $\\Phi_0.$ We recall that the SOHR model reduces to the SOH one in the case in which the repulsive force is set equal to zero. To this aim, we rescale the repulsive force $\\Phi_0$ by\n$$ \\Phi_0 = F_0 \\int_{x \\in {\\mathbb R}^2} \\phi(x) dx $$\nand then we let $F_0$ vary. The repulsive potential $\\phi$ is still given by (\\ref{phi}), so that $\\Phi_0 = F_0 \\pi\/6$.\nThe other numerical parameters are chosen as follows:\n$ d = 0.05,$ $\\alpha = 0,$ $ k_0 = 1\/8, $ $\\mu = 1,$ $ L_x = 10, L_y = 10,$ $ \\Delta x = \\Delta y = 0.15,$ $ \\Delta t = 0.001.$\nThe initial data are those of the vortex problem (\\ref{Vortex}) except that we start with four vortices instead of only one. Periodic boundary conditions in both directions are used.\n\nFigure (\\ref{fig:solution_F5}) displays the solutions for the SOHR system for the density (left) and for the flux direction\n(right) at $T = 1.5s$ with $F_0 = 5.$ Figure (\\ref{fig:solution_F005}) displays the solutions for $F_0 = 0.05.$ The results are almost undistinguishable to those of the SOH model ($F_0=0$) and are not shown for this reason. \nThese figures show that when the repulsive force is large enough, the SOHR model can prevent the formation of high concentrations. By contrast, when this force is small, the SOHR model becomes closer to the SOH one and high concentrations become possible.\n\n\n\n\\begin{figure}[!ht]\n\\subfloat[density $\\rho$ for $F_0 = 5$ \\label{F5_density}]{%\n\\includegraphics[width=0.51\\textwidth]{dens_F501500.png}\n }\n\\hfill\n\\subfloat[$\\Omega$ for $F_0 = 5$ \\label{F5_velo}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_F501500.png}\n }\n\\caption{Solution of the SOHR model for $F_0 = 5.$ Density $\\rho$ (fig \\ref{F5_density} ), flux direction $\\Omega$ (fig.\\ref{F5_velo} ) at $t= 1.5s$.}\n\\label{fig:solution_F5}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\subfloat[ density $\\rho$ for $F_0 = 0.05$ \\label{F005_density}]{%\n\\includegraphics[width=0.51\\textwidth]{dens_F00501500.png}\n }\n\\hfill\n\\subfloat[ $\\Omega$ for $F_0 = 0.05$ \\label{F005_velo}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_F00501500.png}\n}\n\\caption{Solution of the SOHR model for $F_0 = 0.05.$ Density $\\rho$ (fig \\ref{F5_density} ), flux direction $\\Omega$ (fig.\\ref{F5_velo} ) at $t = 1.5s$.}\n\\label{fig:solution_F005}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Comparison between the SOHR and the DLMP model}\n\nIn this final part, we want to compare the SOHR system to the hydrodynamic model proposed by Degond, Liu, Motsch and Panferov in \\cite{Degond_etal_MAA13} (referred to as DLMP model). This model is derived, in a similar fashion as the SOHR model, starting from a system of self-propelled particles which obey to alignment and repulsion. The main difference is that in the DLMP model, the particle velocity is exactly equal to the self propulsion velocity but the particles adjust their orientation to respond to repulsion as well as alignment. The resulting model is of SOH type and is therefore written (\\ref{macro6_intro}), (\\ref{macro7_intro}), but with an increased coefficient in front of the pressure term $P_{\\Omega^\\bot} \\nabla_x \\rho$, this coefficient being equal to $v_0 \\, d (1 + \\frac{d+c_2}{c_1}F_0)$. The initial conditions and numerical parameters are the same as in previous test\n\nIn figure \\ref{fig:solution_DLMP_F5}, we report the density $\\rho$ (left) and the flux direction $\\Omega$ (right) for $F_0 = 5$ for the DLMP model. Comparing figures (\\ref{fig:solution_F5}) with figure \\ref{fig:solution_DLMP_F5}, we observe that the solutions of the SOHR and of the DLMP model are different. The homogenization of the density seems more efficient with the SOHR model than with the DLMP model. This can be attributed to the effect of the nonlinear diffusion terms that are included in the SOHR model but not in the DLMP model. Therefore, the way repulsion is included in the models may significantly affect the qualitative behavior of the solution. In practical situations, when the exact nature of the interactions is unknown, some care must be taken to choose the right repulsion mechanism. \n\n\n\\begin{figure}[!ht]\n\\subfloat[density $\\rho$ \\label{F5DLMP_density}]{%\n\\includegraphics[width=0.51\\textwidth]{dens_F5DLMP01500}\n }\n\\hfill\n\\subfloat[$\\Omega$ \\label{F5DLMP_velo}]{%\n\\includegraphics[width=0.51\\textwidth]{velo_F5DLMP01500}\n }\n\\caption{Solution of the DLMP model for $F_0 = 5.$ Density $\\rho$ (left) and flux direction $\\Omega$ (right) at $t = 1.5s$.}\n\\label{fig:solution_DLMP_F5}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Conclusion}\n\\label{sec:conclu}\nIn this paper, we have derived a hydrodynamic model for a system of self-propelled particles which interact through both alignment and repulsion. In the underlying particle model, the actual particle velocity may be different from the self-propulsion velocity as a result of repulsion interactions with the neighbors. Particles update the orientation of their self-propulsion seeking to locally align with their neighbors as in Vicsek alignment dynamics. The corresponding hydrodynamic model is similar to the Self-Organized Hydrodynamic (SOH) system derived from the Vicsek particle model but it contains several additional terms arising from repulsion. These new terms consist principally of gradients of linear or nonlinear functions of the density including a non-linear diffusion similar to porous medium diffusion. This new Self-Organized Hydrodynamic system with Repulsion (SOHR) has been numerically validated by comparisons with the particle model. It appears more efficient to prevent high density concentrations than other approaches based on simply enhancing the pressure force in the SOH model. In future work, this model will be used to explore self-organized motion in collective dynamics. To this effect, it will be calibrated on data based on biological experiments, such as recordings of collective sperm-cell motion. \n\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}