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+{"text":"\\section{Introduction}\n\n\nIn this work we raise  one generic issue with Maxwell equations in curved space-time. Specifically,\nwe focus on the case of pure electrostatic systems in the noninertial reference frame with a globally time-independent metric tensor with $g_{0i}\\neq0$.\nEvidently, for an observer in a flat space-time and inertial frame the charges in motion do generate both \nelectric and magnetic fields. The issue is what are the fields seen by an observer in the noninertial frame  comoving with the\nelectric charges, so that to this observer  the electric current is a vanishing one. Will he see pure\nelectric field or shall there be some trace of the noninertial motion in the form of a residual magnetic field?\nFrom the point of view of Einstein's general relativity theory such a magnetic field if exists would be \nof geometric origin and will have a peculiar property of not to be subjected to screening by the\nconventional magnetic shields.\n\n\nA focus of this paper will be on the special case of noninertial motion of practical interest: magnetic fields\nin electrostatic systems residing at rest in the curved space-time of the rotating body.\nOur principal statement is  that the full system of Maxwell  equations entails a non-vanishing \nmagnetic field despite zero currents. In terms of the\nmetric properties of the relevant space-time, the effect is due to the nonvanishing off-diagonal components \nof the metric tensor $g_{0i}, \\ i=1,2,3$. In cases of practical interest, this geometric magnetic field is driven by the Earth rotation.\n\n\n\nOur primary motivation was  the recent interest in the impact of the\ngravity induced spin rotations in the ultimate sensitivity searches for\nthe electric dipole moments (EDM) of neutrons and of charged particles in the storage ring experiments \\cite{Silenko2007}-\\cite{Abusaif2019}.  The effect of the Earth\nrotation in the neutron EDM experiments has already received much attention \\cite{Baker2007},  \\cite{Lamoreaux2007},\n\\cite{Serebrov2015}.  The impact of the\ngeometric magnetic field has not been discussed before. As we shall see, this field has a\nstrong dependence on the configuration of the static electric field.   In the  geometry  of the neutron EDM experiments, the geometric\nmagnetic field is a nonuniform one with the nonvanishing gradient. Such a gradient of the magnetic\nfields is of particular concern in the comagnetometry which is crucial in the neutron EDM experiments (\\cite{Abel2019} and references therein) . To this end, the salient feature of the geometric magnetic field is that it changes a sign when the external electric field is flipped. Consequently,\ninteraction of the magnetic moment of the particle with the geometric magnetic field will generate a false signal of EDM. Numerically, the false EDM is still below the sensitivity of the current neutron EDM experiments, but will be in the\nballpark of the ultimate sensitivity neutron EDM experiments aiming atvimproving the existing upper bound by  one order in magnitude and beyond \\cite{Chupp2019}.\n\n\nThe further presentation is organized as follows. A treatment of physics in the noninertial frames demands for a consistent use of Einsten's general relativity (GR) formalism. In  Section II we invoke GR to derive the results for the geometry of the curved spfce-time and dynamics of spinning charged particles in such spaces. \nIn particular, since the space-time is curved, one needs to elucidate the meaning of the  \nmagnetic field. Specifically, we check that the magnetic  field excerts on a charged particle a force which rotates its velocity ${\\bf v}$ (defined in a certain orthonormal basis) preserving its magnitude, and induces the conventional precession of its magnetic moment. Trivial though may they sound, these constraints are important, when the nonvanishing geometrical magnetic field is present.\n\n\nA subject of Section III is a derivation of the geometric magnetic field in the pure electrostatic systems at rest in the noninertial reference frames. We demonstrate that the geometric magnetic field is a salient feature of Maxwell equations in the stationary metrics with $g_{0i}\\neq0$. In the simplest case of the laboratory residing on the rotating Earth, a connection between the geometric magnetic field, which is an axial vector, and the polar external electric field, contains the angular velocity of  Earth's rotation, so that the parity conservation is warranted.  \n\nIn Section IV we comment on three examples of the geometric magnetic field. The most interesting case at present is a search for the EDM of neutrons via precession of the neutron spin in the external electric field. Here the geometric magnetic field happens to have a finite gradient along the electric field and gives rise to the false EDM signal. Although tiny, this false EDM is in the ballpark of future ultimate precision neutron EDM experiments. The second example is the geometric magnetic field in all electric storage rings. Such a ring is discussed as a dedicated proton EDM machine \\cite{Abusaif2019,Anastopoulos2016}. In this case, in contrast to the constant residual magnetic field of the Earth, the geometric magnetic moment has a quadrupole-like behaviour along the circumference of the storage ring. Finally, for the sake of academic completeness, we report an exact solution for the uniformly charged sphere at rest on Earth's surface. \n\nThe major points of this study are summarized in the Conclusions. The Appendix provides a brief survey of the GR formalism our derivations are based upon.\n\n\n \n  \n\\section{Maxwell equations and particle and spin dynamics}\n\n\n\n\\subsection{ Maxwell  equations}\n\n\n The local\ncoordinates are denoted as $x^{\\mu},\\,\\mu=0,1,2,3=(0,i)$, $i=1,2,3$. \nLet the electromagnetic field (2-form) be expressed through 4-potential $A_{\\mu}$ (1-form) in local coordinates as:\n\\begin{gather}\nF_{\\mu\\nu}=\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}.\n\\label{M10}\n\\end{gather}\nThen the homogeneous Maxwell equations\n\\begin{gather}\n\\varepsilon_{\\mu\\nu\\lambda\\rho}\\partial_{\\nu}F_{\\lambda\\rho}=0\n\\label{M20}\n\\end{gather} \nare satisfied automatically. The field (\\ref{M10}) is a holonomic one. \n\nWe have also  the inhomogeneous Maxwell equations in the local coordinates:\n\\begin{gather}\n\\frac{1}{\\sqrt{-g}}\\partial_{\\nu}\\left(\\sqrt{-g}F^{\\nu\\mu}\\right)=4\\pi J^{\\mu},  \n\\nonumber \\\\ \ng\\equiv\\det g_{\\mu\\nu},  \\quad F^{\\mu\\nu}=g^{\\mu\\lambda}g^{\\nu\\rho}F_{\\lambda\\rho}.\n\\label{M30}\n\\end{gather} \n\n\nThe electric and magnetic fields in ONB are defined by the usual rules (see Appendix D):\n\\begin{gather}\n{\\bf E}^{\\alpha}=F^{\\alpha 0}=e^{\\alpha}_{\\mu}e^0_{\\nu}F^{\\mu\\nu}=e^{\\alpha}_ie^0_0F^{i0}+e^{\\alpha}_ie^0_jF^{ij},\n\\label{M40}\n\\end{gather}\n\\begin{gather}\n\\varepsilon_{\\alpha\\beta\\gamma}{\\bf H}^{\\gamma}=-F^{\\alpha\\beta}=-e^{\\alpha}_{\\mu}e^{\\beta}_{\\nu}F^{\\mu\\nu},\n\\quad \\varepsilon_{123}=1.\n\\label{M50}\n\\end{gather}\n\n\n\n\n\n\\subsection{The particle and spin dynamics}\n\n\n\nLet $\\mathop{\\rm d}\\nolimits s\\equiv \\sqrt{g_{\\mu\\nu}\\mathop{\\rm d}\\nolimits x^{\\mu}\\mathop{\\rm d}\\nolimits x^{\\nu}}$ be infinitely small interval  when particle is moving\nalong its  world line. We use the standard  notations $D X^{\\mu}\/\\mathop{\\rm d}\\nolimits s\\equiv u^{\\nu}\\nabla_{\\nu}X^{\\mu}$,\nwhere $u^{\\nu}\\equiv \\mathop{\\rm d}\\nolimits x^{\\nu}\/\\mathop{\\rm d}\\nolimits s$, so that $g_{\\mu\\nu}u^{\\mu}u^{\\nu}=1$. According to the definition and\nEqs. (\\ref{G65}),  (\\ref{G70}) we have\n\\begin{gather}\n\\frac{D X^a}{\\mathop{\\rm d}\\nolimits s}=\\frac{\\mathop{\\rm d}\\nolimits X^a}{\\mathop{\\rm d}\\nolimits s}+\\left(\\gamma^a_{b\\mu}\\frac{\\mathop{\\rm d}\\nolimits x^{\\mu}}{\\mathop{\\rm d}\\nolimits s}\\right)X^b=\\frac{\\mathop{\\rm d}\\nolimits X^a}{\\mathop{\\rm d}\\nolimits s}+\n\\left(\\gamma^a_{bc}u^c\\right)X^b.\n\\label{D5}\n\\end{gather}\nThen the equation of motion of a charged particle in the local coordinates has the form\n\\begin{gather}\nmc\\frac{D u^{\\mu}}{\\mathop{\\rm d}\\nolimits s}=\\frac{q}{c}F^{\\mu\\nu}u_{\\nu}.\n\\label{D10}\n\\end{gather}\n\nEvidently, inside the \"freely falling elevator\" in the  Riemann normal coordinates (\\ref{G160}),\nthe dynamic equations are of the same form as in a flat space-time with Cartesian coordinates:\n\\begin{gather}\n\\frac{\\mathop{\\rm d}\\nolimits \\gamma}{\\mathop{\\rm d}\\nolimits t}=\\frac{q}{mc}{\\bf E}{\\boldsymbol\\beta},\n\\nonumber \\\\\n\\frac{\\mathop{\\rm d}\\nolimits(\\gamma{\\boldsymbol\\beta}^{\\alpha})}{\\mathop{\\rm d}\\nolimits t}=\n\\frac{q}{mc}\n\\left({\\bf E}+[{\\boldsymbol\\beta},\\,{\\bf H}]\\right)^{\\alpha},\n\\nonumber \\\\\nu^a=\\gamma(1,\\,{\\boldsymbol\\beta}),  \\quad {\\boldsymbol\\beta}\\equiv{\\bf v}\/c, \\quad\n\\mathop{\\rm d}\\nolimits s=\\frac{c}{\\gamma}\\mathop{\\rm d}\\nolimits t.\n\\label{D20N}\n\\end{gather}\nHere $\\mathop{\\rm d}\\nolimits t$ is the proper time interval in the laboratory frame of reference in which the particle moves with \nthe velocity ${\\bf v}$.  In the general case \n\\begin{gather}\nu^a=e^a_{\\mu}\\frac{\\mathop{\\rm d}\\nolimits x^{\\mu}}{\\mathop{\\rm d}\\nolimits s}.\n\\label{D22N}\n\\end{gather}\n\n\n A complete description\nof spin precession of relativistic particle with MDM and EDM in the gravity field is found in  works \\cite{Orlov2012}-\\cite{Vergeles2019}.\nLet ${\\bf S}$ be the polarization vector in the particle rest frame. In the Riemann coordinates, the precession angular velocity ${\\bf \\Omega}$ equals\n\\cite{Frenkel1926}-\\cite{Fukuyama2013}\n\\begin{gather}\n\\frac{\\mathop{\\rm d}\\nolimits{\\bf S}}{\\mathop{\\rm d}\\nolimits t}=\\big[{\\boldsymbol\\Omega},\\,{\\bf S}\\big], \n\\nonumber \\\\ \n{\\boldsymbol\\Omega}=\\frac{q}{mc}\\Bigg\\{-\\left(G+\\frac{1}{\\gamma}\\right){\\bf H}+\n\\frac{\\gamma G({\\bf H}{\\boldsymbol\\beta})}{\\gamma+1}{\\boldsymbol\\beta}+\n\\left(G+\\frac{1}{\\gamma+1}\\right)[{\\boldsymbol\\beta},\\,{\\bf E}]+\n\\nonumber \\\\\n+\\eta_{EDM}\\left(-{\\bf E}+\\frac{\\gamma}{\\gamma+1}({\\bf E}{\\boldsymbol\\beta}){\\boldsymbol\\beta}+\n[{\\bf H},\\,{\\boldsymbol\\beta}] \\right)\\Bigg\\}.\n\\label{D30N}\n\\end{gather}\nHere\n\\begin{gather}\n\\mu=(G+1)\\frac{q\\hbar}{2mc}, \\quad d=\\eta_{EDM}\\frac{q\\hbar}{2mc}\n\\nonumber\n\\end{gather}\nare MDM and EDM, correspondingly.\n\n\n\n\nA formulation of the spin precession problem demands the orthonormal basis (ONB).\n It is important that the fields ${\\bf E}$ and ${\\bf H}$ in different ONB differ only by the Lorentz transformation. \nHowever, when describing dynamics in arbitrary ONB, a connection appears in the equations.\n As a result, Eqs. (\\ref{D20N}) take the following form:\n\\begin{gather}\n\\frac{\\mathop{\\rm d}\\nolimits \\gamma}{\\mathop{\\rm d}\\nolimits t}=\\frac{q}{mc}{\\bf E}{\\boldsymbol\\beta}+c(\\gamma_{\\alpha 0c}u^c){\\boldsymbol\\beta}^{\\alpha},\n\\nonumber \\\\\n\\frac{\\mathop{\\rm d}\\nolimits(\\gamma{\\boldsymbol\\beta}^{\\alpha})}{\\mathop{\\rm d}\\nolimits t}=\n\\frac{q}{mc}\n\\left({\\bf E}^{\\alpha}+[{\\boldsymbol\\beta},\\,{\\bf H}]^{\\alpha}\\right)+\nc\\Big((\\gamma_{\\alpha 0c}u^c)+(\\gamma_{\\alpha\\beta c}u^c){\\boldsymbol\\beta}^{\\beta}\\Big).\n\\label{D20}\n\\end{gather}\n\nSince the polarisation vector ${\\bf S}$ is defined in the particle rest frame, the explicit expression for a contribution of the connection to the precession frequency (\\ref{D30N}) for a relativistic particle is too  lengthy to be reproduced here \\cite{Vergeles2019}, we  only cite a result for the nonrelativistic case:\n\\begin{gather}\n{\\boldsymbol\\Omega}^{\\alpha}=-\\frac{q}{mc}\\bigg\\{\\big(G+1\\big){\\bf H}+\n\\eta_{EDM}{\\boldsymbol E}\\bigg\\}^{\\alpha}\n-\\frac{c}{2}\\varepsilon_{\\alpha\\beta\\rho}\n\\gamma_{\\beta\\rho0}.\n\\label{D30}\n\\end{gather}\n\n\n\nThe general conclusion from this brief discussion is \nthat the fields ${\\bf E}$ and ${\\bf H}$ in ONB defined according to\n(\\ref{M40}) and (\\ref{M50}) possess all the dynamic properties of the electric and magnetic fields, correspondingly.\nWe use hereafter the ONB (\\ref{G50}) which is a physically sensible one. \n\n\n\n\\section{Magnetic field in the pure electrostatic system in the nonertial frame }\n\n\n\nLet's consider a noninertial space-time with the stationary metric in the framework of Einstein's general theory of relativity. The stationarity means that all components of the metric\ntensor $g_{\\mu\\nu}(x)$  are time-independent, i.e. $(\\partial\/\\partial x^0)g_{\\mu\\nu}=0$.\nA generic feature of the correspontent metric is nonvanishig off-diagonal elements,\n\\begin{gather}\ng_{0i}\\neq 0.\n\\label{intr10}\n\\end{gather} \nThe outlined coordinate frame is denoted as $K$. In the cases of practical interest treated in Section IV the \nreference frame K is the laboratory frame used in the\ndescription of terrestrial experiments. We use also the inertial reference frame of distant stars $K'$.\nThe  gravitating body and the corresponding frame $K$ rotate w.r.t. to the frame $K'$ with the  angular velocity ${\\boldsymbol\\omega}$. Our electrostatic system resides at rest on the rotating body and  \nthe corresponding electric 4-current, as defined in the frame $K$, is the stationary one:\n\\begin{gather}\n(\\partial\/\\partial x^0)J^{\\mu}(x)=0, \\quad J^0\\neq 0, \\quad J^i=0.\n\\label{intr20}\n\\end{gather}\n\nIn this Section we derive the main result of the paper: {\\it For the stationary metric} (\\ref{intr10})  {\\it in the case of stationary electric 4-current} (\\ref{intr20})  {\\it the Maxwell equations can not be satisfied with the vanishing magnetic field} (see below Eq. (\\ref{H60n})).\n\n\n\nA proof of this property proceeds as follows. Note that any antisymmetric field in the three-dimensional space $\\sqrt{-g}F^{ij}=-\\sqrt{-g}F^{ji}$ can be represented as\n\\begin{gather}\n\\sqrt{-g}F^{ij}=\\varepsilon_{ijk}\\partial_k\\psi+\\left(\\partial_i{\\cal A}_j-\\partial_j{\\cal A}_i\\right).\n\\label{H10n}\n\\end{gather}\nThe decomposition (\\ref{H10n}) would be unique in three-dimensional Euclidean space for fields decreasing at infinity. As well known, in the rotating frames one faces the formal issue of the horizon. However, in all the cases of practical interest, the charge and current distributions are localized way inside the horizon radius. Consequently, the decomposition (\\ref{H10n}) is well defined and unique, and $J_i=0$ entails ${\\cal A}_i=0$.\n\n\n\n\n\nAccording to the rules (\\ref{G64}), the field definition (\\ref{M50}) and  the fact that $\\tilde{e}^i_0=0$, we obtain:\n\\begin{gather}\nF^{ij}=\\tilde{e}^i_a\\tilde{e}^j_bF^{ab}=-\\varepsilon_{\\alpha\\beta\\gamma}\n\\tilde{e}^i_{\\alpha}\\tilde{e}^j_{\\beta}{\\bf H}^{\\gamma}=-\\frac{e^0_0}{\\sqrt{-g}}\n\\varepsilon_{ijk}e^{\\alpha}_k{\\bf H}^{\\alpha}.\n\\label{H20n}\n\\end{gather}\nHere we  used one of  the relations (\\ref{H25n}).\n Making use of (\\ref{H10n}) (with ${\\cal A}_i=0$) in (\\ref{H20n}) yields\n\\begin{gather}\n{\\bf H}^{\\alpha}=-(e^0_0)^{-1}\\tilde{e}^i_{\\alpha}\\partial_i\\psi.\n\\label{H30n}\n\\end{gather}\n\n\nNow we turn to the  homogeneous Maxwell equations (\\ref{M20}).\nWe express the  holonomic field $F_{\\mu \\nu}$  in terms of the electric  and  magnetic fields:\n\\begin{gather}\nF_{i0}=e^a_ie^b_0F_{ab}=e^0_0e^{\\alpha}_iF_{\\alpha0}=\n-e^0_0e^{\\alpha}_i{\\bf E}^{\\alpha}=\\partial_iA_0, \\quad\n \\mbox{or}\n \\quad {\\bf E}^{\\alpha}=-\\frac{\\tilde{e}^i_{\\alpha}}{e^0_0}F_{i0},\n\\label{H40n}\n\\end{gather} \n\\begin{gather}\nF_{ij}=e^a_ie^b_jF_{ab}=-\\varepsilon_{\\alpha\\beta\\gamma}e^{\\alpha}_ie^{\\beta}_j{\\bf H}^{\\gamma}-\n\\left(e^0_je^{\\alpha}_i-e^0_ie^{\\alpha}_j\\right){\\bf E}^{\\alpha}.\n\\label{H50n}\n\\end{gather} \nEq. (\\ref{M20}) with $\\mu=0$ imply the identity $\\varepsilon_{ijk}\\partial_kF_{ij}=0$. Therefore,\napplying the operator $\\varepsilon_{ijk}\\partial_k$ to the Eq. (\\ref{H50n}), using (\\ref{H30n})\nand the identity $\\varepsilon_{ijk}\\partial_kF_{i0}=-\\varepsilon_{ijk}\\partial_k\\left(\ne^0_0e^{\\alpha}_i{\\bf E}^{\\alpha}\\right)=0$ (see (\\ref{H40n})), we\n obtain the equation \n\\begin{gather} \n\\partial_i\\bigg(\\sqrt{-g}\\left(e^0_0\\right)^{-2}g^{ij}\\partial_j\\psi\\bigg)=\n-\\varepsilon_{ijk}e^0_0e^{\\alpha}_i{\\bf E}^{\\alpha}\\partial_k\\left(\\frac{e^0_j}{e^0_0}\\right).\n\\label{H60n}\n\\end{gather} \nRecall that  $e^0_j=g_{0j}\/\\sqrt{g_{00}}\\neq0$  according to (\\ref{G130}) and (\\ref{intr10}). Therefore\nif ${\\bf E}\\neq0$, then (\\ref{H60n}) entails  $\\partial_j\\psi\\neq0$ and,\naccording to Eq. (\\ref{H30n}),  ${\\bf H}\\neq0$ as well. We shall refer to the field (\\ref{H30n}) as the geometric magnetic field $ {\\bf H}_{\\boldsymbol\\omega}$. As we shall see, the geometric magnetic field has its origin in the rotation of the frame $K$, hence the subscript ${\\boldsymbol\\omega}$.\n\n\nNext we look into the equation for the electric field.\nLet's  express $F^{i0}$ in terms of physical fields:\n\\begin{gather} \nF^{i0}=\\tilde{e}^i_a\\tilde{e}^0_bF^{ab}=\\tilde{e}^0_0\\tilde{e}^i_{\\alpha}{\\bf E}^{\\alpha}-\n\\varepsilon_{\\alpha\\beta\\gamma}\\tilde{e}^i_{\\alpha}\\tilde{e}^0_{\\beta}{\\bf H}^{\\gamma}\n=\\frac{1}{e^0_0}\\tilde{e}^i_{\\alpha}{\\bf E}^{\\alpha}-\\frac{1}{\\sqrt{-g}e^0_0}\\varepsilon_{ijk}e^0_j\\partial_k\\psi.\n\\label{H70n}\n\\end{gather} \nThe substitution of the right-hand side of (\\ref{H70n}) into Eq. (\\ref{M30}) with $\\mu=0$ leads to \n\\begin{gather}\n\\partial_i\\left(\\frac{\\sqrt{-g}}{e^0_0}\\tilde{e}^i_{\\alpha}{\\bf E}^{\\alpha}\\right)-\n\\varepsilon_{ijk}\\partial_i\\left(\\frac{e^0_j}{e^0_0}\\right)\\cdot\\partial_k\\psi=\n4\\pi\\sqrt{-g}J^0.\n\\label{H80n}\n\\end{gather}\nHere we have used representation (\\ref{H30n}) and one of the relations (\\ref{H25n}).\n\nThe system of equations  (\\ref{H30n}), (\\ref{H60n}) and (\\ref{H80n})  is complete  and exact. The electric field ${\\bf E}$  is expressed in terms of the potential $A_0$ with the help of relation (\\ref{H40n}).\n\nTo be specific, we describe the curved space-time of the rotating body by the Kerr  metric described in Appendix B.  We report here the iterative solutions for the electromagnetic potentials $A_{0}$ and $\\psi$ and the corresponding fields, treating the angular velocity of the rotating body  ${\\boldsymbol{\\omega}}$ and its gravitational radius $r_g$ as small parameters. Making use of (\\ref{G120})-(\\ref{G150}), we rewrite Eqs.  (\\ref{H30n}), (\\ref{H60n}) and (\\ref{H80n}) keeping the terms of the order of   ${\\boldsymbol{\\omega}}$,\n$r_g$, $r_g{\\boldsymbol{\\omega}}$, ${\\boldsymbol{\\omega}}\\otimes{\\boldsymbol{\\omega}}$. \n\n There emerges a simple hierarchy of corrections in powers of ${\\boldsymbol{\\omega}}$. We are interested in the electrostatic system with the initial potential $A_0^{(0)}$ and the corresponding Minkowski space defined electric field  ${\\boldsymbol{E}}^{(0)}=-\\nabla A_0^{(0)}$. Concerning the geometric magnetic field, the crucial points are Eq. (B3) for the off diagonal $g_{0i}$, Eq. (B5) for $e^0_i$ and Eq. (B6) for $\\tilde{e}^0_{\\alpha}$. The three related  quantities are all proportional to the angular velocity of the rotating body.  Consequently  $e^0_j\/e^0_0=\\mathop{\\rm O}\\nolimits({\\boldsymbol\\omega})$ and, according to (\\ref{H60n}), the expansion for the magnetic potential $\\psi$ starts with the linear term $\\psi^{(1)}=\\mathop{\\rm O}\\nolimits({\\boldsymbol{\\omega}})$. \n \t\n Now we proceed to the electric field. According to the expansion of the Kerr metric in Appendix B, we have\n\\begin{gather}\n\\frac{\\sqrt{-g}\\tilde{e}^i_{\\alpha}}{e^0_0}=\\left(1+\\frac{r_g}{R}\\right)\\delta^i_{\\alpha}\n+\\left(\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^2}{2c^2}\\delta^i_{\\alpha}-\n\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^i[{\\boldsymbol\\omega},\\,{\\bf R}]^{\\alpha}}{2c^2}\\right)\\, ,\n\\label{H90n}\n\\end{gather}\nwhich does not contain the linear term.  Then, equations (\\ref{H80n}) and (\\ref{H90n}) tell that the electric field  acquires the first correction only to the second order in ${\\boldsymbol{\\omega}}$, i.e., $ {\\bf E}^{(1)}=0, \\,{\\bf E}^{(2)}\\neq 0$. In the due turn,  Eq. (\\ref{H60n})  guarantees that the quadratic correction to the magnetic potential vanishes: $\\psi^{(2)}=0$. \n\nGoing back to the magnetic potential $\\psi$, we make use of \n\\begin{gather}\n\\frac{\\tilde{e}^i_{\\alpha}}{e^0_0}=\\delta^i_{\\alpha}+\n\\left(\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^2}{2c^2}\\delta^i_{\\alpha}-\n\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^i[{\\boldsymbol\\omega},\\,{\\bf R}]^{\\alpha}}{2c^2}\\right).\n\\label{H100n}\n\\end{gather} \nand invoke the relations \n\\begin{gather}\n\\frac{\\sqrt{-g}}{\\left(e^0_0\\right)^2}g^{ij}=-\\left(1+\\frac{r_g}{R}\\right)\\delta^{ij}, \\quad\ne^0_0e^{\\alpha}_i=\\delta^{\\alpha}_i+\\mathop{\\rm O}\\nolimits({\\boldsymbol{\\omega}}\\otimes{\\boldsymbol{\\omega}}), \n\\nonumber \\\\\n\\frac{e^0_j}{e^0_0}=-\\Bigg\\{1+\\frac{r_g}{R}\\left(2-\\frac{I R^2_{\\oplus}}{R^2}\\right)\\Bigg\\}\n\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^j}{c},\n\\label{H120n}\n\\end{gather} \nThen Eq. (\\ref{H60n}) takes the final form \n\\begin{gather}\n\\partial_i\\bigg\\{\\left(1+\\frac{r_g}{R}\\right)\\partial_i\\psi\\bigg\\}\n\\nonumber \\\\\n=\\varepsilon_{ijk}{\\bf E}^{(0)i}\n\\partial_j\\Bigg\\{\\left[1+\\frac{r_g}{R}\\left(2-\\frac{I R^2_{\\oplus}}{R^2}\\right)\\right]\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^k}{c}\\Bigg\\}.\n\\label{H130n}\n\\end{gather} \n\nA departure of the metric of the curved space-time from the Minkowski one changes the relationship between the potentail $A_{0}$ and the charge distribution and the relationship between the electric field ${\\bf E}^{(2)}$ and the gradient of $A_0^{(2)}$. Although these second order corrections are unlikely to be of any practical significance in the terrestrial laboratories, we cite them for the sake of completeness. To the zeroth order in ${\\bf \\omega}$, but keeping the terms linear in $r_g$, we have\n\\begin{gather}\n-\\nabla\\bigg\\{\\left(1+\\frac{r_g}{R}\\right)\\nabla A^{(0)}_0\\bigg\\}=4\\pi\\left(1+\\frac{r_g}{R}\\right)J^0,.\n\\label{H160n}\n\\end{gather}\nAfter some algebra, Eq. (\\ref{H80n}) with the account for  Eqs. (\\ref{H90n}), (\\ref{H120n}) yields the equation for the second order correction to the scalar potential,  \n\\begin{gather}\n-\\Delta A^{(2)}_0=\\frac{1}{c^2}\\nabla\\bigg\\{[{\\boldsymbol\\omega},{\\bf R}]^2\\nabla A^{(0)}_0-\n\\left([{\\boldsymbol\\omega},{\\bf R}]\\nabla A_0^{(0)}\\right)[{\\boldsymbol\\omega},{\\bf R}]\\bigg\\}\n-\\frac{2}{c}({\\boldsymbol\\omega}\\nabla\\psi).\n\\label{H170n}\n\\end{gather}\nIn this derivation, to the desired accuracy, we  made use of $e^0_j\/e^0_0=-[{\\boldsymbol\\omega},\\,{\\bf R}]\/c$. \nNote how the second order correction to the electric potential couplles to the first order potential for the geometric magnetic field. \n\nFinally, upon using  Eq. (\\ref{H100n}) and gathering together all second order corrections, the Eq. (\\ref{H40n}) yields the second order correction to the electric field \n\\begin{gather}\n{\\bf E}^{(2)}=-\\nabla A_0^{(2)}-\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^2}{2c^2}\\nabla A_0^{(0)}\n+\\left([{\\boldsymbol\\omega},\\,{\\bf R}]\\nabla A_0^{(0)}\\right)\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]}{2c^2}.\n\\label{H150n}\n\\end{gather} \nApart from the gradient of the second order scalar potential, here emerge corrections quadratic in the angular velocity ${\\bf \\omega}$. The system of equations  (\\ref{H130n})-(\\ref{H150n}) is complete  to the desired accuracy. \n \nThe rotating body of the practical interest is the Earth. It is the case of weak gravity. On the terrestrial surface  \n$r_g\/R_{\\oplus}\\sim 10^{-9}$  and $\\omega R_{\\oplus} \\sim 1.5\\cdot 10^{-6}$. Hence we keep the terms $\\mathop{\\rm O}\\nolimits(|{\\boldsymbol\\omega}|)$ and neglect the former corrections. In this approximation\nequations  (\\ref{G120})-(\\ref{G150}) simplify to  \n\\begin{gather}\ng_{00}=g^{00}=1, \\quad g_{0i}=g^{0i}=-\\frac{\\big[{\\boldsymbol\\omega},\\,{\\bf R}]^i}{c},\n\\nonumber \\\\\ng_{ij}=g^{ij}=-\\delta^{ij}, \\quad g=-1,\n\\nonumber \\\\\ne^0_0=1, \\quad e^0_i=-\\frac{\\big[{\\boldsymbol\\omega},\\,{\\bf R}\\big]^i}{c},\n\\quad e^{\\alpha}_i=\\delta^{\\alpha}_i, \\quad e^{\\alpha}_0=0,\n\\nonumber \\\\\n\\tilde{e}^0_0=1, \\quad \\tilde{e}^0_{\\alpha}=\\frac{\\big[{\\boldsymbol\\omega},\\,{\\bf R}\\big]^{\\alpha}}{c},\n\\quad \\tilde{e}_{\\alpha}^i=\\delta_{\\alpha}^i, \\quad \\tilde{e}^i_0=0,\n\\nonumber \\\\\n\\frac{c}{2}\\varepsilon_{\\alpha\\beta\\rho}\\gamma_{\\beta\\rho 0}={\\boldsymbol\\omega}^{\\alpha}.\n\\label{H90}\n\\end{gather}\nThe second order corrections to the electric potential and electric field can be neglected and we have the familiar ${\\bf E}=-\\nabla A_0$ and  the Poisson equation $\\mathop{\\rm div}\\nolimits{\\bf E}=-\\Delta A_0=4\\pi J^0$, while the Poisson equation for the potential of the geometric magnetic field takes a simple form, \n\\begin{gather}\n\\Delta\\psi=\\frac{2}{c}({\\boldsymbol\\omega}{\\bf E})\\,,\n\\label{H140}\n\\end{gather}\nto be used in the subsequent analysis of terrestrial experiments.\n\n\\section{Manifestations of the geometric magnetic field}\n\n\n\n\\subsection{False EDM signal in the neutron EDM experiments }\n\n\n\n\n\nHere we comment on the possible implications of the geometrical magnetic field for the neutron EDM experiments. The fundamental observable is the change of the Larmor precession frequency \n\\begin{equation}\nf_n =\\frac{1}{\\pi\\hbar}|\\mu_n{\\bf B} +d_n{\\bf E}| \\label{eq:Larmor}\n\\end{equation}\nfrom the parallel to antiparallel uniform fields. The EDM is extracted from the frequency shift\n\\begin{equation} \nd_n = \\frac{\\pi \\hbar \\Delta f}{2|{\\bf E}|} . \n\\label{eq:EDM extraction}\n\\end{equation}\nThe implicit assumption is that flipping the electric field does not change the magnetic one. Our point is that this is not the case with the geometric magnetic field. \n\n\nIn practice the electric field is generated in the plane capacitor with the gap much smaller than the plate size. The relevant solution of the one-dimensional problem for the geometric magnetic field proceeds as follows. In the gap in between the plates we have \n\\begin{gather}\n{\\bf E}_0=(0,\\,0,\\,{\\cal E}_0)=-\\nabla A_0, \\quad A_0=-{\\cal E}_0z.\n\\label{IIiE10}\n\\end{gather}\nNow we solve the Poisson equation (\\ref{H140}) for the magnetic potential, representing the electric field through $A_0(z)$ and integrating by parts:\n\\begin{gather}\n\\psi(z)=\\frac{2{\\boldsymbol\\omega}_z}{c}\\int\\mathop{\\rm d}\\nolimits z'\\Delta^{-1}(z-z'){\\cal E}_0\n=-\\frac{2{\\boldsymbol\\omega}_z}{c}\\int\\mathop{\\rm d}\\nolimits z'\\frac{\\mathop{\\rm d}\\nolimits}{\\mathop{\\rm d}\\nolimits z}\\Delta^{-1}(z-z')A_0(z'),\n\\label{IIiE20}\n\\end{gather}\nwhere $\\Delta^{-1}(z)=|z|\/2$ is the inverse to the Laplace operator. One more differentiation yields \\begin{gather}\n{\\bf H}_{\\boldsymbol\\omega}=-\\nabla\\psi(z)\n=\\left(0,\\,0,\\,\\frac{2{\\boldsymbol\\omega}_z}{c}A_0(z)\\right)=\n\\left(0,\\,0,\\,-\\frac{2{\\boldsymbol\\omega}_z{\\cal E}_0}{c}z\\right)=\n-\\frac{2{\\boldsymbol\\omega}_zz}{c}{\\bf E}_0.\n\\label{IIiE30}\n\\end{gather}\nThe geometric  field ${\\bf H}_{\\boldsymbol\\omega}$ is parallel to the external electric field ${\\bf E}_0$. Its salient feature is the nonvanishing constant gradient\n\\begin{equation}\n\\frac{\\mathop{\\rm d}\\nolimits{\\bf H}_{\\boldsymbol\\omega}}{\\mathop{\\rm d}\\nolimits z} = -\\frac{2{\\boldsymbol\\omega}_z}{c}{\\bf E}_0 \n\\label{eq:Gradient}\n\\end{equation}\nAt the mid plane the geometric field vanishes: ${\\bf H}_{\\boldsymbol\\omega}(0)=0$.\n\nThe crucial feature of the neutron EDM experiments is the comagnetometry: one measures the neutron spin precession frequency with respect to that of the mercury comagnetometer. The mercury atoms are evenly distributed in the volume of the neutron storage cell and the average geometric magnetic field acting on the mercury comagnetometer vanishes: $\\langle {\\bf H}_{\\boldsymbol\\omega}^{(Hg)}\\rangle =\n{\\bf H}_{\\boldsymbol\\omega}(0) =0$.\nThe centre of mass of neutrons differs from that of the mercury by the  offset  $\\langle z \\rangle$, what entails the nonvanishing average geometric magnetic field acting on the magnetic moment of neutrons\n\\begin{equation}\n{\\bf H}_{\\boldsymbol\\omega}^{(n)} = -\\frac{2\\langle z\\rangle{\\boldsymbol\\omega}_z}{c}{\\bf E}_0 .\n\\label{eq:HgeomNeutron}\n\\end{equation}\nThe most important point is that this geometric field changes the sign when the electric field is flipped. The net effect is that the apparent EDM of neutrons, $d_n^{obs}$, as given by the procedure (\\ref{eq:EDM extraction}), will acquire the false component, $d_{n}^{obs} = d_n + d_{false}$, where \n\\begin{equation}\nd_{false} = -\\frac{2\\langle z \\rangle{\\boldsymbol\\omega}_z}  {c}\\mu_n\\, . \\label{eq:dFalse}\n\\end{equation}\nIn the experiment \\cite{Pendlebury2015} the neutron center of mass offset was $\\langle z \\rangle \\simeq 2.8$mm, the more recent experiment reports $\\langle z \\rangle \\simeq 3.9$mm. Taking the former, we find $d_{false} \\approx 2.5\\times  10^{-28}$ e$\\cdot$cm. It is still way below the recently reported best upper bound on the neutron EDM, $d_n = (0.0\\pm 1.1_{stat} \\pm 0.2_{sys})\\times 10^{-26}$e$\\cdot$cm   \n \\cite{Afach2020}, but can become sizeable in the next generation of the neutron EDM experiments aiming at $d_n < 10^{27}$ e$\\cdot$cm  \\cite{Chupp2019}. With the neutron storage cell of height 12 cm, the geometric magnetic field induced spread of the false EDM within the ensemble of stored neutrons can be as large as \n \\begin{equation}\n \\Delta d_{false} = \\pm \\frac{h{\\bf \\omega}_z}{c}\\mu_n \\simeq \\pm 5\\times 10^{-27} {\\text e}\\cdot\\text{cm} \\, . \\label{eq:dFalseSpread}\n \\end{equation}\n \n\n\n\n\n\n\n\\subsection{Geometric  magnetic field as a background in all electric proton  EDM storage rings}\n\nThe principal idea of searches for the proton EDM in the all electric ring, run at the so-called magic energy, is to eliminate the magnetic field acting on the proton magnetic moment. Then the sole rotation of the proton spin will be due to interaction of its EDM with the radial electric field that confines protons in the storage ring, and very ambitious sensitivity  to the proton EDM,\n\\begin{gather}\n\\eta_{EDM}\\sim10^{-15}\\, ,\n\\label{IIE}\n\\end{gather}\nis in sight \\cite{Anastopoulos2016,Abusaif2019}.  Here we comment on implications of the geometric magnetic field for such proton EDM experiments. \n\nThe storage ring is a cylinder capacitor with the gap $d$\nwhich is  much smaller compared to the height of cylinders $h$, which in its turn is  much smaller than radii of cylinders $r_{1,2}=\\rho\\mp d\/2$. In view of $d\\ll h\\ll\\rho$ we neglect the dependence on the vertical coordinate and have the two-dimensional geometry. The beam trajectory is in the midplane of the storage ring at the orbit radius $|{\\bf r}|=\\rho$. The electric field in the gap is given by \n\\begin{gather}\n{\\bf E}_0=-{\\cal E}_0\\frac{\\rho{\\bf r}}{r^2} = -\\nabla A_{0}(r), \\quad\\quad A_0(r) = {\\cal E}_0 \\rho \\ln\\frac{r}{\\rho}\\, .\n\\label{IIE10}\n\\end{gather}\n\nIt is instructive to start with the storage ring located on the North pole. From the viewpoint of distant observer in the reference frame K', the ring rotates with the Earth's rotation angular velocity ${\\bf \\omega}$. The static charges on the two rotating cylinders produce the opposite currents and generate in the gap the magnetic fields of the same sign and magnitude. The net result is the magnetic field\n\\begin{gather}\n{\\bf H}'_{\\boldsymbol\\omega}({\\bf r})=\\frac{1}{c}\\big[{\\bf v}({\\bf r}),\\,{\\bf E}_0({\\bf r})\\big].\n\\label{IIE20}\n\\end{gather}\nAt first sight, it introduces an asymmetry between the  clockwise and anticlockwise beams in the all electric storage ring, parasitic from the viewpoint of searches for the proton EDM. However, it is basically the motional magnetic field and, to the experimenter in the terrestrial laboratory K, it vanishes entirely. Such an exact cancellation only holds at the North and South poles, and at an  arbitrary  latitude it does not work. In the generic case, the result  (\\ref{IIE20}) for the magnetic field  suggests the small parameter $\\eta_{\\boldsymbol\\omega}={|{\\boldsymbol\\omega}|\\rho}\/{c}$, similar to that appearing in Eq. (\\ref{eq:dFalse}). For the storage ring of radius $\\rho \\sim 50$m we have \n\\begin{gather}\n\\eta_{\\boldsymbol\\omega}=\\frac{|{\\boldsymbol\\omega}|\\rho}{c}\\sim 10^{-11}.\n\\label{IIE30}\n\\end{gather}\nwhich is some four orders in magnitude larger than the target value $\\eta_{EDM}\\sim 10^{-15}$.\n\nNow we solve for the geometric magnetic field following the formalism of Section III. The electric field (\\ref{IIE10}) suggests  for the magnetic potential $\\psi$ the Ansatz\n\\begin{gather}\n\\psi=f(r)\\cdot({\\boldsymbol\\omega_t}{\\bf r})\\, ,\n\\label{IIE40}\n\\end{gather}\nwhere ${\\boldsymbol\\omega}_t$ is a projection of the Earth's angular velocity onto the ring plane. A generic solution to Eq. (\\ref{H140}) is\n\\begin{gather}\nf(r)=-\\frac{{\\cal E}_0\\rho}{c}\\left(\\ln\\frac{r}{\\rho}+\\zeta\\right) = -\\frac{A_o(r)}{c} - \\frac{{\\cal E}_0\\rho}{c}\\zeta,\n\\label{IIE70}\n\\end{gather}\nand \n\\begin{gather}\n{\\bf H}_{\\boldsymbol\\omega}^i= \\frac{2{\\boldsymbol\\omega}_t^j}{c}A_{ij}({\\bf r}) =\\Bigg\\{\\frac{A_0}{c}\\delta_{ij}\n+\\frac{{\\cal E}_0\\rho}{c}\\bigg[\\left(\\zeta-1\/2\\right)\\delta_{ij}\n+\\frac12(\\delta_{ij}-2{\\bf n}_i{\\bf n}_j)\\bigg]\\Bigg\\}\n{\\boldsymbol \\omega}_t^j,\n\\label{IIE80}\n\\end{gather}\nwhere ${\\bf n}= {\\bf r}\/r$ .\n\nThe constant $\\zeta$ is fixed by the boundary condition that the electric potential $A_{0}$ vanishes rapidly beyond the capacitor, so that in the integral representation for $\\psi$ one can perform the integration by parts:\n\\begin{gather}\n\\psi({\\bf r})=\\frac{2{\\boldsymbol\\omega}_t^j}{c}\\int\\mathop{\\rm d}\\nolimits^{(2)}y\\,\\Delta^{-1}({\\bf r}-{\\bf y}){\\bf E}^j_0({\\bf y})\n=-\\frac{2{\\boldsymbol\\omega}_t^j}{c}\\int\\mathop{\\rm d}\\nolimits^{(2)}y\\,\\partial_j\\Delta^{-1}({\\bf r}-{\\bf y})A_0(\\bf y), ,\n\\label{IIE90}\n\\end{gather}\nwhere $\\Delta^{-1}({\\bf r})=(1\/2\\pi)\\ln|{\\bf r}|$ is the inverse to the Laplace operator. The resulting equation for the symmetric matrix $A_{ij}({\\bf r})$\n\\begin{gather}\nA_{ij}({\\bf r})=\\int\\mathop{\\rm d}\\nolimits^{(2)}y\\,\\partial_i\\partial_j\\Delta^{-1}({\\bf r}-{\\bf y})A_0({\\bf y}).\n\\label{IIE100}\n\\end{gather}\nentails \n\\begin{gather}\n\\mathop{\\rm tr}\\nolimits A({\\bf r})=A_0({\\bf r}).\n\\label{IIE110}\n\\end{gather}\nHence the expansion of $A_{ij}$ into irreducible tensor structures is of the form  \n\\begin{gather}\nA_{ij}({\\bf r})=\\frac{1}{2}\\delta_{ij}A_0({\\bf r}) + \\sigma({\\bf r})(\\delta_{ij}-\n2{\\bf n}^i{\\bf n}^j) \\, ,\n\\label{IIE120}\n\\end{gather}\nand a comparison to (\\ref{IIE80}) gives immediately\n\\begin{gather}\n\\sigma({\\bf r}) = \\frac{1}{4}{\\cal E}_0\\rho, \\quad \\zeta = \\frac{1}{2}.\n\\label{IIE130}\n\\end{gather}\nOur final result for the geometric magnetic field in the gap of the storage ring is\n\\begin{gather}\n{\\bf H}_{\\boldsymbol\\omega}^i=\\frac{{\\cal E}_0\\rho}{c}\\Bigg\\{\\ln\\left(\\frac{r}{\\rho}\\right)\\cdot\\delta_{ij}+\n\\left(\\frac12\\delta_{ij}-{\\bf n}_i{\\bf n}_j\\right)\\Bigg\\}\n{\\boldsymbol \\omega}_t^j\\simeq  \\frac{{\\cal E}_0\\rho}{2c}\n\\left(\\delta_{ij}-2{\\bf n}_i{\\bf n}_j\\right){\\boldsymbol \\omega}_t^j,\n\\label{IIE130}\n\\end{gather}\nwhere in the last step we neglected $|\\log(r\/\\rho)| < d\/(2\\rho) \\ll 1$.\n\n\nThe background magnetic fields are of prime concern to the planned searches for the proton EDM in the all electric magic storage rings, for a detailed  discussion see the recent monographic document by the CPEDM (Charged Particles EDM) collaboration \\cite{Abusaif2019}. Important virtue of the all electric rings is a cancellation of many systematic effects when one compares spin rotations of simultaneously stored clockwise (CW) and anticlockwise (ACW) rotating protons. The magnetic Lorentz forces split the orbits of the CW and ACW beams. As discussed extensively in \\cite{Abusaif2019}, the modern techniques allow a very strong, but as yet incomplete,  screening of the Earth's magnetic field.  Despite much work, reported in \\cite{Abusaif2019}, the analysis of the magnetic imperfection effects is still in the formative stage.\n\nThe Earth's magnetic field ${\\bf H}_\\oplus$ and the geometric magnetic field ${\\bf H}_{\\bf \\omega}$ do differ markedly. In contrast to the Earth's magnetic field, the  geometric one is not subject to screening by magnetic shields. On the scale of the storage ring, the Earth's magnetic field can be regarded as a uniform one and has the constant projection onto the ring plane. It is pointing along the (magnetic) meridian, which we take for the y-axis : ${\\bf H}_\\oplus ^t = (0, H_\\oplus ^t )$. In contrast to that, the geometric magnetic field has the quadrupole-like behaviour along the particle orbit, ${\\bf H}_{\\bf \\omega} = H_{\\bf \\omega}(\\sin 2\\theta, \\cos 2\\theta) $. The angular position of the particle in a ring,  $\\theta$, is defined by ${\\bf n} = (\\cos \\theta, -\\sin \\theta)$. \n\nIn the all  electric magnetic rings the most dangerous ones are the radial magnetic fields. In the above two cases they are  equal to  $H_\\oplus^{(r)} = ({\\bf n}\\cdot {\\bf H}_\\oplus) = -H_\\oplus \\sin \\theta $  and $H_{\\bf \\omega}^{(r)} = ({\\bf n}\\cdot {\\bf H}_{\\bf \\omega}) = H_{\\bf \\omega} \\sin \\theta $\\, \nAccording to \\cite{Abusaif2019}, to the first approximation the rotation of the proton spin is proportional to the one-turn integral  $\\oint d\\theta H_\\oplus^{(r)} $. .Obviously, both the Earth's  and  geometric magnetic fields share the property \n\\begin{equation}\n\\oint d\\theta H_\\oplus^{(r)}  = \\oint d\\theta H_{\\bf \\omega}^{(r)} =0\\, . \\label{eq:EDMlike}\n\\end{equation}\nThe argument of Ref.  \\cite{Abusaif2019} about vanishing false EDM signal from is then applicable to the geometric magnetic field as well.\n\nThe above consideration is somewhat naive and must be complemented by a consistent  treatment of the spin-orbit coupling, though. Namely, this cancellation of the false EDM effect might become incomplete because of the orbit distortions which are very much distinct in the two cases.  To be on the safe side,  one needs a dedicated analysis of the false spin rotations with simultaneous allowance for the orbit distortions. Furthremore, one needs to pay an attention to  a possible cross talk between the impact of the geometric magnetic field and the residual Earth's magnetic field. It is an important complex issue on its own to be addressed to in the future,  it  goes beyond the scope of the present communication. \n\n\n\\subsection{Geometric magnetic field of the conducting charged sphere}\n\nFor the sake of completeness, we comment on the charged sphere at rest in the  rotating system $K$. Inside the sphere we have ${\\bf E}=0$  and $A_0=\\mathop{\\rm const}\\nolimits$ thereof: \n\\begin{gather}\n{\\bf E}({\\bf r})=\\left\\{\n\\begin{array}{rl}\n\\dfrac{Q{\\bf r}}{r^3},  &  \\mbox{for} \\quad r>a, \\\\  [4mm]\n0, & \\mbox{for} \\quad r<a.\n\\end{array}  \\right.\n\\label{IE10}\n\\end{gather}\nHere $Q$ is the charge of the sphere, and the  radius-vector ${\\bf r}=0$ at the centre of the  sphere. \n\nAccording to (\\ref{H140})\n\\begin{gather}\n\\psi({\\bf r})=-\\frac{\\boldsymbol\\omega}{2\\pi c}\\int\\mathop{\\rm d}\\nolimits^{(3)}x\\frac{1}{|{\\bf r}-{\\bf x}|}{\\bf E}({\\bf x})=\n-\\frac{Q\\cdot ({\\boldsymbol\\omega}{\\bf r})}{3c}\\left\\{\n\\begin{array}{rl}\n\\dfrac{3}{r}-\\dfrac{a^2}{r^3},  &  \\mbox{for} \\quad r>a, \\\\  [4mm]\n\\dfrac{2}{a}, & \\mbox{for} \\quad r<a\\,,\n\\end{array}  \\right.\n\\label{IE30}\n\\end{gather}\nand \n\\begin{widetext}\n\\begin{gather}\n{\\bf H}_{\\boldsymbol\\omega}({\\bf r})=\\left\\{\n\\begin{array}{rl}\n\\dfrac{1}{r^3}\\Big\\{3({\\boldsymbol\\mu}{\\bf n}){\\bf n}-{\\boldsymbol\\mu}\\Big\\}+\n\\dfrac{1}{c}\\big[{\\bf E}({\\bf r}),\\,[{\\boldsymbol\\omega},\\,{\\bf r}]\\big], \\quad {\\bf E}({\\bf r})=\\dfrac{Q{\\bf r}}{r^3} &  \\mbox{for} \\quad r>a, \\\\ [4mm]\n\\dfrac{2Q{\\boldsymbol\\omega}}{3ca}, & \\mbox{for} \\quad r<a.\n\\end{array}  \\right.\n\\label{IE40}\n\\end{gather}\n\\end{widetext}\nHere ${\\bf n}={\\bf r}\/{r}$ and\n\\begin{gather}\n{\\boldsymbol\\mu}=\\frac{Qa^2}{3c}{\\boldsymbol\\omega}.\n\\nonumber\n\\end{gather}\nis the geometric magnetic moment  of  the conducting charged sphere, induced by the Earth's rotation. The second term in (\\ref{IE40}) is the\nfamiliar motional magnetic field in the rotating frame $K$ which is entailed by the electric field in inertial frame $K'$.\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nWe have shown that in the pure electrostatic systems at rest on the rotating bodies there can exist a geometric magnetic field. From the general relativity point of view, it originates from the nonvanishing off-diagonal elements $g_{0i}$ of the metric tensor which are proportional to the angular velocity of rotation of the gravitating body. We presented several specific examples of the geometric field for different configurations of the static electric field on the rotating body described by the Kerr metric. In the configuration of experimental setups used in the terrestrial searches for the EDM of neutrons, the geometric magnetic field changes the sign when the electric field is flipped. Consequently, its interaction with the magnetic moment of the neutron can imitate the neutron EDM and that can become a sizable background in the next generation of the neutron EDM experiments. We found a fairly large background geometric magnetic field in all electric magic storage rings considered a dedicated machine for searches of the proton EDM. The symmetry properties  of the geometric magnetic field suggest strong cancellations of its contribution to the proton spin rotations. Stil, its impact on the signal of EDM remains an open issue - here one needs a dedicated analysis with full allowance for the spin-orbit dynamics in the storage ring. \n \n\n\n\n\\begin{acknowledgments}\n\nWe are grateful to A.Ya. Maltsev, A.A. Starobinsky and  S.S. Vergeles for valuable comments and discussions.\nThis work was carried out as a part of the State Program 0033-2019-0005.\n\n\n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Methods}  \\label{sec:Methods}\n\n{\\noindent \\textbf{Foreshock disturbance properties.}}  Since previous simulation\\cite{riquelme11a} and observational\\cite{oka06a, wilsoniii12c} studies found that magnetosonic-whistler waves\\cite{wilsoniii16a} -- right-hand polarized electromagnetic plasma waves, with density fluctuating in phase with $\\lvert \\mathbf{B}{\\scriptstyle_{o}} \\rvert$ -- can accelerate particles to $>$1 keV, we limited the types of foreshock disturbances to those having a magnetosonic-whistler nature.  We further limited our search to disturbances occurring multiple times per day, those with nonlinear properties, and those capable of forming shock waves, all properties associated with energizing particles.  Therefore, the three foreshock disturbances we examined, short large-amplitude magnetic structures\\cite{lucek08a, mann94a, scholer03b, schwartz92a, wilsoniii13b, wilsoniii16a}, hot flow anomalies\\cite{eastwood08a, omidi07a, omidi13a, omidi14c, schwartz85a, zhang10a, zhang13a}, and foreshock bubbles\\cite{archer15a, omidi10a, turner13a}, are all produced by the interaction between the incident solar wind and suprathermal ($>$100 eV to 100s of keV) ions\\cite{burgess12a, burgess13a, wilsoniii16a}.  Short large-amplitude magnetic structures are short duration ($\\sim$few to 10s of seconds), nonlinear large amplitude ($\\delta B\/B$ $>$ 2), monolithic ``magnetic pulsations'' with spatial scales of $\\sim$1000 km that can exhibit a soliton-like behavior (i.e., large amplitude fluctuations are fast and spatially narrow)\\cite{schwartz92a, mann94a}.  Both hot flow anomalies and foreshock bubbles are localized rarefaction regions surrounded by compression regions that are effectively ``carved out'' by an accumulation of suprathermal ions along a discontinuity in the interplanetary magnetic field.  The difference is that the compression regions for hot flow anomalies are centered on the discontinuity and the discontinuity must interact with the Earth's bow shock, whereas foreshock bubbles form upstream of the discontinuity and the discontinuity need not interact with the Earth's bow shock\\cite{omidi07a, turner13a}.  Both hot flow anomalies and foreshock bubbles are several Earth radii in scale (i.e., $>$10,000 km).\n\n{\\noindent \\textbf{Exclusion of solar source.}}  We first eliminated interplanetary shocks as a possibility by examining the \\emph{Wind} shock database at the Harvard Smithsonian for Astrophysics (Online at \\textcolor{Blue}{\\seqsplit{http:\/\/themis.ssl.berkeley.edu\/index.shtml}}) finding no interplanetary shocks during any of the energetic electron enhancements.\n\n\\indent  We next examine the radio data from the \\emph{Wind} and STEREO spacecraft (Extended Data Fig. \\ref{fig:WindandSTEREOWAVESRadio}).  There are no clear radio bursts or any evidence of significant radio activity on the sun during any of the four THEMIS foreshock passes.  For comparative purposes, we include a date with clear solar radio bursts (Extended Data Figs \\ref{fig:WindandSTEREOWAVESRadio}\\textbf{e}, \\ref{fig:WindandSTEREOWAVESRadio}\\textbf{j}, and \\ref{fig:WindandSTEREOWAVESRadio}\\textbf{o}).  The enhanced radio intensity near $\\sim$200 kHz (Figs \\ref{fig:WindandSTEREOWAVESRadio}\\textbf{f}--\\ref{fig:WindandSTEREOWAVESRadio}\\textbf{i}) is most likely auroral kilometric radiation\\cite{ergun98b}, which would have no effect on particle observations by THEMIS.  Examination of the solid state telescope particle data from \\emph{Wind}\\cite{lin95a} (freely available on CDAWeb, see \\textbf{Data availability} below) \\textcolor{Red}{show} no significant energetic electron enhancements during any of the enhancements observed by THEMIS.  Finally, the electron data \\textcolor{Red}{show} no evidence of forward energy dispersion (i.e., higher energies arrive before lower due to a time-of-flight effect) characteristic solar energetic electrons\\cite{ergun98e} (Fig. \\ref{fig:exampleTIFPs} and Extended Data Figs \\ref{fig:THEMISFBPADExample}\\textbf{k}--\\ref{fig:THEMISFBPADExample}\\textbf{n}).  Note that the energetic ions exhibit slightly larger anisotropies than the electrons with anti-earthward intensities generally dominating (see Extended Data Fig. \\ref{fig:THEMISFBPADExample}).\n\n{\\noindent \\textbf{Exclusion of Earth's bow shock as source.}}  The most common shock acceleration mechanisms cited are diffusive shock acceleration\\cite{malkov01a}, shock drift acceleration\\cite{anagnostopoulos98a, park13a}, and the ``fast Fermi'' mechanism\\textcolor{Red}{\\cite{wu84b, leroy84a}}.  However, for electron acceleration at the Earth's bow shock we can rule out these mechanisms for the following reasons.  First, though diffusive shock acceleration predicts an isotropic particle distribution, it is more efficient for quasi-parallel ($\\theta{\\scriptstyle_{Bn}}$ $<$ 45$^{\\circ}$) shocks with pre-existing upstream electromagnetic fluctuations and the efficiency increases with particle kinetic energy\\cite{caprioli14a, park15a}.  This mechanism also cannot energize electrons below $\\sim$100 keV because their Larmor radii are smaller than the gradient scale lengths of the shock \\textcolor{Red}{ramp\\cite{hobara10a, mazelle10a}} and upstream electromagnetic fluctuations\\cite{wilsoniii16a}.  Further, this mechanism predicts an inverse energy dispersion (i.e., lower energies enhance first)\\cite{anagnostopoulos86a, sarris87a}, which is not observed in the energetic electron enhancements (Fig. \\ref{fig:exampleTIFPs} and Extended Data Fig. \\ref{fig:THEMISFBPADExample}).  Thus, diffusive shock acceleration is generally ignored as a mechanism for energizing electrons from thermal to relativistic energies at the Earth's bow shock.\n\n\\indent  Second, shock drift acceleration predicts anisotropic velocity distributions, perpendicular downstream of shock and field-aligned far upstream of the Earth's bow shock\\cite{anagnostopoulos94b, anagnostopoulos09a}, neither of which are observed (Fig. \\ref{fig:examplePADs} and Extended Data Fig. \\ref{fig:THEMISFBPADExample}).  The mechanism efficiency decreases with increasing ratio of shock speed to $\\cos{\\theta{\\scriptstyle_{Bn}}}$ relative to the particle thermal energy because this increases the minimum energy threshold requirement\\cite{guo14b}, where each interaction with the shock can produce energy gains of factors $>$10 for strong quasi-perpendicular shocks\\cite{ball01a}.  However, any electron distribution observed far upstream (e.g., near the foreshock disturbances) would be highly anisotropic along the magnetic field streaming away from the shock as previously observed\\cite{anderson79a, anderson81a}, inconsistent with the isotropic distributions we observe (Fig. \\ref{fig:examplePADs} and Extended Data Fig. \\ref{fig:THEMISFBPADExample}).  Thus, we can rule out shock drift acceleration at the Earth's bow shock as a source.\n\n\\indent  Third, fast Fermi acceleration assumes electrons undergo a single adiabatic reflection -- particle conserves its magnetic moment, $\\mu$ $=$ $m{\\scriptstyle_{e}} v{\\scriptstyle_{\\perp}}^{2}\/2 B{\\scriptstyle_{o}}$ $\\sim$ constant, during the reflection, where $m{\\scriptstyle_{e}}$ is the electron mass and $v{\\scriptstyle_{\\perp}}$ is the speed perpendicular to $\\mathbf{B}{\\scriptstyle_{o}}$ -- and gains energy proportional to the shock speed divided by $\\cos{\\theta{\\scriptstyle_{Bn}}}$\\cite{savoini10a, wu84b}.  Previous studies\\textcolor{Red}{\\cite{anderson79a, anderson81a, leroy84a}} proposed this mechanism as an explanation for the ``thin sheets'' of highly anisotropic (i.e., field-aligned streaming away from Earth's bow shock) energetic electrons.  To satisfy the condition $\\mu$ $\\sim$ constant, the magnetic gradient scale length must be larger than the particle Larmor radius.  Further, for significant electron energy gains this mechanism requires either very large shock speeds compared to typical electron thermal speeds (i.e., $\\sim$1500--3000 km\/s) or $\\theta{\\scriptstyle_{Bn}}$ $\\gtrsim$ 88$^{\\circ}$.  Since the Earth's bow shock is very slow (i.e., typically $\\sim$100--500 km\/s), the Larmor radii of electrons $\\gtrsim$ few hundred eV are comparable to the shock ramp \\textcolor{Red}{thickness\\cite{hobara10a, mazelle10a}}, and this mechanism can only energize electrons with large pitch-angles, fast Fermi acceleration is not expected to produce energies beyond several 10s of keV\\cite{gosling89b, sarris85a, savoini10a, wu84b}.  We thus rule out the Earth's bow shock as the source of these energetic electron enhancements.\n\n{\\noindent \\textbf{Exclusion of Earth's magnetosphere as source.}}  As discussed in the main article, the observed energetic electron distributions are not highly anisotropic along the magnetic field streaming away from the Earth as previously reported in studies arguing for a magnetospheric source\\cite{anagnostopoulos94b, krimigis78a, kronberg11a, sarris76a, sarris87a}.  These studies observed enhanced geomagnetic activity in association with bursts of energetic electrons, where they argued that substorms -- a fundamental mode of the terrestrial magnetosphere resulting in magnetospheric circulation\/flows and enhanced auroral activity\\cite{angelopoulos08a} -- led to an increased rate of magnetospheric ``leakage.''  During substorms geostationary spacecraft can observe intense and rapid changes in 10s of keV to MeV electron fluxes\\cite{angelopoulos08a}.  One measure of substorm activity can be given by the well known AE indices\\cite{mann08a}.  These previous studies defined enhanced geomagnetic activity as an AE index $>$ 200 nT.\n\n\\indent  Extended Data Fig. \\ref{fig:THEMISAEIndices} shows the AE indices with color-coded bars indicating the time ranges for the example foreshock disturbances in Fig. \\ref{fig:exampleTIFPs}.  The AE index (Extended Data Figs \\ref{fig:THEMISAEIndices}\\textbf{q}--\\ref{fig:THEMISAEIndices}\\textbf{t}) was $<$ 200 nT for $>$1 hour prior to and during the three example disturbances with energetic electron enhancements (Figs \\ref{fig:exampleTIFPs}\\textbf{m}--\\ref{fig:exampleTIFPs}\\textbf{o}).  Thus, we can rule out a magnetospheric source due to the inconsistent pitch-angle distributions and lack of enhanced geomagnetic activity.\n\n\\indent  \\textcolor{Red}{One previous study\\cite{paschmann88a} did find nearly isotropic energetic electrons ($\\sim$70--200 keV) within hot flow anomalies.  However, the authors explicitly state these are not a suprathermal tail of the thermal electrons and that they originated from the magnetosphere.  There are several important differences with our results:  (1) we observe a single power-law from 100s of eV to $\\geq$140 keV in many enhancements, suggesting a common acceleration mechanism;  (2) we always observe ions above $\\sim$10 keV with every foreshock disturbance;  (3) we observe most electron enhancements (8\/10) within short large-amplitude magnetic structures and foreshock bubbles, both of which are disconnected from the bow shock; and (4) the ``magnetic bottle'' model proposed to explain isotropy in the previous study\\cite{paschmann88a} would not work for short large-amplitude magnetic structures because they contract as they evolve, not expand.}\n\n{\\noindent \\textbf{Instrument details.}}  Quasi-static (i.e., finite gain from zero up to Nyquist frequency) vector magnetic field measurements ($\\mathbf{B}{\\scriptstyle_{o}}$) were obtained using the fluxgate magnetometer\\cite{auster08a} at 4 and 128 samples per second.  The data are presented in units of nanotesla [nT] in the geocentric solar ecliptic coordinate basis.\n\n\\indent  Particle data are stored as velocity distribution functions covering $4 \\ \\pi$ steradian over an energy range defined by instrument design.  The electrostatic analyzers\\cite{mcfadden08a, mcfadden08b}, onboard each THEMIS\\cite{angelopoulos08a} spacecraft, detect particles using anodes placed behind microchannel plate detectors and cover an energy range of few eV to over 25 keV.  The number and placement of the anodes determines the poloidal\/latitudinal angular resolution, which is usually $\\Delta \\theta$ $\\sim$ 22.5$^{\\circ}$ for both the electron and ion electrostatic analyzers (Note that $\\Delta \\theta$ can be as low as $\\sim$5$^{\\circ}$ in some ion instrument modes.).  The azimuthal\/longitudinal resolution, $\\Delta \\phi$, is limited by the spacecraft spin rate and instrument design and mode of operation, but is generally $\\sim$11.25$^{\\circ}$.  The energy resolution, $\\Delta E\/E$, is defined by the instrument design and mode of operation but is generally $\\sim$20\\% for both electrostatic analyzers.\n\n\\indent  At higher energies (i.e., $\\sim$30 keV to over 500 keV), data from the electron and ion solid state telescopes\\cite{ni11b, turner13b} onboard each THEMIS spacecraft were used.  Each detector is comprised of two identical telescopes mounted at different angles on the side of the spacecraft body\\cite{angelopoulos08a}.  The angular and energy resolution is usually $\\Delta \\theta$ $\\sim$ 30$^{\\circ}$--40$^{\\circ}$, $\\Delta \\phi$ $\\sim$ 22.5$^{\\circ}$, and $\\Delta E\/E$ $\\sim$ $\\sim$ 30\\%.  For instance, the $\\sim$293 keV energy bin actually includes energetic electrons with $\\sim$249--337 keV energies.\n\n\\indent  All particle data presented herein, except for the high energy ions, \\textcolor{Red}{were} taken \\textcolor{Red}{while in} burst \\textcolor{Red}{mode,} which has a time resolution equal to the spacecraft spin period (i.e., $\\sim$3 seconds).  Even though the high energy ion data were measured in a different mode, the time resolution is still the spin period for the intervals presented.\n\n{\\noindent \\textbf{Unit conversion.}}  The raw data are measured in units of counts, which correspond to the number of events with a pulse height exceeding a defined threshold specific to each instrument.  Conversion to intensity and\/or phase space density requires knowledge of the instruments efficiency\\cite{bordoni71a, goruganthu84a}, deadtime\\cite{meeks08a, schecker92a}, accumulation time, and optical geometric factor\\cite{curtis89a, paschmann98a, wuest07b}.\n\n\\indent  Particle intensity is defined as:  number of particles, per unit area, per unit solid angle, per unit time, per unit energy (e.g., \\# cm$^{-2}$ s$^{-1}$ sr$^{-1}$ eV$^{-1}$).  This unit is not a Lorentz invariant, thus it requires one taking into account the Compton-Getting effect.  Phase space density is defined as:  number of particles, per unit spatial volume, per unit ``velocity volume'' (e.g., \\# s$^{+3}$ cm$^{-3}$ km$^{-3}$).  This unit is a Lorentz invariant under conditions when phase space is incompressible (i.e., when Liouville's theorem reduces to $df\/dt$ $=$ 0), which is true for most cases in in situ space observations\\cite{paschmann98a}.\n\n\\indent  The exact details of the unit conversion can be found in the THEMIS calibration software, called SPEDAS, found at: \\textcolor{Blue}{\\seqsplit{http:\/\/themis.ssl.berkeley.edu\/index.shtml}}.\n\n{\\noindent \\textbf{Reference frames and coordinate systems.}}  All particle data shown herein has been transformed from the spacecraft to the bulk flow reference frame using a relativistically correct Lorentz transformation.  The distributions were converted into units of phase space density prior to any frame transformation, thus any anisotropies due to the Compton-Getting effect\\cite{compton35a, ipavich74a} have been removed.\n\n\\noindent  Reference frame transformations were performed through the following steps:\n\\begin{enumerate}[itemsep=0pt,parsep=0pt,topsep=0pt]\n  \\item  bulk flow velocities were determined from the first velocity moment of the low energy ($<$30 keV) ion velocity distributions\\cite{wilsoniii14a};\n  \\item  all particle distributions were converted from raw counts to phase space density;\n  \\item  particle distributions were then transformed into the bulk flow rest frame using a standard Lorentz velocity transformation;\n  \\item  the particle distributions shown in units of intensity were converted to phase space density prior to the transformation and then back to intensity.\n\\end{enumerate}\n\n\\indent  Direction-dependent spectra, as opposed to omnidirectional averages (i.e., average over all solid angles), called pitch-angle distributions were calculated through the following steps:\n\\begin{enumerate}[itemsep=0pt,parsep=0pt,topsep=0pt]\n  \\item  particle distributions were transformed into the bulk flow rest frame as described above;\n  \\item  construct particle velocity unit vector for the $m^{th}$ particle distribution, $\\hat{\\mathbf{v}}{\\scriptstyle_{i,j,k}}^{m}$, for the $k^{th}$ energy bin from the $i^{th}$ latitude and $j^{th}$ longitude detector look directions described in \\textbf{Instrument details} above;\n  \\item  define the projection angle, $\\alpha{\\scriptstyle_{i,j,k}}^{m}$, between $\\hat{\\mathbf{v}}{\\scriptstyle_{i,j,k}}^{m}$ and the respective orientation unit vector, $\\hat{\\mathbf{u}}^{m}$ (e.g., $\\mathbf{B}{\\scriptstyle_{o}}$\/$\\lvert \\mathbf{B}{\\scriptstyle_{o}} \\rvert$), at the measurement time of the $m^{th}$ particle distribution, where $\\alpha{\\scriptstyle_{i,j,k}}^{m}$ $=$ $\\cos^{-1}{\\left( \\hat{\\mathbf{v}}{\\scriptstyle_{i,j,k}}^{m} \\cdot \\hat{\\mathbf{u}}^{m} \\right)}$; and\n  \\item  define three cuts for the $m^{th}$ particle distribution by averaging data within $\\pm$22.5$^{\\circ}$ of the parallel, perpendicular, and anti-parallel directions defined by $\\hat{\\mathbf{u}}^{m}$.\n\\end{enumerate}\n\n\\noindent  The two relevant directions about which we oriented the particle distributions are the local quasi-static magnetic field vector, $\\mathbf{B}{\\scriptstyle_{o}}$, and the spacecraft-to-Earth unit vector, $\\hat{\\mathbf{e}}{\\scriptstyle_{SC}}$, at the time of each distribution.  Note that the formal definition of a pitch-angle distribution is constructed only with respect to $\\mathbf{B}{\\scriptstyle_{o}}$ but we use the term here for both orientations for brevity.\n\n\\indent  The high time resolution equivalent of the above algorithm involves only a few differences described below.\n\\begin{enumerate}[itemsep=0pt,parsep=0pt,topsep=0pt]\n  \\item  Instead of using a single $\\hat{\\mathbf{u}}^{m}$ for the $m^{th}$ particle distribution, we define $\\hat{\\mathbf{u}}{\\scriptstyle_{i,j,k}}^{m}$ for the $k^{th}$ energy bin from the $i^{th}$ latitude and $j^{th}$ longitude detector look directions of the $m^{th}$ particle distribution.\n  \\item  Now the pitch-angles are defined as $\\alpha{\\scriptstyle_{i,j,k}}^{m}$ $=$ $\\cos^{-1}{\\left( \\hat{\\mathbf{v}}{\\scriptstyle_{i,j,k}}^{m} \\cdot \\hat{\\mathbf{u}}{\\scriptstyle_{i,j,k}}^{m} \\right)}$.\n\\end{enumerate}\n\n\\noindent  The result is a pitch-angle distribution with fewer aliasing effects due to the use of a single $\\hat{\\mathbf{u}}^{m}$ averaged over the duration of the $m^{th}$ particle distribution (Extended Data Fig. \\ref{fig:exampleCompareLTHTBo}).\n\n\\indent  The exact details of the rotations and frame transformations can be found in the additional analysis software at:  \\textcolor{Blue}{\\seqsplit{https:\/\/github.com\/lynnbwilsoniii\/{wind\\_3dp\\_pros}}}.\n\n{\\noindent \\textbf{Particle data presentation.}}  All particle distributions are presented in the bulk flow rest frame in a physically significant coordinate basis, e.g., magnetic field-aligned coordinates.  We define the bulk flow reference frame as described above (in \\textbf{Reference frames and coordinate systems}).  The energy ranges listed above (in \\textbf{Instrument details}) are the measured midpoint kinetic energies in the spacecraft frame of reference.  Energy values will change under any Lorentz transformation.  Thus, the energies for the ion electrostatic analyzer data are not constant in time and show a large variability owing to the large variability of the bulk flow near foreshock disturbances.  In contrast, the low energy electron data above $\\sim$50 eV and all solid state telescope data suffer little energy change under these Lorentz transformations, thus can be approximated as constant in time.  Therefore, we show the low and high energy electron and high energy ion data as stacked line plots vs. time where each line corresponds to a different energy and the low energy ion data are presented as a dynamic energy spectrogram of energy vs. time with the color scale indicating the particle intensity.  Note that while the electron solid state telescope can measure electrons to $>$400 keV, for only the 2008-08-19 event in Fig. \\ref{fig:exampleTIFPs} were significant fluxes observed in energy bins $>$140 keV.\n\n{\\noindent \\textbf{Data availability.}}  The THEMIS data used in this paper is publicly available at: \\\\\n\\textcolor{Blue}{\\seqsplit{http:\/\/themis.ssl.berkeley.edu\/index.shtml}}.\n\n\\noindent  The \\emph{Wind} and STEREO radio data were taken from the S\/WAVES website at: \\\\ \n\\textcolor{Blue}{\\seqsplit{http:\/\/swaves.gsfc.nasa.gov\/data\\_access.html}}.\n\n\\noindent  Solar wind data was taken from the OMNI data products found on CDAWeb at: \\\\\n\\textcolor{Blue}{\\seqsplit{http:\/\/cdaweb.gsfc.nasa.gov}}.\n\n\\noindent  The \\emph{Wind} interplanetary shock list can be found at: \\\\\n\\textcolor{Blue}{\\seqsplit{https:\/\/www.cfa.harvard.edu\/shocks\/wi\\_data\/}}.\n\n{\\noindent \\textbf{Code availability.}}  The THEMIS instrument calibration software, called SPEDAS, can be found at: \\\\ \n\\textcolor{Blue}{\\seqsplit{http:\/\/themis.ssl.berkeley.edu\/index.shtml}};\n\n\\noindent and additional analysis software can be found at: \\\\\n\\textcolor{Blue}{\\seqsplit{https:\/\/github.com\/lynnbwilsoniii\/{wind\\_3dp\\_pros}}}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTaking the Yang-Mills model as the matter part of the equations of Einstein,\none finds a rich spectrum of stationary regular and non-regular solutions.\nThis in contrast\nwith the vacuum solution and Einstein-Maxwell(EM) solutions, where the Schwarzschild\nrespectively the Reissner-Nordstr{\\\"o}m(RN) solution are the only static black hole\nsolutions. In the Einstein-Yang-Mills(EYM) theory one finds not only the embedding of the\nAbelian RN black hole, but in addition there are genuine non-Abelian coloured black\nholes. Their co-existence gives rise to the violation of the no-hair conjecture,\nsince they carry the same (magnetic) charge. For an overview, see Volkov\nand Gal'tsov~\\cite{volk99}.\nAfter the discovery by Bartnik and McKinnon(BMK)~\\cite{bart88}\nof these non-Abelian particle-like and later the non-RN (hairy) black hole solutions\nof the SU(2) EYM theory~\\cite{smol96}, throughout instability\ninvestigations where done by several authors~\\cite{strau190,lav95,strau290,zhou91,breit97}.\nIt turns out that under spherically symmetric perturbations the BMK solution\nis unstable. The nth BMK solution has in fact 2n unstable modes,\ncomparable with the flat space-time 'sphaleron' solution. In the static case,\nit was recently conjectured~\\cite{mav98} that there are hairy black holes\nin the SU(N) EYM-theories with topological instabilities.\nIn order to analyze the stability of the solutions, one usually linearized\nthe field equations~\\cite{strau190,strau290}.\nWe conjectured that the conventional linear analysis is in fact inadequate\nto be applied to the situation where a singularity is formed~\\cite{Slag398}.\nOne better can apply the so-called multiple-scale or two-timing method,\ndeveloped by Choquet-Bruhat~\\cite{choq169,choq277,taub80}. This method\nis particularly useful for constructing uniformly valid approximations to\nsolutions of perturbation problems~\\cite{slag191}.\nAt the threshold of black hole\nformation it is found numerically~\\cite{chop193,chop296,garf97,bizon96}\nthat there is a critical parameter\nwhose value separates solutions containing black holes\nfrom those which do not. For a critical value one observes self-similarity:\nit produces itself (echoes) on progressively finer scales (Choptuik-scaling).\nIt is evident that the collapsing ball of field energy will produce\ngravitational waves, which will be coupled to the  YM-field perturbations.\nDue to the accumulation of echoes, the curvature diverges.\nClose related to this critical behavior is the comparable irregularities which\ncan be found in the Einstein-Skyrme(ES) model and the Einstein-Yang-Mills-Higgs(EYMH)\nmodel. In the latter model, space-like hypersurfaces develop for a critical\nvalue of $\\alpha =M_W\/gM_{pl}$ two distinct regions separated by a long throat,\nwith $M_W$ the YM scale and $M_{pl}=1\/\\sqrt{4\\pi G}$.\n\nThe solutions mentioned above need not be spherically symmetric. Since the\npioneering work of Stachel~\\cite{Sta66}, we know that stationary\ncylindrical symmetric rotating models possess intriguing features.\nFirst, the evolution of the 'classical' cylindrical gravitational wave-solution\nin the vacuum situation can be described in terms of two effects: the usual\ncylindrical reflections and a rotation of the polarization vector of the waves,\nan effect comparable with the Faraday rotation~\\cite{Piran85}.\nSecondly, the time-dependent solutions can be related to the cosmic-string\nsolution interacting with gravitational waves~\\cite{Xan186}.\nUsually the cosmic string solution, with its famous conical structure\n,is found in the U(1)-gauge Einstein-Higgs system.\nIt is remarkable that the cylindrical rotating vacuum solution possesses the\nsame conical structure. Moreover, it is asymptotically flat and free\nof singularities.\nThe physical interpretation can nicely be formulated using the Belinsky-Zakharov\nmethod of integrating the Einstein equations with the help of the Ernst\npotentials~\\cite{Eco88}. Using the C-energy description, one can conclude that the\nsolution represents solitonic gravitational waves interacting with a\nstraight-line cosmic string, where the asymptotic angle deficit depends\non the C-energy of the gravitational waves.\nWhen  a Maxwell field is incorporated, the physical interpretation of the solutions\nis not easily formulated. It is hard to believe that in the electrovac solution\nthe electro-magnetic waves do not contribute to the angle deficit~\\cite{Xan287}.\nMoreover, the mass per unit length of the cosmic string, which is a constant\nin the Xanthopoulos solution, will be determined by the energy-momentum tensor\nof the non-vacuum situation. In stead of 'electrifying' or 'magnetifying' the\nvacuum solutions via the Ernst formulation, one can better consider the coupled\nsystem of Einstein equations and matter field equations.\n\nHere we consider the Einstein-Yang-Mills system on a cylindrical symmetric\nrotating space-time. This situation con formally be obtained from the\naxially symmetric situation by complex substitution of $t\\to iz, z\\to it$~\\cite{kram80}.\nIn the non-rotating cylindrically symmetric situation, it was numerically found\n~\\cite{slag299} that the singular behavior at finite distance of the core as found\nin the supermassive situation, will be pushed to infinity for some critical values\nof the gauge coupling constant and one of the YM components.\n\nWe will use a (2+1)+1 reduction scheme~\\cite{Mae80}\nin order to obtain the suitable parameterization for the Yang-Mills potential.\nIn fact one replaces the YM field by a 2D Abelian one-form, a 2D Kaluza-Klein\ntwo-form, a 2D matrix-valued scalar and a 3D Higgs field~\\cite{gal98}.\n\nIn the Abelian counterpart model it is known that the space-time around\n(spinning) gauge strings could have, besides the usual angle deficit feature,\nexotic properties, such as the violation of causality~\\cite{Jen92,Sol92},\ntime-delay effects~\\cite{Har88}, helical structure\nof time~\\cite{Des92,Let95,Maz86} and frame-dragging~\\cite{Bon91}.\nA class of approximate solutions of the coupled Einstein-scalar-gauge field equations\non the ($r,z$)-plane of an axially symmetric space-time was found~\\cite{slag695}.\nSo a question addressed in this paper will be whether the string features will be\nmaintained in the EYM system.\nIn the general $(r,z)$-dependent situation, a nice overview was given by\nIslam~\\cite{Islam85}. One can proof via the Ernst formulation that for any solution\nof the stationary\nEinstein equations one can generate a corresponding solution of the EM equations.\nIn fact, one introduces two complex potentials, for which one finds two second-order\nelliptic differential equations.\nStarting with the Kerr solution, it is possible to use this correspondence to find\na solution of the EM equations leading to the Kerr-Newman solution.\nThis 'Ernst-route' fails in the EYM system, simply by the fact that one\nof the metric components does not decouple from the other ones.\nIt is also known that the electrovac staticity theorem~\\cite{heus93} does not generalize to the EYM\nsystem. In fact, spinning EYM systems must be electrically charged. So another question\naddressed in this paper is whether electrically and magnetically charged solution will exist\nin the EYM system.\nFinally, this is not the complete story: the investigations can be extended to the\nmodel where the interior solution is  properly matched onto the exterior\nsolution of the string~\\cite{slag496}. This is currently under study by the author.\n\nThe plan of this paper is as follows. In Sec. II we derive the field equation\non an cylindrical symmetric space time. In Sec. III we present the numerical solution.\nIn section IV we present a approximate solution using the multiple-scale method and\nin Sec. V we summarize and analyse our results.\n\n\\section{The field equations of the Einstein Yang-Mills system}\n\nConsider the Lagrangian of the SU(2) EYM system\n\\begin{equation}\nS=\\int d^4x\\sqrt{-g}\\Bigl[\\frac{{\\cal R}}{16\\pi G}-\n\\frac{1}{4}{\\cal F}_{\\mu\\nu}^a{\\cal F}^{\\mu\\nu a}\\Bigr],\n\\end{equation}\nwith the YM field strength\n\\begin{equation}\n{\\cal F}_{\\mu\\nu}^a =\\partial_\\mu A_\\nu^a-\\partial_\\nu\nA_\\mu^a+g\\epsilon^{abc}A_\\mu^bA_\\nu^c,\n\\end{equation}\ng the gauge coupling constant, G Newton's constant, $A_\\mu^a$ the gauge potential,\nand ${\\cal R}$ the curvature scalar. The field equations then become\n\\begin{equation}\nG_{\\mu\\nu}=8\\pi G {\\cal T}_{\\mu\\nu},\n\\end{equation}\n\\begin{equation}\n{\\cal D}_\\mu {\\cal F}^{\\mu\\nu a}=0,\n\\end{equation}\nwith ${\\cal T}$ the energy-momentum tensor\n\\begin{equation}\n{\\cal T}_{\\mu\\nu}={\\cal F}_{\\mu\\lambda}^a{\\cal F}_\\nu^{\\lambda a}\n-\\frac{1}{4}g_{\\mu\\nu}{\\cal F}_{\\alpha\\beta}^a {\\cal F}^{\\alpha\\beta a},\n\\end{equation}\nand ${\\cal D}$ the gauge-covariant derivative,\n\\begin{equation}\n{\\cal D}_\\alpha{\\cal F}\n_{\\mu\\nu}^a\\equiv \\nabla_\\alpha{\\cal F}_{\\mu\\nu}^a+g\\epsilon^{abc}A_\\alpha^b{\\cal F}\n_{\\mu\\nu}^c.\n\\end{equation}\nLet us consider the cylindrical space time\n\\begin{equation}\nds^2=-\\Omega^2(dt^2-dr^2)+f(dz+\\omega d\\varphi)^2+\\frac{r^2}{f}d\\varphi ^2,\n\\end{equation}\nwhere $\\Omega, f$ and $\\omega $ are functions of t and r.\nThe problem is, which ansatz one needs for the YM field.\nIt is often argued that Einsteins gravity on a three dimensional\nspace time shows some similarity with Chern Simons theory with the Poincar\\'e\ngroup being the underlying gauge group. So when trying to find the dimensional\nreduction for the stationary axially symmetric EYM system, one can be guided by\nthe results found by using the (2+1)+1 reduction scheme suggested by Maeda, Sasaki,\nNakamura and Miyama~\\cite{Mae80}, and later worked out by Gal'tsov~\\cite{gal98}.\nOne writes in general\n\\begin{equation}\nds^2=-e^\\psi (dt+v_idx^i)^2+e^{2\\phi-\\psi}(d\\varphi +k_adx^a)^2+g_{ab}dx^adx^b,\n\\end{equation}\nwhere $v_idx^i=\\omega (d\\varphi +k_adx^a)+\\nu_adx^a, k_a$ an one-form, generating\na field strength  $\\kappa_{ab}=\\partial_ak_b-\\partial_bk_a$. The one-form $\\nu_a$ give rise to the\n2D field strength $\\omega_{ab}=\\partial_a\\nu_b-\\partial_b\\nu_a+\\omega\\kappa_{ab}$(a,b=1,2).\nThe YM potential is then written as\n\\begin{equation}\nA_\\mu dx^\\mu ={\\bf a}_adx^a +{\\bf \\Phi}(dt+\\nu_idx^i)+{\\bf \\Psi}(d\\varphi +k_adx^a)\n\\end{equation}\nHere ${\\bf a}$ is the dynamical part of the YM field, parameterized by a 2D complex\nAbelian one-form, ${\\bf \\Phi}$ a 3D Higgs field and ${\\bf \\Psi}$ a matrix-valued\nscalar. In our simplified case of metric (7), we have $\\nu_1=-\\omega k_1, k_2=\\nu_2 =0$ and\nthe YM potential can be as\n\\begin{equation}\nA_\\mu ={\\bf \\Phi}(dz+\\omega d\\varphi )+{\\bf \\Psi}(d\\varphi -k_1dt) +{\\bf a}_adx^a\n\\end{equation}\nThe YM-part of the Lagrangian can then be written as\n\n\\begin{eqnarray}\n{\\cal L}_{YM}=\\Omega^2r\\Bigl[\\Bigl\\{f_{ab}f^{ab}+\\frac{2f}{r^2}(D_a{\\bf\\Psi} +{\\bf\\Phi}\n\\partial_a\\omega -a_a^{'})(D^a{\\bf\\Psi}+{\\bf\\Phi}\\partial^a\\omega\n -a^{' a}) \\Bigr\\} \\cr +\\frac{2}{f}\\Bigl\\{D_a{\\bf\\Phi}D^a{\\bf\\Phi}\n +\\frac{f}{r^2}({\\bf\\Phi}^{'}+g[{\\bf\\Phi},{\\bf\\Psi}])^2\\Bigr\\}\\Bigl],\n\\end{eqnarray}\nwith\n\\begin{equation}\nD_a{\\bf\\Psi}=\\partial_a{\\bf\\Psi}+g[a_a,{\\bf\\Psi}]-k_a{\\bf\\Psi}^{'},\n\\end{equation}\n$f_{ab}$ the 2D field strength\n\\begin{equation}\nf_{ab}=\\partial_a a_b-\\partial_b a_a +g[a_a,a_b]+a_a^{'}k_b-a_b^{'}k_a+{\\bf \\Phi}\\omega_{ab}+\n{\\bf \\Psi}k_{ab},\n\\end{equation}\nand a prime denoting the partial derivative with respect to $\\varphi$.\n\nLet us consider the following parameterization\n\\begin{eqnarray}\na_1=A_2(t,r)\\tau_\\varphi ,\\quad &&a_2=A_1(t,r)\\tau_\\varphi,\\quad\n{\\bf\\Psi}=W_1(t,r)\\tau_r +(W_2(t,r)-1)\\tau_z,\\cr\n&& {\\bf\\Phi}=\\Phi_1(t,r)\\tau_r+\\Phi_2(t,r)\\tau_z,\n\\end{eqnarray}\nwith $\\tau_i$ the usual set of orthonormal vectors.\nFrom the condition $T_{tt}-T_{rr}=0$, we obtain\n\\begin{equation}\n\\partial_t A_1=\\partial_r A_2,\\quad  W_2=\\frac{\\Phi_2 W_1}{\\Phi_1}+\\frac{g-1}{g}.\n\\end{equation}\n\nFrom the Einstein equations and the YM equations we obtain the set of\npartial differential equations for $\\Omega , f , \\omega , \\Phi_1 , \\Phi_2$\nand $W_1$.\nIt follows  from a combination of the YM equations, that $A_1$\nand $A_2$ can be expressed in $\\Phi_1$ and $\\Phi_2$:\n\\begin{eqnarray}\nA_1 =\\frac{\\Phi_1\\partial_r\\Phi_2 -\\Phi_2\\partial_r\\Phi_1}{g(\\Phi_1^2+\\Phi_2^2)},\n\\qquad A_2 =\\frac{\\Phi_1\\partial_t\\Phi_2-\\Phi_2\\partial_t\\Phi_1}{g(\\Phi_1^2+\\Phi_2^2)}\n\\end{eqnarray}\nFor the gauge $\\partial_t A_2=\\partial_r A_1$\nand hence $\\partial_t^2 A_i-\\partial_r^2 A_i =0$, we obtain from the Einstein equations Eq.(3)\n\\begin{eqnarray}\n\\partial_t^2\\Omega -\\partial_r^2\\Omega +\\frac{f^2\\Omega}{4r^2}\\Bigl[(\\partial_t\\omega )^2\n-(\\partial_r\\omega)^2\\Bigr]+\\frac{1}{\\Omega}\\Bigl[(\\partial_r\\Omega )^2\n-(\\partial_t\\Omega)^2\\Bigr]+\n\\frac{\\Omega}{4f^2}\\Bigl[(\\partial_t f)^2-(\\partial_r f)^2\\Bigr]\n+ \\frac{\\Omega}{2fr}\\partial_r f=0,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t^2 f -\\partial_r^2 f +\\frac{1}{f}\\Bigr[(\\partial_r f)^2-(\\partial_t f)^2\n\\Bigr]+\\frac{f^3}{r^2}\\Bigl[(\\partial_r\\omega)^2-(\\partial_t\\omega)^2\\Bigr]\n-\\frac{1}{r}\\partial_r f = \\cr \\frac{8\\pi G}{r^2\\Phi_1^4}\\Bigl[(\\Phi_1^4r^2-f^2\\Phi_2^2W^2)(\n(\\partial_r\\Phi_1)^2-(\\partial_t\\Phi_1)^2) +(\\Phi_1^4r^2-f^2\\Phi_1^2W^2)((\\partial_r\\Phi_2)^2\n-(\\partial_t\\Phi_2)^2) \\cr +f^2\\Phi_1^2(\\Phi_1^2+\\Phi_2^2)\\Bigl((\\partial_tW)^2-(\\partial_rW)^2\n+\\Phi_1^2\\Bigl((\\partial_t\\omega)^2-(\\partial_r\\omega)^2\\Bigr)\\Bigr) \\cr\n+2f^2\\Phi_1^3(\\Phi_1^2+\\Phi_2^2)(\\partial_t\\omega\\partial_tW-\\partial_r\\omega\\partial_rW)\n+2f^2\\Phi_2^2\\Phi_1W(\\partial_rW\\partial_r\\Phi_1-\\partial_tW\\partial_t\\Phi_1)\\cr\n+2f^2\\Phi_1^3\\Phi_2W(\\partial_t\\Phi_2\\partial_t\\omega -\\partial_r\\Phi_2\\partial_r\\omega )\n+2f^2\\Phi_1^2\\Phi_2W(\\partial_tW\\partial_t\\Phi_2-\\partial_rW\\partial_r\\Phi_2)\\cr\n+2f^2\\Phi_1\\Phi_2W^2(\\partial_r\\Phi_2\\partial_r\\Phi_1-\\partial_t\\Phi_2\\partial_t\\Phi_1)\n+2f^2\\Phi_2^2\\Phi_1^2W(\\partial_r\\Phi_1\\partial_r\\omega-\\partial_t\\Phi_1\\partial_t\\omega )\\cr\n+g\\Phi_1^2(\\Phi_1^2r^2-f^2W^2)\\Bigl(2A_1(\\Phi_2\\partial_r\\Phi_1-\\Phi_1\\partial_r\\Phi_2)\n+2A_2(\\Phi_1\\partial_t\\Phi_2-\\Phi_2\\partial_t\\Phi_1)+g(\\Phi_1^2+\\Phi_2^2)(A_1^2-A_2^2)\\Bigl)\\Bigl]\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t^2\\omega -\\partial_r^2\\omega +\\frac{2}{f}(\\partial_tf\\partial_t\\omega\n-\\partial_rf\\partial_r\\omega )+\\frac{1}{r}\\partial_r\\omega\n=\\frac{16\\pi G}{f\\Phi_1^2}\\Bigl[W\\Phi_1 ((\\partial_r\\Phi_2)^2-(\\partial_t\\Phi_2)^2)\\cr\n+\\Phi_2W(\\partial_t\\Phi_2\\partial_t\\Phi_1-\\partial_r\\Phi_2\\partial_r\\Phi_1)\n+\\Phi_2\\Phi_1^2(\\partial_r\\Phi_2\\partial_r\\omega -\\partial_t\\Phi_2\\partial_t\\omega )\n+\\Phi_2\\Phi_1(\\partial_r\\Phi_2\\partial_rW-\\partial_t\\Phi_2\\partial_tW) \\cr\n+\\Phi_1^2(\\partial_r\\Phi_1\\partial_rW-\\partial_t\\Phi_1\\partial_tW)\n+\\Phi_1^3(\\partial_r\\Phi_1\\partial_r\\omega-\\partial_t\\Phi_1\\partial_t\\omega ) \\cr\n+g^2\\Phi_1W(\\Phi_1^2+\\Phi_2^2)(A_1^2-A_2^2)\n+2g\\Phi_1^2W(A_2\\partial_t\\Phi_2-A_1\\partial_r\\Phi_2)+2g\\Phi_1\\Phi_2W(A_1\\partial_r\\Phi_1\n-A_2\\partial_t\\Phi_1)\\Bigr],\n\\end{eqnarray}\nand from the YM equations Eq. (4)\n\\begin{eqnarray}\n\\partial_t^2\\Phi_1-\\partial_r^2\\Phi_1-\\frac{1}{r}\\partial_r\\Phi_1+\\frac{f^2\\Phi_1}{r^2}\n\\Bigl((\\partial_r\\omega)^2-(\\partial_t\\omega)^2\\Bigl)\n+\\frac{1}{f}(\\partial_rf\\partial_r\\Phi_1-\\partial_tf\\partial_t\\Phi_1) \\cr\n+\\frac{f^2}{r^2}(\\partial_r\\omega\\partial_rW-\\partial_t\\omega\\partial_tW)\n+\\frac{gf^2\\Phi_2W}{\\Phi_1r^2}(A_1\\partial_r\\omega - A_2\\partial_t\\omega )\\cr\n+\\frac{g\\Phi_2}{f}(A_1\\partial_rf-A_2\\partial_tf)\n+2g(A_2\\partial_t\\Phi_2-A_1\\partial_r\\Phi_2)-\\frac{gA_1\\Phi_2}{r}+g^2\\Phi_1(A_1^2-A_2^2)=0\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t^2\\Phi_2-\\partial_r^2\\Phi_2 -\\frac{1}{r}\\partial_r\\Phi_2+\\frac{f^2\\Phi_2}{r^2}\n\\Bigl((\\partial_r\\omega)^2-(\\partial_t\\omega)^2\\Bigr)\n+\\frac{1}{f}(\\partial_rf\\partial_r\\Phi_2-\\partial_tf\\partial_t\\Phi_2)\n+\\frac{f^2\\Phi_2}{r^2\\Phi_1}(\\partial_r\\omega\\partial_rW-\\partial_t\\omega\\partial_tW)\\cr\n+\\frac{f^2\\Phi_2W}{r^2\\Phi_1^2}(\\partial_t\\omega\\partial_t\\Phi_1-\\partial_r\\omega\\partial_r\\Phi_1)\n+\\frac{f^2W}{r^2\\Phi_1}(\\partial_r\\omega\\partial_r\\Phi_2-\\partial_t\\omega\\partial_t\\Phi_2)\n+2g(A_1\\partial_r\\Phi_1-A_2\\partial_t\\Phi_1) \\cr +\\frac{g\\Phi_1}{f}(A_2\\partial_tf-A_1\\partial_rf)\n+\\frac{gf^2W}{r^2}(A_2\\partial_t\\omega-A_1\\partial_r\\omega)+g^2\\Phi_2(A_1^2-A_2^2)\n+\\frac{gA_1\\Phi_1}{r}=0\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t^2 W_1-\\partial_r^2 W_1 +\\frac{1}{r}\\partial_rW\n+\\frac{1}{f}(\\partial_tf\\partial_tW-\\partial_rf\\partial_rW)\n+\\frac{\\Phi_1}{f}(\\partial_rf\\partial_r\\omega -\\partial_tf\\partial_t\\omega ) \\cr\n+\\partial_t\\omega\\partial_t\\Phi_1-\\partial_r\\omega\\partial_r\\Phi_1\n+\\frac{2g\\Phi_2}{\\Phi_1}(A_2\\partial_tW-A_1\\partial_rW)\n+g\\Phi_2(A_2\\partial_t\\omega-A_1\\partial_r\\omega) \\cr\n+\\frac{2gW\\Phi_2}{\\Phi_1^2}(A_1\\partial_r\\Phi_1-A_2\\partial_t\\Phi_1)+\n\\frac{2gW}{\\Phi_1}(A_2\\partial_t\\Phi_2-A_1\\partial_r\\Phi_2)  +\\frac{gW\\Phi_2}{f\\Phi_1}\n(A_2\\partial_tf-A_1\\partial_rf)\\cr +g^2W(A_1^2-A_2^2)+\\frac{gA_1\\Phi_2W}{r\\Phi_1}\n-\\frac{16\\pi G}{f}\\Bigl[W\\Bigl((\\partial_t\\Phi_2)^2-(\\partial_r\\Phi_2)^2\\Bigr)\\cr\n+\\Phi_1(\\partial_t\\Phi_1\\partial_tW-\\partial_r\\Phi_1\\partial_rW)\n+\\Phi_1^2(\\partial_t\\omega\\partial_t\\Phi_1-\\partial_r\\omega\\partial_r\\Phi_1) \\cr\n+\\Phi_2(\\partial_t\\Phi_2\\partial_tW-\\partial_r\\Phi_2\\partial_rW)\n+\\frac{W\\Phi_2}{\\Phi_1}(\\partial_r\\Phi_2\\partial_r\\Phi_1-\\partial_t\\Phi_2\\partial_t\\Phi_1)\n+\\Phi_1\\Phi_2(\\partial_t\\Phi_2\\partial_t\\omega-\\partial_r\\Phi_2\\partial_r\\omega) \\cr\n+2gW\\Phi_1(A_1\\partial_r\\Phi_2-A_2\\partial_t\\Phi_2)\n+2gW\\Phi_2(A_2\\partial_t\\Phi_1-A_1\\partial_r\\Phi_1)+g^2\\Phi_1^2W(\\Phi_1^2+\\Phi_2^2)(A_2^2-A_1^2)\\Bigr]=0.\n\\end{eqnarray}\n\n\\section{Numerical solution}\nIn the vacuum situation, the behavior of the cylindrical wave solution is\nwell established~\\cite{Bon294,Tom89}. One can have reflection of incoming\ninto outgoing waves with the same polarization. Further, there is\na non-linear effect describing the interaction between the two different\npolarization $+$ and $\\times$ modes. If an\noutgoing cylindrical wave is linear polarized, its polarization vector\nrotates as it propagates. This effect  is often called 'Faraday' rotation and is given by\n\\begin{equation}\n\\tan\\theta_A=\\frac{A_x}{A_+},\\quad \\tan\\theta_B=\\frac{B_x}{B_+}\n\\end{equation}\nwith\n\\begin{equation}\nA_{+,x}=\\frac{1}{f}(\\partial_tf\\pm\\partial_rf),\\quad B_{+,x}=\\frac{f}{r}(\\partial_t\\omega\n\\pm\\partial_r\\omega)\n\\end{equation}\n\n\nA wave emitted from the axis of symmetry behaves like a shock\nfront and is given by\n\\begin{equation}\nf=w(a^2+1)\/(a^2+w^2),\\quad \\omega=ra(1-w^2)\/w(a^2+1),\n\\quad \\Omega^2 f=b\\sqrt{r}\/\\sqrt{t^2-r^2},\n\\end{equation}\nwith $w=(t-\\sqrt{t^2-r^2})\/r$ and a and b arbitrary constants.\nOne can generate a two-soliton solution and it is found that near the axis\n$\\log (\\Omega^2 f)$, which also describes the 'C-energy',\ndoes not approach zero. So some matter field\nmust act as a possible source of the two-soliton field. This curious\nconclusion can also be formulated in context of the cosmic string with mass\nunit length $m=p\/(2p-2)$, with p a constant originating from the static solution\n$f=r^{2p}, \\Omega =ar^{p^2-p}$, which on his turn is related to the\nangle deficit of the string, given by\n\\begin{equation}\n\\delta\\phi=2\\pi(1-\\frac{1}{\\sqrt{\\Omega^2f}})\n\\end{equation}\nWhen a 'Rosen'-pulse $\\ln f =\\int_0^{t-r}f(\\beta)d\\beta \/ \\sqrt{(t-\\beta)^2-r^2}$ is emitted,\n$\\log (\\Omega^2 f)$ changes by an amount $-k^2$, where k depends solely on $f(\\beta )$.\nSo after the pulse, one has a different string, which let Marder~\\cite{mar58}\nto conclude that the mass per unit length has decreased by an amount\n$\\sim k^2$. Because $\\log (\\Omega^2f)$ is related to the 'C'-energy, this confirms that\nthe waves carry away energy and are evidently generated by a source on the\naxis. So a continuous change in the parameter p  may produce suddenly\nand inexplicable change in physical meaning.\n\n\nFurther, it was emphasized in the vacuum case that there is a reflection of the outgoing\nwave into an ingoing wave and visa versa.\n\n\nFrom the Einstein-Maxwell case~\\cite{Xan186,Xan287} it is known that there exist\nstable open cosmic strings coupled with gravity and electromagnetism.\nWhen gravitational and electromagnetic waves are generated,the angle deficit\nis increased asymptotically and near the axis independent of the coupling\nof the Maxwell field to gravity.\nFurther, it was emphasized that the electromagnetic waves do contribute\nto the rotation of the spacetime by effecting the 'dragging' of the Killing field\n$\\frac{\\partial}{\\partial z}$, but not to the angle deficit.\n\n\nIn our EYM case we solved Eq.(17)-(22) for  the initial values\n\\begin{eqnarray}\n\\Omega(0,r)=C_1r^{p^2-p},\\quad f(0,r)=r^{2p}, \\quad \\Phi_1(0,r)=1-e^{-C_2r},\n\\quad \\Phi_2(0,r)=e^{-C_3r}-1\n\\end{eqnarray}\nFor  $W_1$ we will take as initial value a node number 1\nor node number 2 function, i.e., $W_1(0,r)=-\\tanh(r-1)$, or $W_1 (0,r)=1-e^{-C_4r^2}$.\nHere are $C_i$ some constants.\n\nWe used a 100x100 grid and applied the method of lines with cubic Hermite polynomials.\nThe roundoff error remained below 0.001.\nIn figure 1 we took  $\\omega(0,r)=\\frac{3.5r}{r^2+1}$ and for $W_1$ the\nnode number 2 function. We plotted respectively: $\\Omega$, f, $\\omega$, $\\Phi_1$,\n$W_1$, $\\Phi_2$, the angle deficit, the gravitational puls wave $\\Omega^2f$, $g_{\\varphi\\varphi}$,\nand $W_2$.\nIn figure 2 we took a different initial $\\Omega$ (p=0.1 in stead of 0.25).\nWe obtain a solitonlike solution. The behavior depends  on the initial mass\nper unit length of the string, just as in the vacuum case.\nIn figure 3 we took the node number 1 function for $W_1$. We observe that the angle deficit\nchanges significantly as well as the gravitational outgoing pulse. So the Yang-Mills waves\nhas an impact on the gravitational waves. In figure 4 we took p=0.1 in stead of\np=0.25. We see that the behavior of the solution changes significantly: the angle deficit\nclose to the r-axis increases, while the gravitational puls 'hangs' close to the r-axes.\n\nIn figure 5 we plotted a long-time run for a different initial $\\omega$: an outgoing wave\nis reflected into an ingoing wave.\n\nAn important question is if the solution admits gauge-charges.\nThe gauge-charges $Q_E,Q_M$ (electric and magnetic) are given by\n\n\\begin{eqnarray}\nQ_E=\\frac{1}{4\\pi}\\oint_S\\vert ^*F\\vert,\\quad Q_M=\\frac{1}{4\\pi}\\oint_S\\vert F\\vert,\n\\end{eqnarray}\nwith F the YM field strength, $^*F$ the dual tensor and the integration over\na two-sphere at spatial infinity.\nIn our situation, we obtain for the charges\n\\begin{eqnarray}\nQ_E=\\int r\\Bigl\\{\\Bigl[\\partial_t(W_1+\\omega\\Phi_1)+\ngA_2(W_2-1+\\omega\\Phi_2)+A_2\\Bigr]\\tau_r +\\Bigl[\\partial_t(W_2+\\omega\\Phi_2)\n-gA_2(W_1+\\omega\\Phi_1)\\Bigr]\\tau_z\\Bigr\\}dr\n\\end{eqnarray}\n\\begin{eqnarray}\nQ_M=\\int\\frac{r}{\\Omega^2} \\Bigl\\{\\Bigl[\\frac{1}{f}(\\partial_r\\Phi_1+gA_1\\Phi_2\n-\\frac{f\\omega}{r^2}(\\Phi_1\\partial_r\\omega+\\partial_rW_1+A_1+gA_1(W_2-1))\\tau_r\\Bigr] \\cr\n+\\Bigl[\\frac{1}{f}(\\partial_r\\Phi_2-gA_1\\Phi_1)-\\frac{f\\omega}{r^2}(\\partial_rW_2+\\Phi_2\\partial_r\\omega-gA_1W_1)\\Bigr]\\tau_z)\\Bigr\\}dr,\n\\end{eqnarray}\nwhere we used Eq. (15). So in general the solution will possess gauge charges.\n\n\\section{An approximate wave solution to first order}\n\nIn the appendix we presented the multiple-scale method for the EYM system.\nOn the metric Eq. (7) we have for the tetrad component:\n$l_\\mu =(-1,1,0,0)$ and\n$l^\\mu =\\frac{1}{\\Omega^2}(1,1,0,0)$, which implies that the divergence of the\nnull congruence $l^\\mu$ becomes\n\\begin{equation}\n\\nabla_\\mu l^\\mu =\\frac{1}{\\Omega^2 r},\n\\end{equation}\nas it should be.\nFrom Eq. (A10) we obtain $B_0^a=-B_1^a$, or, in the notation of Eq. (14),\n$\\dot A_1^{(1)}=-\\dot A_2^{(1)}$.\nFrom Eq. (A8) we obtain\n\\begin{eqnarray}\n\\dot h_{01}=-\\frac{1}{2}(\\dot h_{00}+\\dot h_{11}),\\quad\n\\dot h_{12}=-\\dot h_{02},\\quad \\dot h_{03}=-\\dot h_{13},\n\\quad \\dot h_{23}=-\\frac{\\bar g^{22}\\dot h_{22}+\\bar g^{33}\\dot h_{33}}{2\\bar g^{23}}.\n\\end{eqnarray}\nFirst of all we have just as Eq. (15),\n\\begin{eqnarray}\n\\partial_t\\bar A_1=\\partial_r\\bar A_2,\\quad \\bar W_2=\\frac{\\bar \\Phi_2\\bar W_1}{\\bar \\Phi_1}\n+1-\\frac{1}{g}\n\\end{eqnarray}\nFor simplicity we will take $\\Phi_2=0$ and try to find an approximate wave solution to second order.\nFrom the propagation equations Eq. (A14) and (A18) we obtain\n\\begin{eqnarray}\n\\bar\\Phi_1\\partial_\\varphi\\dot \\omega^{(1)}+\\partial_\\varphi\\dot W_1^{(1)}=0,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_\\varphi\\dot W_2^{(1)}=0.\n\\end{eqnarray}\nSo we will take $\\dot\\omega^{(1)}$ and $\\dot W_2^{(1)}$ independent of $\\varphi$.\nFurther,\n\\begin{eqnarray}\n\\partial_t\\dot A_1^{(1)}+\\partial_r\\dot A_1^{(1)}\n=\\ddot A_1^{(2)}+\\ddot A_2^{(2)}-\\frac{g\\bar f\\bar\\Omega^2}{r^2}\\bar W_1\\dot W_2^{(1)},\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\partial_t\\dot\\Phi_1^{(1)}+\\partial_r\\dot\\Phi_1^{(1)}=-\\frac{\\dot\\Phi_1^{(1)}}{2r}\n+\\frac{\\dot f^{(1)}}{2\\bar f}(\\partial_t\\bar\\Phi_1 +\\partial_r\\bar\\Phi_1 )\n+\\frac{\\dot\\Phi_1^{(1)}}{2\\bar f}(\\partial_t\\bar f +\\partial_r\\bar f )\n\\cr +\\frac{\\bar f^2}{r^2}(\\dot\\omega^{(1)}\\bar\\Phi_1+\\frac{1}{2}\\dot W_1^{(1)})\n(\\partial_t\\bar\\omega +\\partial_r\\bar\\omega )\n+\\frac{\\bar f^2\\dot\\omega^{(1)}}{2r^2}(\\partial_t\\bar W_1+\\partial_r\\bar W_1),\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t\\dot W_2^{(1)}+\\partial_r\\dot W_2^{(1)}=-\\frac{1}{2\\bar f}\\dot W_2^{(1)}\n(\\partial_t\\bar f+\\partial_r\\bar f)+\\frac{1}{2r}\\dot W_2^{(1)}+g(\\bar A_1+\\bar A_2)\n\\Bigl(\\dot W_1^{(1)}+\\frac{\\bar\\Phi_1\\dot\\omega^{(1)}}{2}+\\frac{\\dot f^{(1)}\\bar W_1}{2\\bar f}\n\\Bigr)\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\partial_t\\dot W_1^{(1)}+\\partial_r\\dot W_1^{(1)}=\\frac{1}{2r}\\dot W_1^{(1)}\n-\\frac{1}{2\\bar f} (\\dot W_1^{(1)} +\\bar\\Phi_1\\dot\\omega^{(1)})\n(\\partial_t\\bar f+\\partial_r\\bar f)\n-\\bar\\Phi_1(\\partial_t\\dot\\omega^{(1)}+\\partial_r\\dot\\omega^{(1)})-\n\\frac{\\dot f^{(1)}}{2\\bar f}(\\partial_r\\bar W_1+\\partial_t\\bar W_1)\\cr\n-\\frac{1}{2}(\\partial_t\\bar\\omega+\\partial_r\\bar\\omega)(\\dot\\Phi_1^{(1)}+\\frac{\\bar\\Phi_1\n\\dot f^{(1)}}{\\bar f})-\\frac{\\dot\\omega^{(1)}}{2}(\\partial_t\\bar\\Phi_1+\\partial_r\\bar\\Phi_1)\n+\\frac{\\bar\\Phi_1\\dot\\omega^{(1)}}{2r}-g\\dot W_2^{(1)}(\\bar A_1+\\bar A_2)\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t\\dot\\Omega^{(1)}+\\partial_r\\dot\\Omega^{(1)}=-\\frac{\\bar\\Omega\\dot f^{(1)}}{4\\bar f^2}\n(\\partial_t\\bar f+\\partial_r\\bar f)+\\frac{\\dot\\Omega^{(1)}}{\\bar\\Omega}\n(\\partial_t\\bar\\Omega +\\partial_r\\bar\\Omega)+\\frac{\\bar\\Omega\\dot f^{(1)}}{4r\\bar f}\n+\\frac{1}{4\\bar\\Omega}(\\ddot k_{00}+\\ddot k_{11}+2\\ddot k_{01})\\cr\n-\\frac{8\\pi G \\bar\\Omega \\bar f}{r^2}\\Bigl[\\frac{\\dot\\Phi_1^{(1)}}{\\bar f^2}\n(\\partial_t\\bar\\Phi_1+\\partial_r\\bar\\Phi_1)\n+(\\dot\\omega^{(1)}\\bar\\Phi_1+\\dot W_1^{(1)})\\Bigl(\\partial_t\\bar W_1+\\partial_r\\bar W_1\n+\\bar\\Phi_1(\\partial_t\\bar\\omega\n+\\partial_r\\bar\\omega)\\Bigr)-g\\dot W_2^{(1)}\\bar W_1(\\bar A_1+\\bar A_2)\\Bigr]\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\partial_t\\dot f^{(1)}+\\partial_r\\dot f^{(1)}=\\frac{\\dot f^{(1)}}{\\bar f}(\\partial_t \\bar f\n+\\partial_r\\bar f)-\\frac{\\dot f^{(1)}}{2r}+\\frac{\\bar f^3(\\partial_t\\bar\\omega +\n\\partial_r\\bar\\omega )}{r^2}\\dot\\omega^{(1)}-\\frac{8\\pi G\\bar f^2}{r^2}\\Bigl((\n\\dot\\omega^{(1)}\\bar\\Phi_1+\\dot W_1^{(1)})(\\partial_t\\bar W_1+\\partial_r\\bar W_1) \\cr\n+(\\bar\\Phi_1^2\\dot\\omega^{(1)}+\\bar\\Phi_1\\dot W_1^{(1)})(\\partial_t\\bar\\omega\n+\\partial_r\\bar\\omega)\n+\\frac{3r^2\\dot\\Phi_1^{(1)}}{\\bar f^2}(\\partial_t\\bar\\Phi_1+\\partial_r\\bar\\Phi_1)\\Bigr),\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\partial_t\\dot\\omega^{(1)}+\\partial_r\\dot\\omega^{(1)}=\\frac{1}{2r}\\dot\\omega^{(1)}\n-\\frac{\\dot f^{(1)}}{\\bar f}(\\partial_t\\bar\\omega +\\partial_r\\bar\\omega)\n-\\frac{\\dot\\omega^{(1)}}{\\bar f}(\\partial_t\\bar f+\\partial_r\\bar f)\n-\\frac{8\\pi G}{\\bar f}\\Bigl(\\bar\\Phi_1\\dot\\Phi_1^{(1)}(\\partial_t\\bar\\omega\n+\\partial_r\\bar\\omega)\\cr +(\\bar\\Phi_1\\dot\\omega^{(1)}+\\dot W_1^{(1)})(\\partial_t\\bar\\Phi_1\n+\\partial_r\\bar\\Phi_1)+\\dot\\Phi_1^{(1)}(\\partial_t\\bar W_1+\\partial_r\\bar W_1)\n-g\\bar\\Phi_1\\dot W_2^{(1)}(\\bar A_1+\\bar A_2)\\Bigr)\n\\end{eqnarray}\n\n\nThese perturbation equations are linear differential equations of first order in $\\dot h_{\\mu\\nu}$\nand $\\dot B_\\mu^a$ and contain in general  the term  $\\ddot k_{\\mu\\nu}$ and $\\ddot C_\\mu^a$\nIn our special case, we see appear in Eq. (38) a  second order term $\\ddot k_{\\mu\\nu}$ and in\nEq. (34) a term $\\ddot A_i^{(2)}$.\nThe remaining perturbation\nequations will also contain second order term like $\\ddot W_1^{(2)}$.\n\n\n\nFor the background variables $\\bar\\Phi_1, \\bar W_1$ and $\\bar W_2$ we use Eq.(A13).\nThe resulting equations are similar to the equations found in section 2 for $\\Phi_2=0$.\nFrom Eq. (A17) we obtain equations for the background variables $\\bar\\Omega , \\bar f$,\nand $\\bar\\omega$ with back-reaction terms, for example,\n\\begin{eqnarray}\n\\partial_t^2\\bar\\Omega -\\partial_r^2\\bar\\Omega = -\\frac{1}{r}\\partial_r\\bar\\Omega\n+\\frac{1}{\\bar\\Omega}\\Bigl((\\partial_t\\bar\\Omega)^2-(\\partial_r\\bar\\Omega)^2\\Bigr)\n+\\frac{\\bar\\Omega(\\partial_r\\bar f)^2}{2\\bar f^2}-\\frac{\\bar\\Omega\\partial_r\\bar f}{\\bar fr}\n+\\frac{\\bar f^2\\bar\\Omega(\\partial_r\\bar\\omega)^2}{2r^2} \\cr +\\frac{\\bar\\Omega}{2\\tau \\bar f^2r^2}\n\\int[\\bar f^4(\\dot\\omega^{(1)})^2+r^2(\\dot f^{(1)})^2]d\\xi\n+\\frac{8\\pi G\\bar\\Omega}{\\bar fr^2\\tau}\\int\\Bigl[\\bar f^2(\\dot W_1^{(1)}+\\bar\\Phi_1\n\\dot\\omega^{(1)})^2+r^2(\\dot\\Phi_1^{(1)})^2+\\bar f^2(\\dot W_2^{(1)})^2\\Bigr]d\\xi \\cr\n+\\frac{4\\pi G\\bar\\Omega \\bar f}{r^2}\\Bigl[(\\partial_t\\bar W_1+\\bar\\Phi_1\\partial_t\\bar\\omega)^2\n+(\\partial_r\\bar W_1+\\bar\\Phi_1\\partial_r\\bar\\omega)^2+\\frac{r^2}{\\bar f^2}(\n\\partial_r\\bar\\Phi_1^2+\\partial_t\\bar\\Phi_1^2)\\Bigr].\n\\end{eqnarray}\n\nSo we see that a first order HF gravitational and Yang-Mills wave emitted from the\nstring, will change the metric background component $\\bar\\Omega$, just as in\nthe vacuum situation for pure gravitational waves. Moreover, there will be a distortion of the\nshape of the wave during its propagation, which does not occur in the pure Einstein case.\nIn general, this is due to the fact that second order terms appear in the perturbation equations of\n$\\dot h_{\\mu\\nu}$ and $\\dot B_\\mu ^a$.\nNow let us consider, for example, the equation for $\\dot W_2^{(1)}$, Eq. (36). For $A_1=-A_2$, we can solve\n$\\dot W_2^{(1)}$ for $\\bar f=e^{2p}$. For example for $p=1\/4$ we have\n\\begin{equation}\n\\dot W_2^{(1)}=\\sqrt{r}B(t-r)e^{-2\/3r^{1.5}},\n\\end{equation}\nwhich approaches to zero for large r. Here is $B(t-r)$ an arbitrary function of $(t-r)$.\nTogether with Eq.(31) we then have to first order\n\\begin{eqnarray}\n\\dot W_2^{(1)}=1-1\/g+\\sqrt{r}B(t-r)e^{-2\/3r^{1.5}}+...,\n\\end{eqnarray}\nwhich is bounded for large r. One can proceed in the same way for the other first order\nperturbations by imposing some trial solutions for the background variables~\\cite{Slag398}\n\n\\section{Conclusions}\n\nNon-Abelian gravity-coupled solitons and black holes could have\nplayed an important role in gauge theories of elementary particles in the early stage of\nthe universe. Besides the well-known spherically symmetric solutions, axially symmetric\nsolutions are of interest due to the fact that they admit cosmic strings, which could have\nplayed the role of 'seeds' for the large-scale structure of clusters in the universe.\nIt is quite evident that the solution found here is a radiating one and there is an\ninteraction of the solitonlike gravitational waves and the Yang-Mills waves with the\nrotating string. This can be seen from figure 5 where the angle deficit  and $\\Omega$\nis significantly changed by the pulses. This can also be seen from Eq. (41), where the\nterm on the right-hand side represents the back-reaction of the waves on $\\Omega$.\nThe wave-like solutions found in this model is significantly different from the electrovac\nsolution found in the Einstein-Maxwell model.\nThe intervening electro-magnetic waves did not contribute to the angle\ndeficit asymptotically, while in our model the magnetic components $W_i$ has an impact on the\nangle deficit for large r (compare figures 1 and 3). In particularly the number of times the\n$W_i$ crosses the r-axis influences the gravitational wave puls.\nFurther, for a different mass per unit length of the string, a different behavior\nof the solution is obtained (compare figures 3 and 4).\n\nIn order to conclude if the solution is topological stable, we must solve the perturbation\nequations to higher order.\nMoreover, a combination of the results of a throughout numerical investigation with the\nmultiple-scale method pushed to higher orders, will lead to further understanding\nof the rich structure of the EYM system on a rotating space time. Specially the question\nif a electrically and magnetically charged solution ('dyon') of the rotating cosmic string\nexists.\nFurther, bursts of high-frequency gravitational waves might be detectable by the planned\ngravitational wave detectors LIGO\/VIRGO and LISA, even for strings at GUT scale~\\cite{dam00}.\n\n\nThe issues mentioned above are currently under study by the author.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince the seminal work of Mumford on the moduli space $\\mathcal{M}_g$ of smooth curves with $g \\geq 2$ and its compactification $\\overline{\\mathcal{M}}_g$, significant progress has been made in understanding its higher-dimensional analog, namely stable varieties. On a first approximation, those are $\\mathbb{Q}$-Gorenstein varieties with relatively mild singularities, and such that $K_X$ is ample.\nA complete and satisfactory moduli theory for stable varieties of any dimension has been settled due to the work of several mathematicians (see \\cites{KSB, Kol90, Ale94, Vie95, Kol13-mod, Kol13, HK04, AH11, kol08, HX13, kol1, HMX }).\nFurthermore, in this setup, it is well understood what families should be parametrized by the moduli functor.\n\nA natural generalization of $\\mathcal{M}_g$ is given by the moduli space of $n$-pointed curves and its compactification, denoted by $\\mathcal{M}_{g,n}$ and $\\overline{\\mathcal{M}}_{g,n}$, respectively.\nA first attempt to generalize the notion of stable pointed curve is to consider mildly singular pairs $(X,D)$ (specifically, \\emph{semi-log canonical} pairs) where $D$ is a reduced divisor, such that $K_X+D$ is ample.\nThis approach has been worked out Alexeev in dimension 2 \\cites{Ale94, Ale96}, and combining the efforts of several mathematicians (see \\cites{ Ale94, Kol13, kol08, HX13, KP17, kol1, HMX, kol19s }), it has been generalized in higher dimensions.\n\nWhile a stable variety is $\\mathbb{Q}$-Gorenstein, a stable pair might not be, so if one is interested in $\\mathbb{Q}$-Gorenstein pairs, the aforementioned formalism might not give the desired moduli space.\nIn fact, one cannot simply consider the $\\qq$-Gorenstein locus of the moduli of stable pairs, as this could be not proper.\nTherefore, the main goal of this paper is to construct a \\emph{proper} moduli space parametrizing $\\mathbb{Q}$-Gorenstein \\emph{pairs} $(X,D)$.\nAs in the classical case, we require our pairs to have semi-log canonical singularities, but relax the ampleness condition on $K_X+D$,\nrequiring only that $K_X+(1-\\epsilon)D$ is ample for $0< \\epsilon \\ll 1$.\nWe also impose a numerical condition on the intersection number $(K_X + tD)^{\\dim(X)}$, analogous to the genus condition for curves, which we encode via a polynomial function $p(t)$.\nThis condition guarantees the boundedness of our moduli problem.\nThese choices are encoded in the notion of \\emph{$p$-stable pair} (Defintion \\ref{def_p-pair}) and lead to the following\n\\begin{theorem} \\label{thm intro 1}\nFix an integer $n\\in \\mathbb{N}$ and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that assumption (A) holds.\nThen, there is a proper Deligne--Mumford stack $\\mathscr{F}_{n,p,1}$, with projective coarse moduli space, parametrizing $p$-stable ($\\mathbb{Q}$-Gorenstein) pairs $(X;D)$ of dimension $n$ and with polynomial $p(t)$ and reduced boundary.\n\\end{theorem}\nAssumption (A) is a technical assumption (which is vacuous in dimension $\\le 3$ and for the irreducible components generically parametrizing klt pairs) defined in \\S \\ref{term.subs}.\nIn the statement of Theorem \\ref{thm intro 1}, the subscript 1 in $\\mathscr{F}_{n,p,1}$ means that $D$ is a reduced divisor.\n\nWhile it is very natural to consider the divisor $D$ with its reduced structure, the framework of the Minimal Model Program highlights the importance of the use of fractional coefficients for the divisor $D$.\nFor example, in the case of curves, one can replace $\\overline{\\mathcal{M}}_{g,n}$ with a weighted version, where the markings can attain any fractional value in $(0,1]$.\nThis was accomplished by Hassett in \\cite{Has03}, and this approach leads to different compactifications of\n$\\mathcal{M}_{g,n}$.\n\nIt turns out, however, that the construction of the higher dimensional analogs of these weighted moduli spaces is very delicate.\nAmong the many difficulties, the definition of a suitable notion of family of boundary divisors represents a major problem, see \\cite{kol1}*{Ch. 4}. \nNevertheless, over the past decade, there has been significant progress in the development of a moduli theory for higher dimensional stable pairs (see \\cites{ Ale94, Kol13, kol08, HX13, KP17, kol1, HMX }), and this last missing piece has finally been settled by Koll\\'ar in \\cite{kol19s}.\nIn \\emph{loc. cit.} a rather subtle refinement of the flatness condition is introduced, leading to the ultimate notion of family of divisors, that in turn gives a satisfactory treatment of a moduli functor of families of stable pairs, admitting divisors with arbitrary coefficients.\n\nThe difficulty in defining a suitable notion of family of boundary divisors is due to the fact that, in general, a deformation of a pair $(X,D)$ cannot be reduced to a deformation of the total space $X$ and a deformation of the divisor $D$.\nThis na\\\"ive expectation is only true if the coefficients of $D$ are all strictly greater than $\\frac{1}{2}$ (see \\cite{Kol14}), in which case a deformation of $(X,D)$ over a base curve induces a flat deformation of $D$.\nOn the other hand, if smaller coefficients are allowed, the situation becomes much more subtle.\nFor instance, as showed by Hassett (see \\S \\ref{hassett example} for details), by allowing coefficient $\\frac{1}{2}$, it is possible to define a family of stable surface pairs $(\\mathcal{X},\\mathcal{D}) \\rar \\mathbb A ^1$ such that the flat limit of $\\mathcal{D}$ acquires an embedded point along the special fiber.\n\nOne advantage of our formalism is that it allows overcoming the aforementioned difficulties.\nThat is, our setup easily generalizes to the case of any fractional coefficients, preventing the above-mentioned pathologies of families of divisors, and leading to a more transparent definition of the moduli functor in Definition \\ref{Def:functor}.\nIndeed, we generalize $\\mathscr{F}_{n,p,1}$ to an analogous moduli functor $\\mathscr{F}_{n,p,I}$ in Definition \\ref{Def:functor}, allowing arbitrary coefficients on the divisor.\nThe main feature of Definition \\ref{Def:functor} is that the family of boundaries is characterized as a flat, proper, and relatively $S_1$ morphism over the base, thus significantly simplifying the definition of the classical moduli functor.\nThe $S_1$ condition prevents the existence of embedded points, while the polynomial $p(t)$ controls the relevant invariants of the varieties, preventing them from jumping as in Hassett's example. This leads to the following generalization of Theorem \\ref{thm intro 1}:\n\n\\begin{theorem} \\label{thm intro 1 on steroids}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that $I$ is closed under sum: that is, if $a,b \\in I$ and $a+b \\leq 1$, then $a+b \\in I$.\nAssume that assumption (A) holds.\nThen there is a proper Deligne--Mumford stack $\\mathscr{F}_{n,p,I}$ with projective coarse space, parametrizing $p$-stable pairs of dimension $n$ and coefficients in $I$.\n\\end{theorem}\n\n\\begin{Remark}\nIn the proofs of Theorem \\ref{thm intro 1} and Theorem \\ref{thm intro 1 on steroids}, assumption (A) is needed to guarantee that, for every $p$-stable pair $(X;D)$ in $\\mathscr{F}_{n,p,I}$, $K_X+D$ is semi-ample (equivalently, $(X;D)$ admits a good minimal model).\nBy \\cite{HX13}*{Theorem 1.1} and \\cite{HMX}*{Theorem 1.2}, if the general element of an irreducible component of $\\mathscr{F}_{n,p,I}$ is normal and admits a good minimal model, then so does every element of such component.\nIn particular, by the basepoint-free theorem \\cite{KM98}*{Theorem 3.3}, we have that assumption (A) is satisfied for the components of $\\mathscr{F}_{n,p,I}$ that generically parametrize klt pairs.\nThus, Theorem \\ref{thm intro 1} and Theorem \\ref{thm intro 1 on steroids} hold unconditionally for the subfunctor that parametrizes $p$-stable pairs that are generically klt.\n\\end{Remark}\n\nIn \\S\\ref{section morphism} we relate our moduli functor with the moduli functor of stable pairs, and as the main application of the theory of $p$-stable pairs, we obtain a simpler proof of the projectivity of the moduli space of stable pairs, originally proved by Kov\\'acs and Patakfalvi \\cite{KP17}.\n\n\\begin{theorem}[Corollary \\ref{cor proj moduli}] \\label{thm intro 2}\nConsider a proper DM stack $ \\mathcal{K}_{n,v,I}$ which satisfies the following two conditions:\n\\begin{enumerate}\n    \\item for every normal scheme $S$, the data of a morphism $f \\colon S\\to \\mathcal{K}_{n,v,I}$ is equivalent to a stable family of pairs $q \\colon (\\mathcal{Y},\\mathcal{D})\\to B$ with fibers of dimension $n$, volume $v$ and coefficients in $I$; and\n    \\item there is $m_0\\in \\mathbb{N}$ such that, for every $k \\in \\mathbb{N}$, there is a line bundle $\\mathcal{L}_k$ on $\\mathcal{K}_{n,v,I}$ such that, for every morphism $f$ as above, $f^*\\mathcal{L}_k \\cong \\det(q_*(\\omega_{\\mathcal{Y}\/B}^{[km_0]}(km_0\\mathcal{D})))$.\n\\end{enumerate}\nThen, the coarse moduli space of $\\mathcal{K}_{n,v,I}$ is projective. \n\\end{theorem}\n\nAs observed in \\cite{KP17}*{\\S 1.1}, the approach of \\cite{Has03} for proving projectivity of these moduli spaces cannot be adapted to higher dimensions, as certain sheaves are no longer functorial with respect to base change.\nFor this reason, Kov\\'acs and Patakfalvi develop a refinement of Koll\\'ar's ampleness lemma.\nOn the other hand, in the setup of $p$-stable pairs, all the needed sheaves remain functorial with respect to base change.\nIndeed, flatness and the $S_1$ condition guarantee that all notions of pull-back agree.\nThus, we can follow Hassett's strategy and directly apply \\cite{Kol90}.\nIn this way, the projectivity part of Theorem \\ref{thm intro 1 on steroids} is established;\nthen, to deduce Theorem \\ref{thm intro 2}, it suffices to show that the moduli space of $p$-stable pairs naturally admits a \\emph{finite} morphism to the moduli space of stable pairs, and the needed polarization descends with this morphism.\nFinally, we remark that Theorem \\ref{thm intro 2} holds independently of assumption (A), as every stable pair is the stable model of a $p$-stable pair.\n\n\\subsection{Structure of the paper}\n\\label{struttura}\nThe first part of this work is devoted to developing the notion of $p$-stable pair and extending several statements from pairs to $p$-stable pairs.\nIn particular, in \\S \\ref{preliminaries} we set the key definitions and properties, while in \\S \\ref{section_boundedness} we extend the boundedness results of \\cite{HMX18} to the context of $p$-stable pairs.\nIn doing so, a key technique is to relate a $p$-stable pair $(X,D)$ to its ample model $\\operatorname{Proj} \\bigoplus_{m\\ge 0} H^0(X,\\mathcal{O}_{X}(m(K_X + \\Delta +D))$, which is a stable pair.\nIn the framework of nef and big log canonical divisors, the abundance conjecture is fully known only in dimension up to 3 \\cite{KMM94}, while in higher dimensions we only know the klt case by the basepoint-free theorem.\nTherefore, some technical statements are phrased conditionally to the existence of the needed ample models;\nthe specific condition is referred to as assumption (A), which is defined in \\S \\ref{term.subs}.\nThis also justifies the presence of assumption (A) in Theorem \\ref{thm intro 1} and Theorem \\ref{thm intro 1 on steroids}.\nOn the other hand, any stable pair is the ample model of a $p$-stable pair.\nThus, the application to Theorem \\ref{thm intro 2} holds unconditionally.\n\nIn \\S \\ref{section functor} and \\S \\ref{section_properness}, we analyze the moduli functor $\\mathscr{F}_{n,p,I}$.\nIn particular, in \\S \\ref{section functor} we show that $\\mathscr{F}_{n,p,I}$, which a priori is only a category fibered in groupoids over $\\mathrm{Sch}\/k$, is a Deligne--Mumford stack.\nThen, in \\S \\ref{section_properness} we show that $\\mathscr{F}_{n,p,I}$ is proper.\nThus, \\S \\ref{section functor} and \\S \\ref{section_properness} settle Theorem \\ref{thm intro 1 on steroids}, except for the projectivity part.\n\nIn \\S \\ref{section morphism}, we make a connection between $\\mathscr{F}_{n,p,I}$ and the moduli space of stable pairs.\nIn particular, we show that every irreducible component of the moduli space of stable pairs is dominated by a component of $\\mathscr{F}_{n,p,I}$ (for a suitable choice of invariants), and this morphism is finite.\n\nIn \\S \\ref{section projectivity ksba} we conclude our work.\nWe use the ampleness lemma to show that the coarse space of $\\mathscr{F}_{n,p,I}$ is projective, and we use the results from \\S \\ref{section morphism} to descend the analogous statement to the moduli space of stable pairs.\n\n\\subsection*{Acknowledgements} We thank Jarod Alper, Dori Bejleri, Christopher Hacon, S\\'andor Kov\\'acs, Joaqu\\'in Moraga, and Zsolt Patakfalvi for helpful discussions.\n\n\n\n\\section{Preliminaries} \\label{preliminaries}\n\n\\subsection{Terminology and conventions}\n\\label{term.subs}\nThroughout this paper, we will work over the field of complex numbers.\nFor the standard notions in the Minimal Model Program (MMP) that are not addressed explicitly, we direct the reader to the terminology and the conventions of~\\cite{KM98}.\nSimilarly, for the relevant notions regarding non-normal varieties, we direct the reader to \\cite{Kol13}.\nA variety will be an integral separated scheme of finite type over $\\mathbb{C}$.\n\nThroughout the paper, when we say ``assume that assumption (A) holds'' we mean \"assume that one of the two following conditions hold\": \n\\begin{itemize}\n    \\item[(i)] the dimension of all the varieties appearing as general fibers of the families\/morphisms is at most 3; or\n    \\item[(ii)]  if $(X,D)$ is an lc pair such that $K_X + (1-\\epsilon)D$ is ample for $0<\\epsilon \\ll 1$, then $K_X + D$ is semi-ample.\n\\end{itemize}\n\n\\begin{Remark}\nWe remark that assumption (A) is a weakening of the Abundance Conjecture, which is known up to dimension 3.\nMoreover, assumption (A) is known to hold for klt pairs $(X,D)$ (it is the basepoint-free theorem).\nFinally, a similar statement holds if $D$ (rather than $K_X + D$) contains an ample divisor: it is proved in \\cite{Hu17} that if $D\\ge A$ where $A$ is an ample $\\mathbb{R}$-divisor, if $K_X + D $ is nef it is also semi-ample.\n\\end{Remark}\n\n\\subsection{Contractions}\nA \\emph{contraction} is a projective morphism $f\\colon X \\rar Z$ of quasi-projective varieties with $f_\\ast  \\O X. = \\O Z.$. \nIf $X$ is normal, then so is $Z$.\n\n\\subsection{Divisors}\nLet $\\mathbb{K}$ denote $\\mathbb{Z}$, $\\qq$, or $\\mathbb{R}$. We say that $D$ is a \\emph{$\\mathbb{K}$-divisor} on a variety $X$ if we can write $D = \\sum \\subs i=1. ^n d_i P_i$ where $d_i \\in \\mathbb{K}$, $n \\in \\mathbb{N}$ and the $P_i$ are prime Weil divisors on $X$ for all $i=1, \\ldots, n$. \nWe say that $D$ is $\\mathbb{K}$-Cartier if it can be written as a $\\mathbb{K}$-linear combination of $\\mathbb{Z}$-divisors that are Cartier.\nThe \\textit{support} of a $\\mathbb{K}$-divisor $D=\\sum_{i=1}^n d_iP_i$ is the union of the prime divisors appearing in the formal sum $\\mathrm{Supp}(D)= \\sum_{i=1}^n P_i$.\n\nIn all of the above, if $\\mathbb{K}= \\mathbb{Z}$, we will systematically drop it from the notation.\n\nGiven a prime divisor $P$ in the support of $D$, we will denote by $\\mu_P (D)$ the coefficient of $P$ in $D$.\nGiven a divisor $D = \\sum \\mu_{P_i}(D) P_i$ on a normal variety $X$, and a morphism $\\pi \\colon X \\to Z$, we define\n\\[\nD^v \\coloneqq \\sum_{\\pi(P_i) \\subsetneqq Z} \\mu_{P_i}(D) P_i, \\\nD^h \\coloneqq \\sum_{\\pi(P_i) = Z} \\mu_{P_i}(D) P_i.\n\\]\n\nLet $D_1$ and $D_2$ be divisors on $X$.\nWe write $D_1 \\sim_{\\mathbb K ,Z} D_2$ if there is a $\\mathbb{K}$-Cartier divisor $L$ on $Z$ such that $D_1 - D_2 \\sim _{\\mathbb K}f^\\ast L$.\nEquivalently, we may also write $D_1 \\sim_{\\mathbb{K}} D_2\/Z$, or $D_1 \\sim_{\\mathbb{K}} D_2$ over $Z$.\nIf $\\mathbb{K}=\\mathbb{Z}$, we omit it from the notation.\nSimilarly, if $Z= \\mathrm{Spec}(k)$, where $k$ is the ground field, we omit $Z$ from the notation.\n\n\nLet $\\pi \\colon X \\rar Z$ be a projective morphism of normal varieties.\nLet $D_1$ and $D_2$ be two $\\mathbb{K}$-divisors on $X$.\nWe say that $D_1$ and $D_2$ are numerically equivalent over $Z$, and write $D_1 \\equiv D_2\/Z$, if $D_1 . C = D_2 . C$ for every curve $C \\subset X$ such that $\\pi(C)$ is a point.\nIn case the setup is clear, we just write $D_1 \\equiv D_2$, omitting the notation $\/Z$.\n\n\\subsection{Non-normal varieties and pairs} There are two important generalizations of the notion of normal variety.\n\\begin{Def}\nAn $S_2$ scheme is called \\emph{demi-normal} if its codimension 1 points are either regular or nodal.\\end{Def}\nRoughly speaking, the notion of demi-normal schemes allows extending the notion of log canonical singularities to non-normal varieties, allowing for a generalization of the notion of stable curve to higher dimensions.\nWe refer to \\cite{kol1}*{\\S 1.8} for more details.\n\n\\begin{Def}A finite morphism of schemes $X' \\rar X$ is called a \\emph{partial seminormalization} if $X'$ is reduced and, for every point $x \\in X$,  the induced map $k(x) \\rar k(\\mathrm{red}(g^{-1}(x)))$ is an isomorphism.\nThere exists a unique maximal partial seminormalization, which is called \\emph{the seminormalization} of $X$.\nA scheme is called \\emph{seminormal} if the seminormalization is an isomorphism.\\end{Def}\n\n\nThis is an auxiliary lemma, it is probably well known. We include it for completeness.\nWe refer to \\cite{Kol13}*{\\S 10.2} for the details about seminormality.\n\n\\begin{Lemma}\\label{lemma:connected:fivers:vs:cohomol:connected:fibers}\nLet $p \\colon X\\to Y$ be a proper surjective morphism with connected fibers, with $Y$ seminormal and $X$ reduced.\nThen $p_*\\mathcal{O}_X = \\mathcal{O}_Y$.\n\\end{Lemma}\n\n\\begin{Remark}\\label{remark:cohom:connected:implies:one:can:take:global:sections:after:pull:back}\nObserve that, in situation of Lemma \\ref{lemma:connected:fivers:vs:cohomol:connected:fibers}, by the projection formula it follows that, for any line bundle $L$ on $Y$, we have $H^0(Y,L) =H^0(X,p^*L)$.\n\\end{Remark}\n\n\\begin{proof}\nWe can take the Stein factorization $X\\xrightarrow{q} Z\\xrightarrow{a} Y$.\nSince $X$ is reduced, then $Z$ is reduced.\nObserve that $q_*\\mathcal{O}_X = \\mathcal{O}_Z$, so the desired result follows if we can show that $a$ is an isomorphism.\n\nSince the composition $p=a \\circ q$ has connected fibers and $q \\colon X\\to Z$ is surjective, then $a \\colon Z\\to Y$ has connected fibers.\nSince $a$ is finite, it is injective.\nIt is also surjective since $p$ is surjective, so $a$ is a bijection.\nThen, we can take the seminormalization $Z^{sn}\\to Z$ and consider the composition $Z^{sn}\\to Z \\to Y$.\nThis is a bijective morphism since it is a composition of bijections, and it is proper since $a$ is proper and $Z^{sn}\\to Z$ is proper.\nThen, the composition $Z^{sn} \\to Y$ is an isomorphism, since the source and the target are seminormal.\n\nNow, on the topological space given by $Z$ we have the following morphsims of sheaves:\n$$\\mathcal{O}_Y\\xrightarrow{a^\\#} \\mathcal{O}_Z\\xrightarrow{b} \\mathcal{O}_{Z^{sn}}.$$\nThe composition is surjective, so $b$ is surjective. Moreover, since $Z$ is reduced, the map $\\mathcal{O}_Z\\xrightarrow{b} \\mathcal{O}_{Z^{sn}}$ is also injective. Then it is an isomorphism, so $Z\\cong Z^{sn}\\cong Y$.\n\\end{proof}\n\n\\begin{Lemma}\\label{lemma:canonical:model:of:slc:pairs:has:connected:fibers}\nLet $(X,D_X)$ and $(Y,D_Y)$ be semi-log canonical pairs, and let $(X^n,D_X^n+\\Delta_X)$ and $(Y^n,D_Y^n+\\Delta_Y)$ denote the respective normalizations.\nAssume we have the following diagram:\n\\[\n\\xymatrix{\\Delta_X^n \\ar[r] \\ar[d]_\\gamma & (X^n, D_X^n+\\Delta_X)\\ar[r]^{\\quad n_X} \\ar[d]^\\alpha & (X,D_X)\\ar[d]^\\beta \\\\\\Delta^n_Y \\ar[r] & (Y^n, D^n_Y+\\Delta_Y) \\ar[r]^{\\quad n_Y} & (Y,D_Y)}\n\\]\nwhere the following properties are satisfied:\n\\begin{itemize}\n    \\item $\\alpha$ induces a bijection between the irreducible components of $X^n$ and $Y^n$ and it is birational on each irreducible component; and\n    \\item the fibers of $\\alpha$ and $\\gamma$ are connected.\n\\end{itemize}\nThen the fibers of $\\beta$ are connected.\n\nFurthermore, if we assume that\n\\begin{itemize}\n    \\item $\\gamma$ induces a bijection between the irreducible components of $\\Delta_X^n$ and $\\Delta_Y^n$ and it is birational on each such component; and\n    \\item $\\alpha^*(\\K Y^n. + D^n_Y + \\Delta_Y) = \\K X^n. + D^n_X + \\Delta_X$.\n\\end{itemize}\nThen, we have $\\beta^*(K_{Y} +D_Y) =  K_{X} +D_X$.\n\\end{Lemma}\n\n\\begin{proof}\nLet $\\tau_X$ be the natural involution induced on $\\Delta_X^n$.\nSince $n_X$ is determined as the quotien by a finite equivalence relation, by \\cite{Kol13}*{Lemma 9.10}, we can write $\\mathcal{O}_X$ as the kernel of the following morphism:\n\\begin{equation} \\label{eq_ses_normalization}\n0\\to \\mathcal{O}_X \\to (n_{X})_*\\mathcal{O}_{X^n} \\xrightarrow{g_X} (n_{X})_*\\mathcal{O}_{\\Delta^n_X},\n\\end{equation}\nwhere $g_X$ is the difference between the pull-back to $\\Delta^n_X$ and the pull-back to $\\Delta^n_X$ composed with $\\tau_X$.\nWe push forward the sequence \\eqref{eq_ses_normalization} via $\\beta$ and we get \n$$\n0 \\to \\beta_*  \\mathcal{O}_X \\to (\\beta \\circ n_X)_*  \\mathcal{O}_{X^n} \\xrightarrow{\\beta_*g}(\\beta \\circ {n_X})_* \\mathcal{O}_{\\Delta^n_X}.\n$$\nSince both $\\alpha$ and $\\gamma$ have connected fibers and the diagram in the statement commutes, the sequence above becomes \n$$\n0\\to \\beta_* \\mathcal{O}_X \\to (n_{Y})_*\\mathcal{O}_{Y^n} \\xrightarrow{g_Y} (n_{Y})_*\\mathcal{O}_{\\Delta^n_Y},\n$$\nwhere $g_Y$ is the difference between the pull-back to $\\Delta^n_Y$ and the pull-back to $\\Delta^n_Y$ composed with the involution $\\tau_Y$.\nSince $\\beta_*$ is left exact, we conclude that $\\beta_*\\mathcal{O}_X = \\mathcal{O}_Y$, as we can again apply \\cite{Kol13}*{Lemma 9.10}; thus, $\\beta$ has connected fibers.\n\nNow, we are left with proving that $\\beta^*(K_Y + D_Y) = K_X + D$.\nAssume the additional hypotheses in the statement\nIt suffices to check that these two divisors agree in codimension 1.\nBy the additional assumptions, $\\beta$ is an isomorphism generically on the double loci of $X$ and $Y$.\nThus, we are left with checking that the desired statement holds in a neighborhood of the generic point of any closed subscheme of codimension one not contained in the double locus.\nThen, the desired statement follows since the normalization $Y^n\\to Y$ is an isomorphism on the generic point of every divisor $E$ intersecting the smooth locus of $Y$, and $\\alpha^*(K_{Y^n} +D^n_Y+\\Delta_Y) =  K_{X^n} +D_X^n+\\Delta_X$.\n\\end{proof}\n\n\n\\subsection{Divisorial sheaves}\\label{divisorial sheaves} We begin this subsection with the following definition:\n\n\\begin{Def}Let $X$ be a scheme.\nA sheaf $\\mathfrak F$ on $X$ is called \\emph{divisorial sheaf} if it is $S_2$ and there is a closed subscheme $Z \\subset X$ of codimension at least 2 such that $\\mathfrak F | \\subs X \\setminus Z.$ is locally free of rank 1.\\end{Def}\n\n\\begin{Def}[\\cite{HK04}*{Section 3}]Let $f:X\\to S $ a flat morphism of schemes, and $\\mathcal{E}$ a coherent sheaf on $X$. We say that $\\mathcal{E}$ is relatively $S_n$ if it is flat over $S$ and its restriction to each fiber is $S_n.$\n\\end{Def}\n\nNow, let $X$ be a demi-normal scheme, and let $\\mathrm{Weil}^*(X)$ denote the subgroup of $\\mathrm{Weil}(X)$ generated by the prime divisors that are not contained in the conductor of $X$.\nThen, there is an identification between $\\mathrm{Weil}^*(X)\/\\sim$ and the group of divisorial sheaves, where $\\sim$ denotes linear equivalence.\nThe identification is defined as follows.\nBy the definition of $\\mathrm{Weil}^*(X)$ and the demi-normality of $X$, for every element $B \\in \\mathrm{Weil}^*(X)$,there is a closed subset $Z \\subset X$ of codimension at least 2 such that $B| \\subs X \\setminus Z.$ is a Cartier divisor.\nThen, the corresponding divisorial sheaf is defined as $j_* \\O X \\setminus Z. (B| \\subs X \\setminus Z.)$, where we have $j \\colon X \\setminus Z \\rar X$.\n\nConsider a flat family $f \\colon X\\to B$ with $S_2$ fibers, and assume that there is an open subset $i \\colon U\\subseteq X$ such that $U_b$ is big for every $b\\in B$.\nThen, for every locally free sheaf $\\mathcal{G}$ on $U$, we can consider $i_*\\mathcal{G}$ on $X$.\nFrom \\cite{HK04}*{Corollary 3.7}, this is a reflexive sheaf on $X$ (a priori, it is not $S_2$ relatively to $B$, see \\S \\ref{hassett example}).\n\nIf $X$ is demi-normal, its canonical sheaf is a divisorial sheaf, as $X$ is Gorenstein in codimension 1.\nBy the above identification, we can then choose a canonical divisor $K_X$ such that $\\O X. (K_X) \\cong \\omega_X$.\nObserve that this construction can be carried out in families.\nIndeed, if $f \\colon X\\to B$ is a flat morphism with demi-normal fibers of dimension $n$, there is an open locus $i \\colon U\\subseteq X$ that has codimension 2 along each fiber, on which $f$ is Gorenstein.\nThen we can define $\\omega_{X\/B} \\coloneqq i_*\\omega_{U\/B}$.\nObserve that this agrees with the $(-n)$-th cohomology of the relative dualizing complex.\nIndeed, the latter is $S_2$ (see \\cite{LN18}*{Section 5}).\n\nLet $X$ be demi-normal, and consider two divisorial sheaves $L_1$ and $L_2$.\nThen, their \\emph{reflexive tensor product} is defined as $L_1 \\hat{\\otimes} L_2 \\coloneqq (L_1 \\otimes L_2)^{**}$ and it is a divisorial sheaf itself.\nIf we have $L_1 \\cong \\O X. (D_1)$ and $L_2 \\cong \\O X. (D_2)$, we have $L_1 \\hat{\\otimes} L_2 \\cong \\O X. (D_1+D_2)$.\nThe \\emph{$m$-fold reflexive power $L^{[m]}$} is defined as the $m$-fold self reflexive tensor product.\n\nFor more details, we refer to \\cite{kol1}*{\\S 3.3}, \\cite{Kol13}*{5.6} and \\cite{HK04}.\n\n\n\\subsection{Boundedness}\nLet $\\mathfrak{D}$ be a set of projective pairs.\nThen, we say that $\\mathfrak{D}$ is {\\it log bounded} (resp. {\\it log birationally bounded}) if there exist a variety $\\mathcal{X}$, a reduced divisor $\\mathcal{E}$ on $\\mathcal{X}$, and a projective morphism $\\pi \\colon \\mathcal{X} \\rar T$, where $T$ is of finite type, such that $\\mathcal{E}$ does not contain any fiber of $\\pi$, and, for every $(X,B) \\in \\mathfrak{D}$, there are a closed point $t \\in T$ and a morphism (resp. a birational map) $f_t \\colon \\mathcal{X}_t \\rar X$ inducing an isomorphism $(X,\\Supp(B)) \\cong (\\mathcal{X}_t,\\mathcal{E}_t)$ (resp. such that $\\Supp(\\mathcal B _t)$ contains the strict transform of $\\Supp(B)$ and all the $f_t$ exceptional divisors).\n\nA set of projective pairs $\\mathfrak{D}$ is said to be {\\it strongly log bounded} if there is a quasi-projective pair $(\\mathcal{X},\\mathcal{B})$ and a projective morphism $\\pi \\colon \\mathcal{X} \\rar T$, where $T$ is of finite type, such that $\\Supp(\\mathcal{B})$ does not contain any fiber of $\\pi$, and for every $(X,B) \\in \\mathfrak D$, there is a closed point $t \\in T$ and an isomorphism $f \\colon X \\rar \\mathcal{X}_t$ such that $f_*B=\\mathcal{B}_t$.\n\nA set of projective pairs $\\mathfrak{D}$ is {\\it effectively log bounded} if it is strongly log bounded and we may choose a bounding pair $\\pi \\colon (\\mathcal{X},\\mathcal{B}) \\rar T$ such that, for every closed point $t \\in T$, we have $(\\mathcal{X}_t,\\mathcal{B}_t) \\in \\mathfrak D$.\n\n\\subsection{Index of a set}\nGiven a finite subset $I \\subseteq \\mathbb{Q}^n$, we define the \\emph{index} of $I$ to be the smallest positive rational number $r$ such that $rI\\subseteq \\mathbb{Z}^n$.\n\n\\subsection{Stable pairs and $p$-stable pairs} \\label{p-pairs section}\nLet $(X,D)$ denote a projective semi-log canonical pair, where $D$ has coefficients in $\\qq$.\nWe say that $(X,D)$ is a \\emph{stable pair} if $K_X + D$ is ample.\n\\begin{Def}\\label{def_p-pair}\nConsider a polynomial $p(t)\\in \\mathbb{Q}[t]$ and a set $I\\subseteq (0,1]$.\nA \\emph{$p$-pair} with polynomial $p(t)$ and coefficients in $I$ is the datum of a semi-log canonical pair $(X,\\Delta)$ and a $\\qq$-Cartier $\\qq$-divisor $D$ on $X$ satisfying the following properties:\n\\begin{itemize}\n    \\item $(X,D+\\Delta)$ is semi-log canonical;\n    \\item $p(t) = (K_X+tD+\\Delta)^{\\dim(X)}$; and\n    \\item the coefficients of $D$ and $\\Delta$ are in $I$.\n\\end{itemize}\nIf moreover there is $\\epsilon_0>0$ such that for every $0<\\epsilon \\le\\epsilon_0$ the pair $(X,(1-\\epsilon)D+\\Delta)$ is stable, the $p$-pair $(X,\\Delta;D)$ is called a \\emph{$p$-stable pair}. Finally, if $\\mathcal{A} \\coloneqq \\bigoplus_{m\\ge 0} H^0(X,\\mathcal{O}_{X}(md(K_X + \\Delta +D))$ is finitely generated, where $d$ is the Cartier index of $K_X + \\Delta + D$, we will call $\\operatorname{Proj}(A)$ the \\emph{ample model} of $(X,\\Delta; D)$.\\end{Def}\n\nFor brevity, we denote the datum of a $p$-pair by $(X,\\Delta;D)$, where the polynomial $p(t)$ and the set of coefficients $I$ are omitted in the notation.\nIn case $\\Delta=0$, we then write $(X;D)$.\n\\begin{Remark}\nFor example, if $(X,0;D)$ is an slc $p$-stable pair, and $\\nu:X^n\\to X$ is the normalization of $X$ with conductor $\\Delta$, then $(X^n,\\Delta;\\nu_*^{-1}D)$ is a $p$-stable pair.\n\\end{Remark}\n\n\\begin{Notation}\\label{notation_Dsc}\nIf $I$ is a finite set and $r$ is its index, given a $p$-pair $(X,\\Delta;D)$ with polynomial $p(t)$ and coefficients in $I$ we denote by $D^{sc}$ as the subscheme of $X$ defined by the reflexive sheaf of ideals $\\O X.(-rD)$.\n\\end{Notation}\n\n\n\\begin{Remark}\\label{remark_KX+D+delta_is_nef}\nObserve that if $(X,\\Delta;D)$ is a $p$-stable pair, the $\\qq$-divisor $K_X+D+\\Delta$ is nef since it is limit of ample $\\qq$-divisors.\nFurthermore, $K_X+D+\\Delta$ is big, as it is the sum of an ample $\\qq$-divisor and an effective $\\qq$-divisor.\n\\end{Remark}\n\n\\begin{Lemma} \\label{lemma_convex_combination}\nLet $X$ be a projective variety, and let $D_1$ and $D_2$ be two nef $\\mathbb{R}$-Cartier $\\mathbb{R}$-divisors.\nAssume that for some $t \\in (0,1)$, the divisor $tD_1 + (1-t)D_2$ is ample.\nThen, $cD_1 + (1-c)D_2$ is ample for every $c \\in (0,1)$.\n\\end{Lemma}\n\n\\begin{proof}\nArguing by contradiciton, there exists $c_0 \\in (0,1)$ such that $c_0D_1+(1-c_0)D_2$ is not ample.\nNotice that, as $D_1$ and $D_2$ are nef, then so is $c_0D_1+(1-c_0)D_2$.\nThen, by Kleiman's criterion (see \\cite{urb}*{Proposition 4.4} for a reference including $\\mathbb{R}$-divisors), there exists a class $\\gamma \\in \\overline{NE}(X) \\setminus \\{ 0 \\}$ such that $(c_0D_1+(1-c_0)D_2).\\gamma = 0$.\nAs $D_1$ and $D_2$ are nef, we have $D_1 . \\gamma \\geq 0$ and $D_2 . \\gamma \\geq 0$.\nIf $D_1 . \\gamma = D_2 . \\gamma =0$, it follows that $(tD_1+(1-t)D_2).\\gamma =0$, and $(tD_1+(1-t)D_2)$ is not ample by Kleiman's criterion.\nThen, $D_i . \\gamma>0$ for some $i=1,2$.\nWithout loss of generality, we may assume $D_1 . \\gamma > 0$.\nThen, as $(c_0D_1+(1-c_0)D_2).\\gamma = 0$, it follows that $D_2 . \\gamma <0$, which is impossible, as $D_2$ is nef.\n\\end{proof}\n\n\\begin{Lemma} \\label{ample model p-pair}\nLet $(X,\\Delta;D)$ be a $p$-stable pair, and assume that $f \\colon X \\rar Y$ is its ample model.\nThen, $\\mathrm{Ex}(f) \\subset \\operatorname{Supp}(D)$.\n\\end{Lemma}\n\n\\begin{proof}\nBy assumption, there exists $\\epsilon>0$ such that $(X,(1-\\epsilon)D + \\Delta)$ is a stable pair.\nThus, if $C$ is an irreducible curve that is not contained in $\\operatorname{Supp}(D)$, we have that\n\\[\n(\\K X. + D + \\Delta).C = (\\K X. + (1-\\epsilon)D + \\Delta).C + \\epsilon D .C >  0,\n\\]\nsince the first summand is positive by ampleness and the second is non-negative as $C$ is not contained in $\\operatorname{Supp}(D)$.\nThus, the curves contracted by $f$ are contained in $\\operatorname{Supp}(D)$ and the claim follows.\n\\end{proof}\nThe following lemma is the main technical tool that we will use in Sections \\ref{section_boundedness} and \\ref{section_properness}:\n\\begin{Lemma}\\label{lemma_bound_epsilon_lc_case}\nLet $n$ and $M$ be natural numbers.\nThen, there exists $\\epsilon_0 > 0$, only depending on $n$ and $M$, such that the following holds.\nLet $(Y,\\Delta_Y;D_Y)$ be a normal $p$-stable pair of dimension $n$, and assume that $K_Y+D_Y+\\Delta_Y$ induces a morphism $\\pi \\colon Y \\rar X$ to the canonical model $(X,D_X+\\Delta_X)$ of $(Y,D_Y+\\Delta_Y)$.\nFurther assume that the Cartier index of $K_X + D_X + \\Delta_X$ is less than $M$.\nThen, for every $0 < \\epsilon < \\epsilon_0$, the pair $(Y,(1-\\epsilon)D_Y+\\Delta_Y)$ is stable.\n\\end{Lemma}\n\n\\begin{proof}\nBy definition of $p$-stable pair and of canonical model,\nfor every $\\pi$-exceptional curve $C$, we have $(K_Y+D_Y+\\Delta_Y).C=0$ and $(K_Y+(1-\\epsilon)D_Y+\\Delta_Y).C>0$ for $0 < \\epsilon \\ll 1$.\nFor every such curve $C$, the function $t\\mapsto (K_Y+tD_Y+\\Delta_Y).C$ is linear and not identically 0, so it has at most one zero. \nIn particular, for every $\\epsilon >0$ we have \n\\begin{equation} \\label{eq_intersection0}\n(K_Y+(1-\\epsilon)D_Y+\\Delta_Y).C>0.    \n\\end{equation}\nHence, if we choose $\\epsilon =1$, we have\n\\begin{equation} \\label{eq_intersection1}\n(K_Y+\\Delta_Y).C>0\n\\end{equation}\nfor every $\\pi$-exceptional curve.\n\nBy the same argument, for every curve $C'$ such that $(K_Y + \\Delta_Y).C' \\geq 0$, it follows that\n\\begin{equation} \\label{eq_mori_cone}\n    (K_Y  + (1-\\eta) D_Y + \\Delta_Y).C' >0\n\\end{equation}\nfor every $\\eta \\in (0,1)$.\nNow, define\n\\[\na\\coloneqq \\frac{1}{2M},\\quad \\epsilon_0 \\coloneqq \\displaystyle{\\frac{a}{2n+a+1}}.\n\\]\n\n\n\nSince $K_X+D_X+\\Delta_X$ is ample with Cartier index bounded by $M$, for every curve $\\Gamma$ that is not $\\pi$-exceptional,\nby the projection formula, we have\n\\begin{equation} \\label{eq_intersection2}\n\\frac{1}{2M} = a<(K_X+D_X+\\Delta_X).p_*\\Gamma = (K_Y+D_Y+\\Delta_Y).\\Gamma.\n\\end{equation}\nFrom \\cite{Fuj11}*{Theorem 1.4}, for every $(K_Y+\\Delta_Y)$-negative extremal ray $R$, we may find a curve $\\Gamma$ generating $R$ such that\n\\begin{equation} \\label{eq_bdd_negative_curves}\n-2n\\le (K_Y+\\Delta_Y).\\Gamma<0.\n\\end{equation}\nBy \\eqref{eq_intersection1}, any such curve is not $\\pi$-exceptional.\nThen, by \\eqref{eq_intersection2} and \\eqref{eq_bdd_negative_curves}, for any such $\\Gamma$, we have\n\\begin{equation} \\label{eq_extremal_curves}\n    (K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y).\\Gamma= (1-\\epsilon_0)(K_Y + D_Y + \\Delta_Y).\\Gamma + \\epsilon_0 (K_Y + \\Delta_Y).\\Gamma>(1-\\epsilon_0)a-2n\\epsilon_0 > 0.\n\\end{equation}\n\n\nNow, consider the cone of curves $\\overline{NE}(Y)$ and its decomposition given by the cone theorem associated to the pair $(Y,\\Delta_Y)$ \\cite{Fuj11}*{Theorem 1.4}.\nThen, by \\eqref{eq_mori_cone}, we have that $K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y$ is positive on $\\overline{NE}(Y)_{K_Y + \\Delta_Y \\geq 0}$.\nThus, every $(K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y)$-negative extremal ray is also a $(K_Y +  \\Delta_Y)$-negative extremal ray.\nThen, by \\eqref{eq_extremal_curves}, we have that $(K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y)$ is positive on $R$.\nThus, by the cone theorem \\cite{Fuj11}*{Theorem 1.4}, the log canonical pair $(Y,(1-\\epsilon_0)D_Y + \\Delta_Y)$ has no negative extremal rays; thus, $K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y$ is nef.\n\nNow, by the definition of $p$-stable pair, some convex combination of $K_Y+(1-\\epsilon_0)D_Y+\\Delta_Y$ and $K_Y+D_Y+\\Delta_Y$ is ample.\nThen, the claim follows by Lemma \\ref{lemma_convex_combination}.\n\\end{proof}\n\n\\begin{Lemma}\\label{lemma_bound_for_epsilon}\nFix an integer $n\\in \\mathbb{N}$, a volume $v\\in \\mathbb{Q}_{>0}$, and a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$.\nAssume that assumption (A) holds.\nThere is $0<\\epsilon_0$, only depending only on $n$ and $v$, such that, for every $p$-stable pair $(Y,\\Delta_Y;D_Y)$ of dimension $n$, coefficients in $I$, and polynomial $p(t)$ with $p(1) = v$, the pair $(Y,(1-\\epsilon)D_Y+\\Delta_Y)$ is stable for every $0<\\epsilon < \\epsilon_0$.\n\\end{Lemma}\n\n\\begin{proof}\nWithout loss of generality, we may assume from now on that $1 \\in I$.\nLet $Y^n\\to Y$ be the normalization of $Y$.\nLet $(Y^n,\\Delta_Y^n)$ be the pair induced by $(Y,\\Delta_Y)$, and let $D_Y^n$ denote the pull-back of $D_Y$ to $Y^n$.\nNotice that $Y^n$ possibly has more than one connected component.\nThen, we have:\n\\begin{enumerate}\n     \\item $(Y^n,D_Y^n+\\Delta_Y^n)$ has still coefficients in $I$;\n     \\item $K_{Y^n} + D_Y^n+\\Delta_Y^n$ is nef and big; and\n     \\item $(Y^n,\\Delta_Y^n;D_Y^n)$ is a $p$-stable pair.\n\\end{enumerate}\nFrom \\cite{HMX14}*{Theorem 1.3},\nthere are finitely many possibilities to write $v$ as $v = \\sum v_i$, where $v_i>0$ is the volume of a log canonical pair of general type with coefficients in $I$.\nSo from (1), there is a finite set $\\mathscr{S}=\\{v_1,...,v_k\\}$ such that the volume of each connected component of $Y^n$ is in $\\mathscr{S}$. \n\nBy assumption (A), if $(Z,D)$ is a log canonical pair with $K_Z+D$ big and nef, there is a canonical model $(Z^c,D^c)$ of $(Z,D)$, and by construction $\\vol(Z,D) = \\vol(Z^c,D^c)$.\nIn particular, it follows from (2) that the irreducible components of $(Y^n,D_Y^n+\\Delta_Y^n)$ admit a canonical model, and the volumes of these models are contained in $\\mathcal{S}$ as well.\nFrom \\cite{HMX}, there is an $M>0$ such that every stable pair with coefficients in $I$, dimension $n$, and volume in the finite set $\\mathscr{S}$ has Cartier index less than $M$.\nSince we can check ampleness after passing to the normalization, and the Cartier indexes of each irreducible component of the normalization are bounded, the thesis follows from Lemma \\ref{lemma_bound_epsilon_lc_case}.\n\\end{proof}\n\n\n\\begin{Lemma} \\label{lift stable pair slc}\nLet $(Y,D_Y)$ be a stable pair.\nThen, there exists a $p$-stable pair $(X;D)$ having $(Y,D_Y)$ as ample model.\n\\end{Lemma}\nObserve in particular that from Lemma \\ref{ample model p-pair}, $X$ is normal if and only if $Y$ is normal.\n\n\\begin{proof}\nWe consider the semi-canonical modification $\\pi \\colon X \\rar Y$ of $(Y,0)$, in the sense of \\cite{Fuj15}.\nWe observe that, in order to consider a semi-canonical modification, $K_Y$ does not need to be $\\qq$-Cartier.\nSuch modification exists by \\cite{Fuj15}*{Theorem 1.1} and the fact that $Y$ demi-normal.\nBy \\cite{Fuj15}*{Definition 2.6}, $\\pi \\colon X \\rar Y$ has the following properties:\n\\begin{itemize}\n    \\item $\\pi$ is an isomorphism around every generic point of the double locus of $X$;\n    \\item this procedure is compatible with taking the normalizations of $X$ and $Y$, see \\cite{Fuj15}*{Lemma 3.7.(2)}.\n    In particular, $X$ is normal if so is $Y$, and, in general, $\\pi$ establishes a bijection between irreducible components of $X$ and $Y$; and\n    \\item $K_X$ is $\\qq$-Cartier and $\\pi$-ample.\n\\end{itemize}\nNow, set $K_X + D = \\pi^*(K_Y+D)$.\nSince $K_Y+D$ is ample, for $0 < \\epsilon \\ll 1$, we have that $\\epsilon K_X + (1-\\epsilon(K_X+D)=K_X + (1-\\epsilon)D$ is ample.\nSince both $K_X$ and $K_X+D$ are $\\qq$-Cartier, then so is $D$.\nLastly, as $X$ and $(Y,D_Y)$ are semi-log canonical, to conclude it suffices to show that $D \\geq 0$.\n\nNow, let $(X^\\nu,D^\\nu + \\Delta^\\nu)$ and $(Y^\\nu,D_Y^\\nu + \\Delta^\\nu_Y)$ denote the respective normalizations, where $\\Delta^\\nu$ and $\\Delta_Y^\\nu$ denote the double loci.\nThus, it suffices to show that $D^\\nu \\geq 0$.\nSince this can be checked by considering one irreducible component of $X^\\nu$ at the time, by abusing notation, we may assume that $X^\\nu$ and $Y^\\nu$ are irreducible.\nBy construction, we have\n\\[\n\\K X^\\nu. + (1-\\epsilon)D^\\nu + \\Delta^\\nu \\sim_{\\qq}-\\epsilon D^\\nu \\sim_{\\qq} \\epsilon(\\K X^\\nu. + \\Delta^\\nu) \/Y^\\nu,\n\\]\nand $\\K X^\\nu. + \\Delta^\\nu$ is relatively ample over $Y^\\nu$.\nThen, since we have $D^\\nu_Y \\geq 0$, by the negativity lemma \\cite{KM98}*{Lemma 3.39}, it follows that $D^\\nu \\geq 0$.\nThis concludes the proof.\n\\end{proof}\n\n\\subsection{Families of pairs} \nWe now recall the main definitions of families of pairs from \\cite{kol1}*{Chapter 4} and \\cite{kol19s}. A \\emph{family of pairs} $f \\colon (X,D) \\rar S$ over a reduced base $S$ is the datum of a morphism $f \\colon X \\rar S$ and an effective $\\qq$-divisor $D$ on $X$, such that the following conditions hold:\n\n\\begin{itemize}\n    \\item $f$ is flat with reduced fibers of pure dimension $n$;\n    \\item the fibers of $\\Supp(D) \\rar S$ are either empty or of pure dimension $n-1$; and\n    \\item $f$ is smooth at the generic points of $X_s \\cap \\Supp(D)$ for every $s \\in S$.\n\\end{itemize}\n\nFurthermore, we say that a family of pairs is \\emph{well defined} if it also satisfies the following property:\n\n\\begin{itemize}\n    \\item $mD$ is Cartier locally around the generic point of each irreducible component of $X_s \\cap \\Supp(D)$ for every $s \\in S$, where $m \\in \\mathbb{N}$ is a sufficiently divisible natural number clearing the denominators of $D$.\n\\end{itemize}\n\nThis latter condition guarantees that $mD$ is Cartier on a big open set $U \\subset X$ with the property that $U \\cap X_s$ is a big open set of $X_s$ for every $s \\in S$.\nThis guarantees that we have a well-defined notion of pull-back of $D$ under any possible base change $S' \\rar S$, as we can pull back $mD|_U$, take its closure in $X \\times_S S'$, and then divide the coefficients by $m$.\n\nThere is a more general definition of families of divisors, over possibly non-reduced bases due to K\\'ollar in \\cite{kol19s}. We will not report it here since we will not need it, we refer the interested reader to \\emph{loc. cit.}\n\nA well defined family of pairs $f \\colon (X,D) \\rar S$ over a reduced base $S$ is called \\emph{locally stable} if, for every base change $g \\colon (X_C,D_C) \\rar C$ where $C$ is the spectrum of a DVR with closed point $0$, $(X_C,D_C+X_0)$ is a semi-log canonical pair. \nThen, a family $f \\colon (X,D) \\rar S$ is called \\emph{stable} if it is locally stable, $f$ is proper, and $\\K X\/S. + D$ is $f$-ample.\n\n\n\\subsection{Families of $p$-pairs}\n\nLet us fix a positive integer $n$, a polynomial $p(t)\\in \\mathbb{Q}[t]$, and a finite set of coefficients $I \\subset (0,1] \\cap \\qq$.\nLet $r$ be the index of $I$.\n\n\\begin{Def}\\label{Def:functor}\nLet $\\mathscr{F}_{n,p,I}$ be the category fibered in groupoids over $\\operatorname{Sch}\/k$ whose fibers over a scheme $B$  consists of:\n\\begin{itemize}\n    \\item a flat and proper morphism $f \\colon \\mathcal{X}\\to B$ of relative dimension $n$;\n    \\item a flat and proper morphism $\\mathcal{D}\\to B$ of relative dimension $n-1$ and relatively $S_1$;\n    \\item a closed embedding $i \\colon \\mathcal{D}\\to \\mathcal{X}$ over $B$; and\n    \\item for every point $p\\in B$, the fiber $(\\mathcal{X}_b;\\frac{1}{r}\\mathcal{D}_b) $ is $p$-stable with polynomial $p(t)$ and coefficients in $I$.\n\\end{itemize}\nFor every morphism $B'\\to B$, we denote by  $j_{B'} \\colon \\mathcal{X}\\times_B B'\\to \\mathcal{X}$ the second projection, and by $\\mathcal{X}'$ the fiber product $\\mathcal{X}\\times_B B'$.\nSimilarly, for every point $p\\in B$, we denote by $\\mathcal{D}_b \\coloneqq \\operatorname{Spec}(k(b))\\times_B\\mathcal{D}$.\nIf we denote by $\\mathcal{I}_\\mathcal{D}$ the ideal sheaf of $\\mathcal{D}$, we require that for every $B'\\to B$ and every $m$, the data above is such that\n\\begin{enumerate}\n    \\item the natural morphism $(j_{B'}^*(\\mathcal{I}_\\mathcal{D}^{[m]}))\\to (j_{B'}^*(\\mathcal{I}_\\mathcal{D}))^{[m]}$ is an isomorphism; and\n    \\item the natural morphism $(j_{B'}^*(\\omega_{\\mathcal{X}\/B}^{[m]}))\\to (\\omega_{\\mathcal{X}'\/B'})^{[m]}$ is an isomorphism. \n\\end{enumerate}\nWe will denote by $(\\mathcal{X};\\mathcal{D})\\to B$ an object of $\\mathscr{F}_{n,p,I}$ over $B$, and we will call it a \\emph{family of p-stable pairs}.\n\\end{Def}\n\n\\begin{Def}\nWe will call \\emph{$p$-stable morphism} the data of a flat and proper morphisms $f \\colon \\mathcal{X}\\to B$ and a closed embedding $i \\colon \\mathcal{D}\\to \\mathcal{X}$ that satisfies the four bullet points of Definition \\ref{Def:functor}.\nIf there is no ambiguity we will still denote it with $(\\mathcal{X};\\mathcal{D})\\to B$.\n\\end{Def}\n\nConditions as (1) and (2) in Definition \\ref{Def:functor} are usually referred to as \\emph{Koll\\'ar's condition}. \nNow, we add a series of remarks and technical statements that are relevant in this context.\nWe keep the notation of Definition \\ref{Def:functor}.\n\\begin{Remark}\nObserve that, with the notation of Definition \\ref{Def:functor}, $(\\frac{1}{r}\\mathcal{D}_b)^{sc} = \\mathcal{D}_b$.\n\\end{Remark}\n\\begin{Remark} \\label{remark_ideal_flat}\nThe ideal sheaf $\\mathcal{I}_\\mathcal{D}$ is flat over $B$.\nThis follows from the fact that $\\O \\mathcal{X}.$ and $\\O \\mathcal{D}.$ are flat, by considering the associated long exact sequence of $\\operatorname{Tor}$.\n\\end{Remark}\n\n\\begin{Remark}\nSince $f$ is flat and by condition (3) its fibers are $S_2$, $f$ is relatively $S_2$.\nThen, since $\\mathcal{D}\\to B$ is flat and relatively $S_1$, it follows from \\cite{Kol13}*{Corollary 2.61} that $\\mathcal{I}_\\mathcal{D}$ is relatively $S_2$.\n\\end{Remark}\n\n\\begin{Remark}\nBy condition (3), there exists an open subset $V\\subseteq \\mathcal{X}$ whose restriction to any fiber is a big open subset, such that $\\mathcal{D}_b$ is a Cartier divisor on $V_b$ for every $b \\in B$.\nThen, by flatness, we may apply \\cite{stacks-project}*{Tag 062Y}, and we conclude that $\\mathcal{D}$ is Cartier along $V$.\nThen, by Remark \\ref{remark_ideal_flat} and \\cite{HK04}*{Proposition 3.5}, $\\O \\mathcal{X}.$ and $\\mathcal{I}_\\mathcal{D}$ are reflexive.\nIn particular, we have $(\\mathcal{I}_\\mathcal{D})^{[1]} = \\mathcal{I}_\\mathcal{D}$, and $\\mathcal{I}_\\mathcal{D}$ satisfies the conditions in \\cite{kol1}*{Definition 3.51}.\n\\end{Remark}\n\n\\begin{Remark}\nObserve that in Definition \\ref{Def:functor}, the divisor $\\mathcal{D}_b$ in point (3) was defined via a fiber product.\nHowever, according to our conventions (see \\S \\ref{divisorial sheaves}) it should also correspond to a Weil divisor on $\\mathcal{X}_b$.\nThis is true since $\\mathcal{D}\\to B$ is relatively $S_1$, so $\\mathcal{D}_b$ is $S_1$ so its ideal sheaf in $\\mathcal{X}_b$ is $S_2$ from \\cite{Kol13}*{Corollary 2.61}. \n\\end{Remark}\n\n\n\n\n\\begin{Remark}\nThe sheaves $\\mathcal{I}_\\mathcal{D}^{[m]}$ are ideal sheaves of $\\mathcal{O}_\\mathcal{X}$.\nIndeed, we can again consider an open subset $V\\subseteq \\mathcal{X}$ whose restriction to any fiber is a big open subset and such that $\\mathcal{D}$ is a Cartier divisor on $V$.\nThen, if we denote by $i \\colon V\\to \\mathcal{X}$ the inclusion of $V$, by \\cite{HK04}*{Corollary 3.7} we have\n\\[\n\\mathcal{I}_\\mathcal{D}^{[m]} = i_*(\\mathcal{O}_V(-m\\mathcal{D}|_{V})).\n\\]\nThen, the inclusion \n$\\mathcal{O}_V(-m\\mathcal{D}|_{V})\\hookrightarrow \\mathcal{O}_V$ can be pushed forward via $i$, to have an inclusion $\\mathcal{I}_\\mathcal{D}^{[m]}\\hookrightarrow i_*(\\mathcal{O}_V)=\\mathcal{O}_\\mathcal{X}$, where the last equality follows from \\cite{HK04}*{Proposition 3.5} since $\\mathcal{X}\\to B$ is $S_2$.\n\\end{Remark}\n\\begin{Notation}\\label{Notation_mD_for_pstable_fam}\nIf $f:(\\mathcal{X};\\mathcal{D})\\to B$ is a family of p-stable pairs, we denote by $m\\mathcal{D}$ the closed subscheme of $\\mathcal{X}$ with ideal sheaf $\\mathcal{I}^{[m]}_\\mathcal{D}$. \n\\end{Notation}\n\n\n\\begin{Remark}\\label{remark:bc:implies:flat}\nBy \\cite{AH11}*{Proposition 5.1.4}, the sheaves $\\mathcal{I}_\\mathcal{D}^{[m]}$ and $\\omega_{\\mathcal{X}\/B}^{[m]}$ are flat over $B$ for every $m\\in \\mathbb{Z}$.\n\\end{Remark}\n\n\\begin{Lemma}\nThe morphism $m\\mathcal{D}\\to B$ is flat with $S_1$ fibers (i.e., the fibers have no embedded points).\\end{Lemma}\n\\begin{proof}\nTo check that $m\\mathcal{D}\\to B$ is flat it suffices to check that for every closed point $\\operatorname{Spec}(k(b))\\to B$ we have $\\operatorname{Tor}^1(k(b),\\mathcal{O}_{m\\mathcal{D}}) = 0$. We pull back the exact sequence\n\\[\n0\\to \\mathcal{I}_{\\mathcal{D}}^{[m]}\\to \\mathcal{O}_{\\mathcal{X}} \\to \\mathcal{O}_{m\\mathcal{D}}\\to 0\n\\]\nvia $j_{b} \\colon \\mathcal{X}_b\\to \\mathcal{X}$, and we obtain\n\\[\n0 = \\operatorname{Tor}^1(k(b),\\mathcal{O}_{\\mathcal{X}}) \\to \\operatorname{Tor}^1(k(b),\\mathcal{O}_{m\\mathcal{D}}) \\to j_{b}^*\\mathcal{I}_{\\mathcal{D}}^{[m]}\\to \\mathcal{O}_{\\mathcal{X}_{b}} \\to \\mathcal{O}_{(m\\mathcal{D})_{b}}\\to 0.\n\\]\nHowever, $j_{b}^*\\mathcal{I}_{\\mathcal{D}}^{[m]} \\cong (j_{b}^*\\mathcal{I}_{\\mathcal{D}})^{[m]}$, so in particular it is a torsion free sheaf of rank 1.\nThen, the map $j_{b}^*\\mathcal{I}_{\\mathcal{D}}^{[m]}\\to \\mathcal{O}_{\\mathcal{X}_{b}}$ is injective, so $\\operatorname{Tor}^1(k(b),\\mathcal{O}_{m\\mathcal{D}})=0$ as desired.\n\nFinally, $\\mathcal{I}_{\\mathcal{D}_p}^{[m]}$ is $S_2$ and $\\mathcal{O}_{\\mathcal{X}_p}$ is $S_2$ from condition (2), so $\\mathcal{O}_{(m\\mathcal{D})_p}$ is $S_1$ from \\cite{Kol13}*{Corollary 2.61}.\\end{proof}\n\n\\begin{Remark}\\label{remark_for_m_big_enough_kollar_comndition_guarantees_cartier}\nThere is an $m>0$, which does not depend on $B$, such that $\\omega_{\\mathcal{X}\/B}^{[m]}$ and $\\mathcal{I}_\\mathcal{D}^{[m]}$ are locally free on $\\mathcal{X}$.\nIndeed, we will prove in Theorem \\ref{thm_boundedness} that there is an $m$ such that for every fiber $(X;D)$ of $f$, the sheaves $\\omega_X^{[m]}$ and $\\mathcal{O}_X(-mD)$ are Cartier. Since our family is bounded (see Theorem \\ref{thm_boundedness}), we can choose such an $m$ which does not depend on the basis $B$.\nThen, by conditions (1) and (2) and Remark \\ref{remark:bc:implies:flat}, we may apply  \\cite{stacks-project}*{Tag 00MH}, which implies that $\\omega_{\\mathcal{X}\/B}^{[m]}$ and $\\mathcal{I}_\\mathcal{D}^{[m]}$ are locally free since they restrict to locally free sheaves along each fiber.\n\\end{Remark}\n\nNow, we specify the morphisms in the fibered category $\\mathscr{F}_{n,p,I}$ over a morphism $f \\colon T\\to B.$\nLet $\\alpha = ((\\mathcal{Y},\\mathcal{D}_\\mathcal{Y})\\to T)$ be an element of $\\mathscr{F}_{n,p,I}(T)$, and let $\\beta = ((\\mathcal{X};\\mathcal{D}_\\mathcal{X})\\to B)$ be an element of $\\mathscr{F}_{n,p,I}(B)$.\nAn arrow $\\alpha \\to \\beta$ is the data of two morphisms $(g,h)$ which fit in a diagram like the one below, where all the squares are fibered diagrams:\n\\[\n\\xymatrix{\\mathcal{D}_\\mathcal{Y}\\ar[r]^g\\ar[d] & \\mathcal{D}\\ar[d] \\\\ \\mathcal{Y}\\ar[d] \\ar[r]^h & \\mathcal{X}\\ar[d] \\\\ T\\ar[r]^f & B.}\n\\]\n\n\\begin{Oss}\nObserve that the only morphisms over the identity $\\operatorname{Id} \\colon B\\to B$ are isomorphisms.\nThus, $\\mathscr{F}_{n,p,I}$ is fibered in groupoids.\n\\end{Oss}\n\n\\subsection{Hassett's example} \\label{hassett example}\nIn this subsection, we present a well-known example due to Hassett that is helpful to keep in mind to navigate the rest\nof the paper.\nSee also \\cite{KP17}*{Subsection 1.2} or \\cite{kol1}. \n\nConsider the DVR $R = \\operatorname{Spec}(k[t]_{(t)})$, let $\\eta$ (resp. $p$) be the generic (resp. closed) point of $\\operatorname{Spec}(R)$, and let $\\mathcal{X}=\\mathbb{P}^1\\times \\mathbb{P}^1 \\times \\operatorname{Spec}(R)$.\nConsider $C$ a smooth member of $\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(1,2)$, and let $\\mathcal{Y}$ be the blow-up of $C$ in the special fiber of $\\mathcal{X}$.\nThen, if we compose the blow-down $\\mathcal{Y}\\to \\mathcal{X}$ with the projection $\\mathcal{X}\\to \\operatorname{Spec}(R)$, we get a family of surfaces $\\mathcal{Y}\\to \\operatorname{Spec}(R)$ where the generic fiber is a copy of $\\mathbb{P}^1\\times \\mathbb{P}^1$, while the special fiber is a surface with two irreducible components.\nOne irreducible component of the special fiber is isomorphic to $\\mathbb{P}^1\\times \\mathbb{P}^1$\n(the proper transform of the special fiber of $\\mathcal{X}\\to \\operatorname{Spec}(R)$), and the other one is the exceptional divisor $F$.\nThe surface $F$ is the projectivization of the normal bundle of $C$.\nSince $C\\cong \\mathbb{P}^1$ and the normal bundle of $C$ in $\\mathcal{X}$ is isomorphic to $\\mathcal{O}_{\\mathbb{P}^1}(0) \\oplus \\mathcal{O}_{\\mathbb{P}^1}(4)$, we have that $F$ is isomorphic to the Hirzebruch surface $\\mathbb{F}_4$.\nWe denote by $\\Delta\\subseteq \\mathbb{F}_4$ the preimage of the double locus of the central fiber on $\\mathbb{F}_4$.\n\nWe consider a divisor on $\\mathbb{P}^1\\times \\mathbb{P}^1$ consisting of five irreducible components, three general members of $\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(1,2)$ (which we denote by $C_1, C_2, C_3$), $C$, and a smooth member $G$ of $\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(2,0)$.\nWe consider a deformation of $C_1+C_2+C_3+C + D$ in $\\mathcal{X}$ given by the trivial deformation of $C$ and $D$, and we deform $C_i$ to $C$ for every $i$.\nWe denote by $\\mathcal{D}_\\mathcal{X}$ the total space of this deformation, and by $\\mathcal{D}$ its proper transform in $\\mathcal{Y}$.\nMore explicitly, if $C$ is the zero locus of a global section $g\\in H^0(\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(C))$ and $\\varphi_1, \\varphi_2, \\varphi_3$ are generic sections of $H^0(\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(C))$, the deformation we consider is $\\mathcal{D}_\\mathcal{X} = V(g(t\\varphi_1 - g)(t\\varphi_2 - g)(t\\varphi_3 - g))$.\n\n\\begin{center}\n\\includegraphics[scale=0.22]{Y_and_div.jpg}\\\\ Figure 1: the family $(\\mathcal{Y},c\\mathcal{D})\\to \\operatorname{Spec}(R).$\n\\end{center}\n\nNow, we construct the ample model of $(\\mathcal{Y},c\\mathcal{D})\\to \\operatorname{Spec}(R)$, with the coefficient $c$ in a neighbourhood of $\\frac{1}{2}$.\nFirst, we introduce some notations.\nWe denote $K_{\\mathcal{Y}\/\\operatorname{Spec}(R)}+c\\mathcal{D}$ by $\\mathcal{L}(c)$, the irreducible component of the central fiber isomorphic to $\\mathbb{P}^1\\times \\mathbb{P}^1$ by $\\mathscr{P}$, and the two rulings of $\\mathscr{P}$ by $f_1$ and $f_2$, with the convention that $C = f_1+2f_2$.\nSince the generic fiber $(\\mathcal{Y},c\\mathcal{D})_\\eta$ is stable for every $c$ with $|c-\\frac{1}{2}| \\ll 1$, we just need to control the intersection pairings on the special fiber.\n\nLet $c=\\frac{1}{2} + \\epsilon$, where $|\\epsilon| \\ll 1$.\n\\begin{center}\n\\includegraphics[scale=0.20]{special_fiber.jpg} \\\\ Figure 2: special fiber of the family $(\\mathcal{Y},c\\mathcal{D})\\to \\operatorname{Spec}(R)$.\n\\end{center}\nOne can check that:\n\\begin{enumerate}\n\\item when $\\epsilon > 0$, the divisor $\\mathcal{L}(\\frac{1}{2} + \\epsilon)$ is nef.\nIt is ample on $F$, positive on $f_2$ and 0 on $f_1$;\n\\item when $\\epsilon = 0$, the divisor $\\mathcal{L}(\\frac{1}{2})$ is nef.\nIt is 0 on $\\mathscr{P}$ and on $\\Delta$; and\n\\item when $\\epsilon < 0$, the divisor $\\mathcal{L}(\\frac{1}{2}+\\epsilon)$ is  not nef.\nOn $\\mathscr{P}$, it is negative on $f_2$ and 0 on $f_1$, while on $F $ it is negative on $\\Delta$.\n\\end{enumerate}\nTherefore, we can explicitly describe the special fiber of the ample model of $(\\mathcal{Y},c\\mathcal{D})$ over $\\operatorname{Spec}(R)$. We denote by $(\\mathcal{Z}^+,c\\mathcal{D}^+)$ (resp. $(\\mathcal{Z}^0, c\\mathcal{D}^0)$, $(\\mathcal{Z}^-,c\\mathcal{D}^-)$) the ample model of $(\\mathcal{Y},c\\mathcal{D})\\to \\operatorname{Spec}(R)$ when $0<\\epsilon\\ll 1$ (resp. $\\epsilon=0$, $-1 \\ll \\epsilon < 0$):\n\\begin{enumerate}\n\\item when $\\epsilon > 0$, to construct the canonical model we contract the ruling $f_1$.\nSince via this contraction the map $C\\subseteq \\mathbb{P}^1\\times \\mathbb{P}^1 \\to \\mathbb{P}^1$ is a $2:1$ ramified cover of $\\mathbb{P}^1$, the special fiber is the push-out of the following diagram:\n$$\\xymatrix{\\mathbb{P}^1\\cong \\Delta \\ar[r] \\ar[d]^{2:1} \\ar[r] & F\\ar[d]_\\pi \\\\ \\mathbb{P}^1\\ar[r] & \\mathcal{Z}^+_p}$$\n\\item when $\\epsilon = 0$, to construct the canonical model, we contract $\\mathscr{P}$ and $\\Delta$.\nThe special fiber is isomorphic to $\\mathbb{F}_4$ with the section $\\Delta$ contracted, which is the projectivization of the cone over a rational quartic curve; and\n\\item When $\\epsilon < 0$, we perform a divisorial contraction to make the divisor nef.\nWe contract the ruling $f_2$, and the special fiber is $\\mathbb{F}_4$.\n\\end{enumerate}\nIn particular, there are morphisms $\\pi^+ \\colon \\mathbb{F}_4\\to \\mathcal{Z}^+_p$,  $\\pi^0 \\colon \\mathbb{F}_4\\to \\mathcal{Z}^0_p$ and $\\pi^- \\colon \\mathbb{F}_4\\to \\mathcal{Z}^-_p$ from $\\mathbb{F}_4$ to the special fibers of $\\mathcal{Z}^+$, $\\mathcal{Z}^0$ and $\\mathcal{Z}^-$.\nThe divisors $\\mathcal{D}^+_p$, $\\mathcal{D}^0_p$ and $\\mathcal{D}^-_p$ can be described as follows.\nFirst, recall that $\\mathbb{F}_4$ is the projectivization of a vector bundle on $\\mathbb{P}^1$, so it has a relative $\\mathcal{O}_{\\mathbb{F}_4}(1)$.\nWe denote a generic section of $\\mathcal{O}_{\\mathbb{F}_4}(1)$ by $h$.\nThen, the following holds:\n\n\\begin{enumerate}\n\\item when $\\epsilon > 0$, the divisor $\\mathcal{D}^+_p$ is the image via $\\pi^+$ of four generic fibers of $F$, together with a divisor linearly equivalent to $4h$. All the components have coefficient $\\frac{1}{2} + \\epsilon$;\n\\item when $\\epsilon = 0$, the divisor $\\mathcal{D}^0_p$ is the image via $\\pi^0$ of four generic fibers with a divisor linearly equivalent to $4h$.\nAll the components have coefficient $\\frac{1}{2}$; and\n\\item when $\\epsilon < 0$, the divisor $\\mathcal{D}^-_p$ consists of four generic fibers with a divisor linearly equivalent to $4h$, four generic fibers, and $\\Delta$.\nAll the components have coefficient $\\frac{1}{2} + \\epsilon$, with the exception of $\\Delta$, which has coefficient $1+2\\epsilon$.\n\\end{enumerate}\n\n\nRecall now that the flat limit of $\\mathcal{D}_{\\eta}\\subseteq \\mathcal{Z}^0_\\eta$ in $\\mathcal{Z}^0_p$ is \\emph{not} $S_1$, since it has an embedded point (see \\cite{KP17}*{Subsection 1.2}). However, $(\\mathcal{Z}^-;\\mathcal{D}^-)\\to \\operatorname{Spec}(R)$ is a $p$-stable morphism with coefficients $I = \\{\\frac{1}{2}, 1\\}$ (see Proposition \\ref{prop:flat:limit:is:S1}), so in particular the flat limit of $\\mathcal{D}_\\eta$ in $\\mathcal{Z}_0^-$ does \\emph{not} have an embedded point on the special fiber.\n\n\n\\subsection{$p$-stable morphisms with constant part}\nFor proving that $\\mathscr{F}_{n,p,I}$ is bounded, it will be useful to introduce the following definition.\n\n\\begin{Def}\nAssume that $S$ is reduced.\nA \\emph{locally $p$-stable morphism with constant part} $f \\colon (\\mathcal{X},\\Omega;\\mathcal{D}) \\rar S$ over $S$ and with coefficients in $I$ is the datum of a proper morphism $f \\colon \\mathcal{X} \\rar S$, a closed subscheme $\\mathcal{D}$ on $\\mathcal{X}$, and an effective $\\qq$-divisor $\\Omega$ on $X$, such that the following conditions hold:\n\\begin{itemize}\n    \\item $f \\colon (X,\\Omega) \\rar S$ is a proper, locally stable family of pairs;\n    \\item $\\mathcal{D} \\rar S$ is flat and relatively $S_1$ (namely, with no embedded points) with fibers of pure dimension $n-1$; and\n    \\item for every $s \\in S$, there is a $p$-pair $(X,\\Delta;D)$  with coefficients in $I$ such that $X=\\mathcal{X}_s$, $\\Delta=\\Omega_s$, and $D^{sc}=\\mathcal{D}_s$.\n\\end{itemize}\nFurthermore, if we have a polynomial $p(t) \\in \\qq[t]$ and for every $s \\in S$, $(\\mathcal{X}_s;\\mathcal{D}_s)$ has polynomial $p(t)$, we say that it is a family of $p$-pairs with polynomial $p(t)$.\n\\end{Def}\n\n\\begin{Notation}\nWe say that a locally $p$-stable morphism with constant part (with coefficients in $I$ and polynomial $p(t)$) is a \\emph{$p$-stable morphism with constant part} (\\emph{with coefficients in $I$ and polynomial $p(t)$}) if $\\K \\mathcal{X}\/S. + \\frac{1}{r} \\mathcal{D}$ (respectively, $\\K \\mathcal{X}\/S. + \\frac{1}{r} \\mathcal{D} + \\Omega$) is $f$-ample.\nHere, recall the $r$ denotes the index of $I$.\n\\end{Notation}\n\n\\begin{Prop} \\label{prop_well_defined_p-pairs}\nLet us fix a set of coefficients $I \\subset (0,1] \\cap \\qq$, a polynomial $p$ and an integer $n$.\nLet $r$ denote the index of $I$.\nLet $f \\colon (\\mathcal{X},\\Omega;\\mathcal{D}) \\rar S$ be a $p$-stable morphism with constant part $\\Omega$ over a reduced base $S$.\nThen $f \\colon (\\mathcal{X}, \\frac{1}{r}\\mathcal{D}+\\Omega) \\rar S$ is a well defined family of pairs.\n\\end{Prop}\n\\begin{proof}\nBy definition $\\mathcal{D} \\rar S$ is flat with fibers of pure dimension $n-1$.\nThus, it follows that the fibers of $\\mathrm{Supp}(\\mathcal{D} + \\Omega) \\rar S$ are either empty or of pure dimension $n-1$.\nThus, to show that $f \\colon (\\mathcal{X}, \\frac{1}{r}\\mathcal{D} + \\Omega) \\rar S$ is a family of pairs, we are left with showing that $f$ is smooth at the generic points of $X_s \\cap \\operatorname{Supp}(\\mathcal{D} + \\Omega)$ for every $s \\in S$.\n\nBy assumption, this is the case for all the generic points of $X_s \\cap \\operatorname{Supp}(\\mathcal{D} + \\Omega)$ arising from $\\Omega$.\nThus, we may focus on the contribution of $\\mathcal{D}$.\nBut then, since each fiber $(\\mathcal{X}_s,\\Omega_s;\\mathcal{D}_s)$ is a $p$-pair, it follows that the generic points of $\\mathcal{D}_s$ are contained in the smooth locus of $X_s$ by the semi-log canonical condition.\nThus, $f \\colon (\\mathcal{X}, \\frac{1}{r}\\mathcal{D} + \\Omega) \\rar S$ is a family of pairs.\n\nTo conclude, we need to show that the family is well defined.\nTo this end, it suffices we focus on $\\mathcal{D}$, as $\\Omega$ satisfied the needed conditions by definition.\nAs argued in \\cite{kol1}*{Definition 3.66}, there is a big open subset $U \\subset \\mathcal{X}$ such that every point of $U$ is either smooth or nodal, and $U \\cap \\mathcal{X}_s$ has codimension at least 2 in $\\mathcal{X}_s$ for every $s \\in S$.\nFor every $s \\in S$, $\\mathcal{D}_s$ does not contain any irreducible component of the double locus.\nThus, the intersection between $\\mathcal{D}$ and $U$ has codimension at least 2 in every fiber.\nLet $V$ denote the open set obtained by removing this intersection from $U$.\nThen, as the scheme theoretic restriction $\\mathcal{D}_s$ is an integral Weil divisor, it is a Carter divisor along $V_s$.\nThen, by \\cite{stacks-project}*{Tag 062Y}, $\\mathcal{D}$ is a Cartier divisor along $V$.\nThus, the claim follows.\n\\end{proof}\n\n\\begin{Prop} \\label{prop_stable_family_p-pairs}\nFix an integer $n\\in \\mathbb{N}$, a polynomial $p(t)\\in \\mathbb{Q}[t]$, and a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$.\nAssume that assumption (A) holds, and let $\\epsilon_0>0$ be as in Lemma \\ref{lemma_bound_for_epsilon}.\nLet $f \\colon (\\mathcal{X},\\Omega;\\mathcal{D}) \\rar S$ be a $p$-stable morphism with constant part, coefficients in $I$, and polynomial $p(t)$, over a reduced base $S$.\nThen $f \\colon (\\mathcal{X}, \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega) \\rar S$ is a stable family of pairs for every $0 < \\epsilon < \\epsilon_0$ rational.\nIn particular, $\\mathcal{D}$ is $\\qq$-Cartier.\n\\end{Prop}\n\n\\begin{proof}\nFix $\\epsilon$ as in the statement.\nThen, by Proposition \\ref{prop_well_defined_p-pairs}, $f \\colon (\\mathcal{X}, \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega) \\rar S$ is a well defined family of pairs.\nBy assumption, the self-intersection $(\\K \\mathcal{X}_s. + \\frac{1-\\epsilon}{r}\\mathcal{D}_s + \\Omega_s)^n$ is independent of $s \\in S$, as it is $p(1-\\epsilon)$.\nSince $f \\colon (\\mathcal{X}, \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega) \\rar S$ is a well defined family of pairs, we may find a big open subset $U \\subset \\mathcal{X}$ such that $U_s$ has codimension at least 2 for every $s \\in S$ and $\\K \\mathcal{X}\/S. + \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega$ is $\\qq$-Cartier along $U$.\nThen, $\\K \\mathcal{X}\/S. + \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega$ is $\\qq$-Cartier by \\cite{kol1}*{Theorem 5.7}.\nThe claim follows by \\cite{kol1}*{Theorem-Definition 4.45} and the fact that the argument was independent of $\\epsilon \\in (0,\\epsilon_0) \\cap \\qq$.\n\\end{proof}\n\n\n\\subsection{Existence of good minimal models}\nLet $(X,\\Delta)$ be a log canonical pair, and let $f \\colon X \\rar S$ be a projective morphism such that $\\K X. + \\Delta$ is $f$-pseudo-effective.\nThen, it is expected that $X$ admits a good minimal model over $S$.\nThat is, $X$ admits a birational contraction $\\phi \\colon (X,\\Delta) \\dashrightarrow (X',\\Delta')$ over $S$ to a log canonical pair $(X',\\Delta')$, such that $\\Delta'$ is the push-forward of $\\Delta$ to $X'$, $\\phi$ is $(K_X+\\Delta)$-negative, and $\\K X'. + \\Delta'$ is semi-ample over $S$.\n\nHere, we collect a technical statement that shows the existence of relative good minimal models under certain assumptions.\n\n\\begin{Lemma} \\label{lemma minimal models}\nLet $(X,\\Delta)$ be a log canonical pair, and let $f \\colon X \\rar S$ be a projective morphism such that $\\K X. + \\Delta$ is $f$-pseudo-effective.\nAssume that $f$ has relative dimension at most 3 and that every log canonical center of $(X,\\Delta)$ dominates $S$.\nThen, up to extracting some log canonical places of $(X,\\Delta)$, $(X,\\Delta)$ admits a relative good minimal model over $S$.\nFurthermore, if $\\K X. + \\Delta$ is nef over $S$, then it is semi-ample.\n\\end{Lemma}\n\n\\begin{Remark}\nThe assumption on the relative dimension in Lemma \\ref{lemma minimal models} is only needed to guarantee the existence of a good minimal model (possibly after extracting some log canonical center).\nIf the existence of a good minimal model (possibly after extracting some log canonical places) is known by other means, then the nefness of $K_X+\\Delta$ still implies its semi-ampleness, regardless of the relative dimension of $f$.\n\\end{Remark}\n\n\\begin{proof}\nLet $\\pi \\colon X' \\rar X$ be a log resolution of $(X,\\Delta)$, and let $\\pi^*(K_X+\\Delta)= \\K X'. + \\Delta' + E' + \\Gamma' - F'$.\nHere $\\Delta'$ denotes the strict transform of $\\Delta$, the divisors $E'$, $\\Gamma'$, and $F'$ are effective, $\\pi$-exceptional, and share no common components.\nFurthermore, $\\Gamma'$ is reduced, while the coefficients of $E'$ are in $(0,1)$.\nLet $\\Xi'$ denote the reduced $\\pi$-exceptional divisor, and fix a rational number $0 < \\epsilon \\ll 1$.\nThen, $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ is dlt, and it has the same pluricanonical ring as $(X,\\Delta)$.\nFurthermore, by the addition of $\\epsilon (\\Xi' - \\Gamma')$, every $\\pi$-exceptional divisor that is not in $\\Gamma$ is in the relative stable base locus of $\\K X'. +\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma')$.\nFinally, by assumption and the choice of $\\epsilon$, every log canonical center of $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ dominates $S$.\n\nSince the fibers have dimension at most three, by \\cite{KMM94} and \\cite{HMX18}*{Theorem 1.9.1}, it follows that $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ has a relative good minimal model over a non-empty open subset $U \\subseteq S$.\nThen, as by assumption there are no vertical log canonical centers, it follows from \\cite{HX13}*{Theorem 1.1} that $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ has a relative good minimal model over $S$.\nSince every $\\pi$-exceptional divisor that is not in $\\Gamma'$ is in the relative stable base locus, any such divisor is contracted on the minimal model.\nThis shows the first part of the claim.\n\nNow, assume that $K_X + \\Delta$ is relatively nef.\nThen, $(X,\\Delta)$ is a relative weak log canonical model for $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ in the sense of \\cite{HMX}.\nThen, we conclude by \\cite{HMX}*{Lemma 2.9.1} that $(X,\\Delta)$ is a relatively semi-ample model.\nNotice that \\cite{HMX}*{Lemma 2.9.1} is phrased for projective pairs.\nOn the other hand, by \\cite{HX13}*{Corollary 1.2}, one can first take a projective closure of $\\tilde X$ over a compactification $\\overline{S}$ of $S$, and take a projective relative good minimal model.\nThen, by adding the pull-back of some divisor on $\\overline{S}$, we can regard the relative good minimal model as a projective minimal model.\nThus, it follows that $K_{X} + \\Delta$ is relatively semi-ample over $S$, and it defines a morphism to the relative ample model.\n\\end{proof}\n\n\n\n\\section{Boundedness}\\label{section_boundedness}\nThe goal of this section is to prove that, if we fix a set of coefficients $I$, a polynomial $p$ and a dimension $n$, the corresponding set of $p$-stable pairs is effectively log-bounded. \n\\begin{Lemma} \\label{lemma_nef_generalizes}\nLet $f \\colon Y \\rar B$ be a flat and proper morphism, and let $L$ be a line bundle on $Y$.\nThen, the locus of points $b \\in B$ such that $L_b$ is nef is closed under generalization.\n\\end{Lemma}\n\n\\begin{proof}\nWithout loss of generality, we may assume that $B=\\operatorname{Spec}(R)$ and $R$ is a DVR.\nLet also $0$ (resp. $\\eta$) be the closed (resp. generic) point of $\\operatorname{Spec}(R)$.\n\nAssume by contradiction that $L$ is nef on the special fiber, but it is not nef on the generic one.\nThen, there is an irreducible curve $C'_\\eta\\subseteq Y_\\eta$ that is negative for $L_\\eta$.\nLet $C_\\eta \\to C_\\eta'$ denote the normalization of $C'_\\eta$.\nBy semistable reduction, up to replacing $\\operatorname{Spec}(R)$ with a ramified cover, we can compactify  $C_\\eta\\to Y_\\eta$ to a family of curves $C\\to \\operatorname{Spec}(R)$ where the special fiber is nodal and maps to $Y$.\nThe situation is described by the following diagram:\n\\[\n\\xymatrix{C_\\eta \\ar[r] \\ar[d] & C \\ar[d] \\\\ Y_\\eta \\ar[r] \\ar[d] & Y \\ar[d] \\\\ \\eta \\ar[r] & \\operatorname{Spec}(R).}\n\\]\nLet $G$ denote the pull-back of $L$ to $C$.\nBy assumption, $G_\\eta$ has negative degree, while $G_0$ has non-negative degree on each irreducible component of $C_0$.\nThen we can consider the cohomology groups $H^1(C_\\eta, G_{\\eta}^{\\otimes n})$ and $H^1(C_0,G_0^{\\otimes n})$.\nBy Riemann--Roch, $H^1(C_\\eta, G_{\\eta}^{\\otimes n})$ grows linearly in $n$ and is unbounded, while $H^1(C_0,G_0^{\\otimes n})$ is bounded.\nThis contradicts the upper-semicontinuity theorems for cohomology and base change.\n\\end{proof}\n\n\n\\begin{Prop}\\label{prop:nef:open}\nFix an integer $n\\in \\mathbb{N}$.\nAssume that assumption (A) holds.\nConsider a locally stable family of stable pairs of relative dimension $n$ $(\\mathcal{X},\\mathcal{D})\\to B$ over a reduced scheme $B$, with $\\mathcal{D}$ being $\\mathbb{Q}$-Cartier.\nAssume that there is an $0<\\epsilon_0 < 1$ such that  $(\\mathcal{X},(1-\\epsilon_0)\\mathcal{D})\\to B$ is stable.\nThen, the set of points $p\\in B$ such that $K_{\\mathcal{X}\/B} + \\mathcal{D}$ is nef is open.\n\\end{Prop}\n\n\\begin{proof}\nIt is well known that a subset is open if and only if it is constructible and stable under generalization, see \\cite{stacks-project}*{Tag 0542}.\nBy Lemma \\ref{lemma_nef_generalizes}, the set of interest is stable under generalization.\nThus, we are left with showing that it is constructible.\n\nRecall that, if $f \\colon B'\\to B$ is a morphism between schemes of finite type, $f(B')$ is a constructible subset of $B$.\nIn particular, we can replace $B$ with a resolution, and assume that $B$ is smooth and irreducible.\n\n{\\bf Step 1:} In this step, we reduce the proof to the case when $\\mathcal{X}$ and its general fiber are normal.\n\nIf $\\nu \\colon Z^n\\to Z$ is the normalization of a scheme and $L$ is a line bundle on $Z$, then $L$ is nef if and only if $\\nu^*L$ is nef.\nTo reduce to the case when $\\mathcal{X}$ is normal, we will produce a simultaneous normalization of the family $\\mathcal{X} \\to B$.\nThis will be achieved by stratifying $B$.\n\n\nWe begin by considering the generic point $\\eta\\in B$, and we spread out the normalization $\\mathcal{Y}'$ of $\\mathcal{X}_\\eta$, together with its conductor $\\Gamma_{\\mathcal{Y}}$, along an open subset $U$ of $B$ to get a pair $(\\mathcal{Y},\\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} +\\Gamma_{\\mathcal{Y}})$.\nUp to shrinking $U$, we can assume that $(\\mathcal{Y},\\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} +\\Gamma_{\\mathcal{Y}}) \\to U$ is the normalization fiber by fiber, with conductor $\\Gamma_{\n\\mathcal{Y}}$.\nWe proceed then by Noetherian induction.\nAt the end of this procedure, up to stratifying $B$, we get a family of (possibly with multiple irreducible components) log canonical pairs $(\\mathcal{Y},\\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} +\\Gamma_{\\mathcal{Y}}) \\to B$.\nFurthermore, we assume that $B$ is smooth: this can be achieved by resolving $B$ and pulling back the family, or by stratifying $B$.\n\nIn the above procedure, it may be that, over an irreducible component of $B$, the family $\\mathcal{Y} \\rar B$ is not irreducible.\nOn the other hand, by construction, each irreducible component of $(\\mathcal{Y},\\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} +\\Gamma_{\\mathcal{Y}}) \\to B$ provides a family of irreducible log canonical pairs.\nThen,\n$(K_{\\mathcal{X}\/B} + \\mathcal{D} + \\Omega)|_{{\\mathcal{X}}_p}$ is nef if and only if so is its pull-back to each irreducible component of $\\mathcal{Y}_p$.\nSince $\\mathcal{Y} \\rar B$ has finitely many irreducible components, the set \n\\[\n\\{p \\in B : (K_{\\mathcal{X}\/B} + \\mathcal{D} + \\Omega)|_{{\\mathcal{X}}_p}\\text{ is nef}\\}\n\\]\nis locally closed if and only if so are the sets corresponding to the same property for each irreducible component of $\\mathcal{Y} \\rar B$.\nThus, in the following, we may assume that $\\mathcal{Y} \\rar B$ is a morphism with connected fibers.\n\n{\\bf Step 2:} We conclude the proof by addressing the case when $\\mathcal{X} \\rar B$ has normal fibers.\nWe will use \\cite{HMX}*{Theorem 1.9.1} to finish the proof.\n\nConsider an irreducible component, which we will still denote by $\\pi \\colon (\\mathcal{Y},  \\mathcal{D} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}) \\rightarrow B$,\nand assume that  for a point $0\\in B$, the divisor $K_{\\mathcal{Y} _0}+ \\mathcal{D} _{0} + \\Omega_0 + \\Gamma_0$ is nef.\nIn the following, we will perform several stratifications of $B$.\nWe will always assume that $0$ (or some other member of our moduli problem, which will be then denoted by $0$) is still a point in the new stratum, and we will disregard the strata not containing members of our moduli problem. \n\nUp to a stratification (which we omit in the notation), we may assume that $\\pi$ admits a log resolution in family.\nCall it $\\rho \\colon (\\mathcal{Y}',  \\mathcal{D}' + \\Omega' + \\Gamma' + \\mathcal{F}) \\rightarrow B$, where $\\mathcal{F}$ is the reduced exceptional divisor, and the remaining divisors are the strict transforms of $\\mathcal{D}_\\mathcal{Y}$, $\\Omega_\\mathcal{Y}$ and $\\Gamma_\\mathcal{Y}$, respectively.\nThen, $(\\mathcal{Y}_0,  \\mathcal{D}_0 + \\Omega_0 + \\Gamma_0)$ is a minimal model for $(\\mathcal{Y}'_0,  \\mathcal{D}'_0 + \\Omega'_0 + \\Gamma'_0 + \\mathcal{F}_0)$.\n\nBy our assumptions on the existence of good minimal models for projective pairs, $(\\mathcal{Y}_0,  \\mathcal{D}_0 + \\Omega_0 + \\Gamma_0)$ is a good minimal model for $(\\mathcal{Y}'_0,  \\mathcal{D}'_0 + \\Omega'_0 + \\Gamma'_0 + \\mathcal{F}_0)$.\nThus, by \\cite{HMX18}*{Theorem 1.9.1}, $\\rho \\colon (\\mathcal{Y}',  \\mathcal{D}' + \\Omega' + \\Gamma' + \\mathcal{F}) \\rightarrow B$ admits a relative good minimal model over $B$, and it admits a relative ample model, which computes the ample model fiberwise.\nBy construction, this is the relative ample model of $\\pi \\colon (\\mathcal{Y},  \\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}) \\rightarrow B$.\nCall this model $\\sigma \\colon (\\mathcal{Y}^c,  \\mathcal{D}^c + \\Omega^c + \\Gamma^c) \\rightarrow B$.\n\nLet $p \\colon \\mathcal{W} \\rightarrow \\mathcal{Y}$ and $q \\colon \\mathcal{W} \\rightarrow \\mathcal{Y}^c$ be a common resolution.\nUp to a stratification, we may assume that this induces a resolution fiberwise, and the exceptional divisors of $p$ and $q$ are flat over $B$.\nIn particular, every fiber $\\mathcal{W}_0$ intersects all the exceptional divisors of $p $ and $q$.\nThen, we have\n\\begin{equation} \\label{eq_discr_constructible}\np^*(K_{\\mathcal{Y}} + \\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}) \\geq q^*(K_{\\mathcal{Y}^c} + \\mathcal{D}^c + \\Omega^c + \\Gamma^c).\n\\end{equation}\nOn the other hand, $K_{\\mathcal{Y}_0} + \\mathcal{D}_0 + \\Omega_0 + \\Gamma_0$ is the pull-back of $K_{\\mathcal{Y}^c_0} + \\mathcal{D}^c_0 + \\Omega^c_0 + \\Gamma_0^c$ under the contraction $\\mathcal{Y}_0 \\rightarrow \\mathcal{Y}^c_0$, so\n\\begin{equation} \\label{eq_dirsc2_constructible}\n(p^*(K_{\\mathcal{Y}} + \\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}))|_{\\mathcal{W}_0} = (q^*(K_{\\mathcal{Y}^c} + \\mathcal{D}^c + \\Omega^c + \\Gamma^c))|_{\\mathcal{W}_0} .\n\\end{equation}\nFrom how the resolution $\\mathcal{W}$ and the stratification of $B$ are constructed, the fiber $\\mathcal{W}_0$ intersects all the $p$ and $q$ exceptional divisors.\nIn particular, by \\eqref{eq_dirsc2_constructible}, \\eqref{eq_discr_constructible} has to be an equality.\nThus, $K_{\\mathcal{Y}} + \\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}$ is relatively nef over the stratum of $0 \\in B$.\nThis shows that\n\\[\n\\{p \\in B : (K_{\\mathcal{X}\/B} + \\mathcal{D} + \\Delta)|_{{\\mathcal{X}}_p}\\text{ is nef}\\}.\n\\]\nis locally closed and it concludes the proof.\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm_boundedness}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that assumption (A) holds.\nThen, the set of $p$-stable pairs $(X,\\Delta;D)$ of dimension $n$, polynomial $p(t)$ and coefficients in $I$ is effectively log bounded.\n\\end{theorem}\n\\begin{proof}\nWe proceed in several steps.\n\n{\\bf Step 1:} In this step, we show that the $p$-stable pairs of interest are log bounded.\n\nFrom Lemma \\ref{lemma_bound_for_epsilon}, there is an $\\epsilon_0>0$ such that, for every $p$-stable pair $(X,\\Delta;D)$ as in the statement, $(X,(1-\\epsilon_0)D + \\Delta)$ is a stable pair.\nThen, from \\cite{HMX}, there is a bounding family $(\\mathcal{X},\\mathcal{E})\\to B$ of stable pairs of volume $p(1-\\epsilon_0)$, coefficients in the finite set $(1-\\epsilon_0)I\\cup I \\cup \\{1\\}$ and dimension $n$.\n\n{\\bf Step 2:} In this step, we show that the $p$-stable of pairs of interest are strongly log bounded.\nFurthermore, we may choose the family to be locally stable.\n\nSince the set of coefficients involved is finite, up to taking finitely many copies of the family, we may find divisors $\\mathcal{D}$ and $\\Omega$ supported on $\\mathcal{E}$ such that $(1-\\epsilon_0)\\mathcal{D}$ restricts to $(1-\\epsilon_0)D$ fiberwise, and $\\Omega$ restricts to $\\Delta$ fiberwise.\nBy \\cite{kol1}*{Lemma 4.48}, up to replacing $B$ with a finite disjoint union of locally closed subsets, we can further assume that both $(\\mathcal{X},(1-\\epsilon)\\mathcal{D} + \\Omega)\\to B$ and\n$(\\mathcal{X},\\mathcal{D} + \\Omega)\\to B$ are locally stable.\nIn particular, $K_{\\mathcal{X}\/B} + t\\mathcal{D} + \\Omega$ is $\\mathbb{Q}$-Cartier for any $t$.\nFurthermore, by flatness, up to disregarding some irreducible components of $B$, we can assume that for every $b\\in B,$ $p(t) = (K_{\\mathcal{X}\/B} + t\\mathcal{D} + \\Omega)^{\\dim(X)}$.\nFinally, up to stratifying $B$, we may assume that each irreducible component of $B$ is smooth;\nin particular, $K_B$ is well defined, and it follows that the pairs are strongly log bounded. \n\n{\\bf Step 3:} In this step we finish the proof.\n\nBy construction, for the choice of $t = 1-\\epsilon_0$, $K_{\\mathcal{X}\/B} + (1-\\epsilon_0)\\mathcal{D} + \\Omega$ is ample on the general fibers.\nThus, up to removing some proper closed subset of $B$, we may assume that $(\\mathcal{X},(1-\\epsilon_0)\\mathcal{D} + \\Omega)\\to B$ is a stable family.\nThus, to conclude the proof it suffices to use Proposition \\ref{prop:nef:open}, which guarantees that the set $\n\\{p \\in B : (K_{\\mathcal{X}\/B} + \\mathcal{D} + \\Omega)|_{{\\mathcal{X}}_p}\\text{ is nef}\\}$ is open.\n\\end{proof}\n\n\n\n\n\\section{The moduli functor}\n\\label{section functor}\n\n\nThe goal of this section is to prove that $\\mathscr{F}_{n,p,I}$ is an algebraic stack. We begin by the following proposition:\n\n\n\\begin{Prop}\nThe fibered category $\\mathscr{F}_{n,p,I}$ is a stack.\n\\end{Prop}\n\n\\begin{proof}\nSince our argument follows the same strategy in \\cite{Alp21}*{Proposition 1.4.6}, we only sketch the salient steps here.\nThe role that in \\emph{loc. cit.} is the one of $\\omega_\\mathscr{C}^{\\otimes 3}$, for us is $L \\coloneqq \\mathcal{O}_{X}(m(K_X + (1-\\epsilon)D))$, where $\\epsilon$ and $m$ are chosen such that $L$ is very ample with $h^i(X,L)=0$ for $i \\geq 1$ and such that $H^0(X,L)\\to H^0(D^{sc},L|_{D^{sc}})$ is surjective.\nThese $m$ and $\\epsilon$ can be chosen uniformly by Theorem \\ref{thm_boundedness}. \n\nThe fact that isomorphisms are a sheaf in the \\'etale topology of $\\mathscr{F}_{n,p,I}$ follows from descent as in \\cite{Alp21}*{Proposition 1.4.6}.\n\nFor proving that $\\mathscr{F}_{n,p,I}$ satisfies descent, we begin by the following observation.\nConsider an object $f \\colon (\\mathcal{X},\\mathcal{D})\\to B$ of $\\mathscr{F}_{n,p,I}(B)$, and pick $m$ and $\\epsilon$ as before. \nThen, up to replacing $m$ with some uniform multiple, from Remark \\ref{remark_for_m_big_enough_kollar_comndition_guarantees_cartier} and from cohomology and base change, we can assume that $\\mathcal{G} \\coloneqq \\omega_{\\mathcal{X}\/B}^{[m]}\\otimes \\mathcal{I}_\\mathcal{D}^{[-\\frac{m(1-\\epsilon)}{r}]}$ is relatively very ample, and $f_*\\mathcal{G}$ is a vector bundle over $B$.\nIndeed, by the deformation invariance of $\\chi(\\mathcal{X}_b,\\mathcal{G}_b)$ and the vanishing of $H^i(X,L)$ for $i \\geq 1$, it follows that the sections of $H^0(\\mathcal{X}_b,\\mathcal{G}_b)$ are deformation invariant, and hence $f_*\\mathcal{G}$ is a vector bundle.\nThen, on every affine open trivializing $f_*\\mathcal{G}$, the morphism $\\mathcal{X} \\rar \\mathbb{P}(f_*\\mathcal{G})$ can be identified with $\\mathcal{X} \\rar B \\times \\mathbb{P} (H^0(\\mathcal{X}_b,\\mathcal{G}_b))$, and the latter is an embedding as it is an embedding over $B$ fiber by fiber, by Nakayama's lemma.\nThis gives an embedding $\\mathcal{X}\\hookrightarrow \\mathbb{P}(f_*\\mathcal{G})$, and composing it with $\\mathcal{D}\\hookrightarrow \\mathcal{X}$ an embedding $\\mathcal{D}\\hookrightarrow \\mathbb{P}(f_*\\mathcal{G})$.\nNow the proof is analogous to the one in \\cite{Alp21}*{Proposition 1.4.6}.\n\\end{proof}\n\n\n\n\n\n\\begin{theorem}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that assumption (A) holds.\nThen $\\mathscr{F}_{n,p,I}$ is an algebraic stack.\n\\end{theorem}\n\n\\begin{proof}Let $r$ denote the index of $I$.\nWe will proceed in several steps.\n\n\n{\\bf Step 1:} In this step, we fix some invariants and consider a suitable Hilbert scheme parametrizing (among others) the total spaces of the $p$-pairs of interest.\n\nBy Lemma \\ref{lemma_bound_for_epsilon}, we may find a rational number $\\epsilon \\in (0,1)$ such that $K_X+(1-\\epsilon)D$ is ample for every $p$-stable pair $(X;D)$ with polynomial $p(t)$ and coefficients in $I$.\nWithout loss of generality, we may assume that $\\epsilon = \\frac{1}{k}$ for a suitable $k \\in \\mathbb{N}$.\nBy Theorem \\ref{thm_boundedness}, we may consider a family $\\pi \\colon (\\mathcal{X};\\mathcal{D})\\to B$ of $p$-stable pairs with coefficients in $I$ and of dimension $n$ that is effectively log bounding for our moduli problem.\nBy stratification of $B$, we may assume that $B$ is smooth, $\\pi$ and $\\pi|_{\\mathcal{D}}$ are flat, and that $\\pi|_{\\mathcal{D}}$ has $S_1$ fibers.\nFurthermore, by Proposition \\ref{prop_stable_family_p-pairs}, $\\pi$ also induces a family of stable pairs.\nIn particular, we can also regard $\\mathcal{D}$ as a divisor, not only as a subscheme, and we will be free to take its multiples.\nSimilarly, any natural multiple $k\\mathcal{D}$ can be regarded as a subscheme, by considering the vanishing locus of $\\O \\mathcal{X}.(-k\\mathcal{D})$.\nThen, there is $m>0$ such that, for every $p$-stable pair $(X;D)$ in our moduli problem, we have:\n\\begin{enumerate}\n    \\item $mD$ and $m(1-\\epsilon)D$ are Cartier;\n    \\item $mK_X$ is Cartier; and\n    \\item $m(K_X+(1-\\epsilon)D)$ is a very ample line bundle that embeds $X$ into $\\mathbb{P}(H^0(X,\\mathcal{O}_X (m(K_X+(1-\\epsilon)D)))$.\n\\end{enumerate}\nFurthermore, up to taking a multiple, we may assume that $\\frac{m}{r}$ and $\\frac{m(1-\\epsilon)}{r}$ are integers, and\n\\begin{enumerate}\n    \\item $\\frac{m}{r} \\mathcal{D}$ and $\\frac{m(1-\\epsilon)}{r} \\mathcal{D}$ are Cartier;\n    \\item $mK_{\\mathcal{X}\/B}$ is Cartier; and\n    \\item $m(K_{\\mathcal{X}\/B}+\\frac{1-\\epsilon}{r}\\mathcal{D})$ is a relatively very ample line bundle.\n\\end{enumerate}\n\nBy boundedness and upper semi-continuity of the space of global sections, $h^0(X,\\mathcal{O}_X (m(K_X+(1-\\epsilon)D))$ attains finitely many values.\nThen, we have finitely many polynomials $q_1,\\ldots,q_l$ such that the fibers of $\\pi$ have Hilbert polynomial $q_i$ for some $i$, for the relatively ample line bundle $\\mathcal{O}_\\mathcal{X}(m(\\K \\mathcal{X}\/B.+\\frac{1-\\epsilon}{r}\\mathcal{D}))$.\nWe consider a union of Hilbert schemes for the polynomials $q_i$, and we denote such a union with $\\mathscr{H}_0$.\nOver $\\mathscr{H}_0$, we have a universal family $f_0 \\colon \\mathscr{X}\\subseteq \\mathscr{H}_0\\times \\mathbb{P}^N\\to \\mathscr{H}_0$, where the fibers are closed subschemes of $\\mathbb{P}^N$ with Hilbert polynomial $q_i$ for some $i$.\nHere, we observe that $N$ may actually attain finitely many distinct values, as we are assuming that each $X$ is embedded with a full linear series; on the other hand, we will work on one Hilbert scheme at the time, thus, by abusing notation, we will simply write $N$.\n\n{\\bf Step 2:} In this step, we highlight the strategy for the construction of the moduli functor.\n\nWe will construct our moduli functor as a subfunctor of a suitable relative Hilbert scheme, modulo the action of $\\mathrm{PGL}_{N+1}$.\nFor this reason, we will shrink $\\mathscr{H}_0$ to cut the locus of interest for our moduli problem.\nIn doing so, we have to guarantee that this locus is locally closed and has a well-defined scheme structure.\nIf we shrink to an open subset, there is no ambiguity in the scheme structure.\nOn the other hand, if we need to consider a closed or locally closed subset, we need to show this choice has a well-defined scheme structure, which will be functorial in nature.\nFinally, we need to guarantee that, at each step, the locus we consider is invariant under the action of $\\mathrm{PGL}_{N+1}$.\n\n{\\bf Step 3:} In this step, we cut the locus parametrizing demi-normal schemes.\n\nSince being $S_2$ is an open condition for flat and proper families \\cite{EGAIV}*{Theorem 12.2.1}, and since small deformations of nodes are either nodes or regular points, up to shrinking $\\mathscr{H}_0$ we may assume that the fibers of $f_0$ are $S_2$ and nodal in codimension one.\nThat is, the fibers are demi-normal.\nSince being demi-normal is independent of the embedding into projective space, this locus is stable under the action of $\\mathrm{PGL}_{N+1}$.\n\n{\\bf Step 4:} In this step, we cut the locus parametrizing varieties embedded with a full linear series.\n\nLet $(X;D)$ be a $p$-pair in our moduli problem, and let $I_X$ denote its ideal sheaf in $\\pr N.$.\nThen, by assumption we have $\\O {\\mathbb P ^N}.(1)|_X \\cong \\O X. (m(\\K X. + (1-\\epsilon)D))$ and the higher cohomologies of both sheaves vanish.\nThus, if we consider the short exact sequence\n\\[\n0 \\rar I_X \\otimes \\O {\\mathbb P ^N}.(1) \\rar \\O {\\mathbb P ^N}.(1) \\rar \\O {\\mathbb P ^N}.(1)|_X \\rar 0,\n\\]\nit provides the following long exact sequence of cohomology groups\n\\[\n0 \\rar H^0(\\pr N.,I_X \\otimes \\O {\\mathbb P ^N}.(1)) \\rar H^0(\\pr N.,\\O {\\mathbb P ^N}.(1)) \\rar H^0(X,\\O {\\mathbb P ^N}.(1)|_X) \\rar H^1(\\pr N.,I_X \\otimes \\O {\\mathbb P ^N}.(1))=0,\n\\]\nwhere the vanishing of $H^1(\\pr N.,I_X \\otimes \\O {\\mathbb P ^N}.(1))$ follows from the surjectivity of the map $H^0(\\pr N.,\\O {\\mathbb P ^N}.(1)) \\rar H^0(X,\\O {\\mathbb P ^N}.(1)|_X)$ and the vanishing of the higher order cohomologies of $\\O {\\mathbb P ^N}.(1)|_X$.\nBy definition of Hilbert scheme, $\\mathscr{X} \\rar \\mathscr{H}_0$ is flat, and the subschemes parametrized correspond to a flat quotient sheaf of $\\O \\mathscr{X}.$.\nThus, as we have a short exact sequence of sheaves where the last two terms are flat over the base, then so is the first term, which is the family of ideal sheaves of the subschemes of interest.\nThus, upper semi-continuity of the dimension of cohomology groups applies, and we may shrink $\\mathscr{H} _0$ to the open locus parametrizing varieties $Y$ with $H^1(\\pr N.,I_Y \\otimes \\O {\\mathbb P ^N}.(1))=0$.\nThis guarantees that, for every such variety $Y$, any automorphism of $Y$ preserving $\\O {\\mathbb P ^N}.(1)|_Y$ is induced by an automorphism of $\\pr N.$.\nSince the vanishing of $H^1(\\pr N.,I_Y \\otimes \\O {\\mathbb P ^N}.(1))=0$ is invariant under the action of $\\mathrm{PGL}_{N+1}$, then so is the locus we cut.\n\n{\\bf Step 5:} In this step, we introduce a relative Hilbert scheme, in order to parametrize the boundaries of the $p$-pairs of interest.\n\nProceeding as in Step 1, for every $p$-pair $(X;D)$ in our moduli problem, we may consider the Hilbert polynomial of $rD$ with respect to $\\mathcal{O}_X(m(\\K X.+(1-\\epsilon)D))$.\nHere, recall that $\\mathcal{D}$ corresponds to $rD$ fiberwise, hence the choice of Hilbert polynomial for $rD$ rather than for $D$.\nBy effective log boundedness and generic flatness of $\\mathcal{D}$, there exist finitely many such polynomials.\nAs before, we will deal with one Hilbert polynomial at the time, and omit this choice from the notation.\n\nNow, $f$ is projective over $\\mathscr{H}_0$.\nIn particular, if we pull back the ample line bundle $\\mathcal{O}_{\\mathscr{H}_0\\times \\mathbb{P}^N}(1)$ to get a relatively very ample line bundle $\\mathscr{G}$ on $\\mathscr{X}$, we can take the relative Hilbert scheme for the morphism $f$, the line bundle $\\mathscr{G}$, and the polynomial determined by $\\mathcal{D}$ (see \\cite{ACH11}*{Chapter IX}).\nThis gives a scheme $\\mathscr{H}_1\\to \\mathscr{H}_0$, together with an universal family $\\mathscr{D}\\subseteq \\mathscr{X}_1 \\coloneqq \\mathscr{X}\\times_{\\mathscr{H}_0} \\mathscr{H}_1$.\nThen, as $\\mathscr{G}$ corresponds to $\\mathcal{O}_X(m(\\K X.+(1-\\epsilon)D))$ on the elements of our moduli problem, every $p$-pair of interest appears as a fiber of this family.\n\n\n\n{\\bf Step 6:} In this step, we shrink $\\mathscr{H}_1$ to an open subset such that $\\mathscr{D} \\cap \\mathrm{Sing}(\\mathscr{X}_1 \\rar \\mathscr{H}_1)$ has codimension at least 2 along each fiber and such that the ideal sheaf $\\mathscr{I}_\\mathscr{D}$ of $\\mathscr{D}$ is relatively $S_2$.\n\nFor every $p$-pair $(X;D)$, $\\operatorname{Supp}(D)$ does not contain any component of the double locus of $X$.\nThus, by upper semi-continuity of the dimension of the fibers of a morphism, we may shrink $\\mathscr{H}_1$ to an open subset such that $\\mathscr{D} \\cap \\mathrm{Sing}(\\mathscr{X}_1 \\rar \\mathscr{H}_1)$ has codimension at least 2 along each fiber.\n\nNow, by Step 3, all the fibers of $\\mathscr{X}_1 \\rar \\mathscr{H}_1$ are demi-normal.\nThus, we may find an open subset $V \\subset \\mathscr{X}_1$ such that the following properties hold:\n\\begin{enumerate}\n    \\item for every $s \\in \\mathscr{H}_1$, $V_s$ is a big open set in $\\mathscr{X}_{1,p}$;\n    \\item $\\mathscr{D} \\cap \\mathrm{Sing}(\\mathscr{X}_1 \\rar \\mathscr{H}_1) \\subset \\mathscr{X}_1 \\setminus V$; and\n    \\item the fibers of $V \\rar \\mathscr{H}_1$ have at worst nodal singularities.\n\\end{enumerate}\nThen, $V \\rar \\mathscr{H}_1$ is a Gorenstein morphism, so, by \\cite{stacks-project}*{Tag 0C08}, $\\omega_{\\mathscr{X}_1\/\\mathscr{H}_1}$ is an invertible sheaf along $V$.\nFurthermore, by \\cite{stacks-project}*{Tag 062Y}, the ideal sheaf $\\mathscr{I}_\\mathscr{D}$ is locally free along $V$.\nFinally, up to shrinking $\\mathscr{H}_1$, we may assume that $\\mathscr{I}_\\mathscr{D}$ is relatively $S_2$.\n\nIndeed, $\\mathscr{X}_1$ is flat over $\\mathscr{H}_1$, as $\\mathscr{X}$ is flat over $\\mathscr{H}_0$, and $\\mathscr{D}$ is flat over $\\mathscr{H}_1$ by definition of relative Hilbert scheme.\nThus, $\\mathscr{I}_\\mathscr{D}$ is flat over $\\mathscr{H}_1$, as it is the kernel of a surjection of flat sheaves.\nThen, we conclude, as being $S_2$ is an open condition for flat and proper families \\cite{EGAIV}*{Theorem 12.2.1}.\nNotice that, by \\cite{HK04}*{Proposition 3.5}, we have $\\mathscr{I}_\\mathscr{D}=\\mathscr{I}_\\mathscr{D}^{[1]}$.\n\nNotice that in this step we shrank $\\mathscr{H}_1$ twice, and both times the process is invariant under the natural action of $\\mathrm{PGL}_{N+1}$, as the locus is characterized by properties of the fibers.\n\n{\\bf Step 7:} In this step, we cut $\\mathscr{H}_1$ to a locally closed subset $\\mathscr{H}_2$ to ensure that $\\mathscr{I}_\\mathscr{D}^{[n]}$ and $\\omega_{\\mathscr{X}_1\/\\mathscr{H}_1}^{[n]}$ are flat and $S_2$ over $\\mathscr{H}_1$ for $0 \\leq n \\leq m$, and that $\\mathscr{I}_\\mathscr{D}^{[\\frac{m(1-\\epsilon)}{r}]}$, $\\mathscr{I}_\\mathscr{D}^{[\\frac{m}{r}]}$, and $\\omega_{\\mathscr{X}_1\/sH_1}^{[m]}$ are invertible sheaves.\n\nNow, we consider the sheaves $\\omega_{\\mathscr{X}_1\/\\mathscr{H}_1}^{[n]}$ and $\\mathscr{I}_\\mathscr{D}^{[n]}$ for every $0\\le n \\le m$.\nThese are defined as \n$i_*(\\omega_{\\mathscr{X}_1 \/ \\mathscr{H}_1}|_V^{\\otimes n})$ and\n$i_*(\\mathscr{I}_\\mathscr{D}|_V^{\\otimes n})$, respectively, where $V$ is the open set defined in Step 6 and $i \\colon V\\to \\mathscr{X}_1$ is the natural inclusion.\n\nThere is a stratification into functorial locally closed subsets of $\\mathscr{H}_1$, which we denote by $C_i\\subseteq \\mathscr{H}_1$, where the above sheaves are flat over $C_i$ (see \\cite{Mum16}*{Lecture 8}).\nNotice that, as the stratification is functorial, it is respected by the natural action of $\\mathrm{PGL}_{N+1}$.\nThen, as at the end of Step 6, up to replacing the $C_i$ with an open subset, we may assume that the above sheaves are relatively $S_2$.\nHence, the above sheaves are reflexive by \\cite{HK04}*{Proposition 3.5}.\nAgain by \\cite{HK04}*{Corollary 3.8} and the existence of the big open set $V$, the formation of the above sheaves commutes with base change for every $0 \\leq n \\leq m$.\nFurthermore, by Remark \\ref{remark_for_m_big_enough_kollar_comndition_guarantees_cartier}, the sheaves $\\mathscr{I}_\\mathscr{D}^{[\\frac{m(1-\\epsilon)}{r}]}$, $\\mathscr{I}_\\mathscr{D}^{[\\frac{m}{r}]}$,  and $\\omega_{\\mathscr{X}_1\/sH_1}^{[m]}$ are invertible over each component $C_i$.\n\n\nConsidering the union of the $C_i$'s produces another base $\\mathscr{H}_2$, with a family $f_2 \\colon \\mathscr{X}_2 \\to \\mathscr{H}_2$ and a closed subset $\\mathscr{D}_2 \\subset \\mathscr{X}_2$ as the one over $\\mathscr{H}_1$, but such that $\\mathscr{I}_{\\mathscr{D}_2}^{[\\frac{m(1-\\epsilon)}{r}]}$, $\\mathscr{I}_{\\mathscr{D}_2}^{[\\frac{m}{r}]}$, and $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[m]}$ are Cartier and the formation of $\\mathscr{I}_{\\mathscr{D}_2}^{[n]}$ and $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[n]}$ commutes with base change for every $0\\leq n \\leq m$.\nThen, the formation of $\\mathscr{I}_{\\mathscr{D}_2}^{[n]}$ and $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[n]}$ commutes with base change for every $n$, since we can write $n=km+b$ for $0\\le b <m$, and we have\n$\\mathscr{I}_{\\mathscr{D}_2}^{[n]}=\\mathscr{I}_{\\mathscr{D}_2}^{[b]} \\otimes (\\mathscr{I}_{\\mathscr{D}_2}^{[\\frac{m}{r}]})^{\\otimes kr}$ and $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[n]}=\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[b]} \\otimes (\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[m]})^{\\otimes k}$.\n\n{\\bf Step 8:} In this step, we shrink $\\mathscr{H}_2$ to an open subset parametrizing semi-log canonical pairs.\n\n\nBy the reductions in the previous steps, for every $k \\geq 1$, the ideal sheaf $\\mathscr{I}_{\\mathscr{D}_2}^{[k]}$ determines a family of generically Cartier divisors, in the sense of \\cite{kol1}*{Definition 4.25}.\nIndeed, this is obtained by dualizing the inclusion $\\mathscr{I}_{\\mathscr{D}_2}^{[k]} \\rar \\O \\mathscr{X}_2.$, as observed in \\cite{kol1}*{Definition 4.25}.\nThen, as observed in \\cite{kol1}*{4.25.5}, a family of generically Cartier divisors induces a well-defined family of divisors.\nAbusing notation, we denote by $\\mathscr{D}_2$ the family of divisors corresponding to $\\mathscr{I}_{\\mathscr{D}_2}$.\n\nNow, recall that taking the reduced structure does not change the topology.\nThus, in the remainder of this step, we may assume that $\\mathscr{H}_2$ is reduced.\nThen, $(\\mathscr{X}_2,\\frac{1}{r}\\mathscr{D}_2) \\rar \\mathscr{H}_2$ is a well defined family of pairs, and by \\cite{kol1}*{Corollary 4.49}, there is an open locus where the fibers of $(\\mathscr{X}_2,\\frac{1}{r}\\mathscr{D}_2) \\rar \\mathscr{H}_2$ are semi-log canonical.\nSince we shrunk $\\mathscr{H}_2$ according to a property of the fibers, this respects the natural action of $\\mathrm{PGL}_{N+1}$.\nAlso, since we took an open subset of $\\mathscr{H}_2$, this choice is not affected by considering the reduced structure of $\\mathscr{H}_2$.\n\n{\\bf Step 9:} In this step, we shrink $\\mathscr{H}_2$ to an open subset parametrizing the $p$-stable pairs in our moduli problem.\n\nRecall that taking the reduced structure does not change the topology.\nThus, in this step, we may take the reduced structure of $\\mathscr{H}_2$ if necessary.\nThen, by Proposition \\ref{prop:nef:open}, up to shrinking $\\mathscr{H}_2$ and possibly discarding some components, we can assume that $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[m]}\\otimes \\mathscr{I}_{\\mathscr{D}_2}^{[-\\frac{m}{r}]}$ is nef along each fiber.\nSimilarly, since being ample is an open condition, we can assume that $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[m]} \\otimes \\mathscr{I}_{\\mathscr{D}_2}^{[-\\frac{m(1-\\epsilon)}{r}]}$ is relatively ample.\nNow, recall that we have chosen $\\epsilon = \\frac{1}{k}$ for a chosen fixed $k \\in \\mathbb N$.\n\nNotice that, the two former conditions guarantee that, for every $j \\geq k$, we have that $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[rjm]} \\otimes \\mathscr{I}_{\\mathscr{D}_2}^{[-m(j-1)]}$ is relatively ample, as we have $mj > m(j-1) \\geq mj(1-\\frac{1}{k})$.\nThen, we may fix $n+1$ such values $j_1,\\ldots,j_{n+1}$ and disregard all components of $\\mathscr{H}_2$ but the ones where, over $s \\in \\mathscr{H}_2$, $(\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[rj_im]} \\otimes \\mathscr{I}_{\\mathscr{D}_2}^{[-m(j_i-1)]})^n$ has prescribed value.\nBy flatness, these self-intersections are locally constant, and thus this condition is open.\nFor each $i$, the self-intersection is prescribed by $p(1-\\frac{1}{j_i})$, up to the rescaling factor given by $rj_im$.\nSince we are prescribing $n+1$ values of a polynomial of degree $n$, this guarantees that all the fibers correspond to $p$-stable pairs $(X;D)$ with $(K_X+tD)^n=p(t)$.\n\nAs in the previous steps, we shrunk $\\mathscr{H}_2$ according to properties of the fibers of $\\mathscr{X}_2 \\rar \\mathscr{H}_2$, thus this preserves the natural action of $\\mathrm{PGL}_{N+1}$.\nAlso, since we took an open subset of $\\mathscr{H}_2$, this choice is not affected by considering the reduced structure of $\\mathscr{H}_2$.\n\n\n{\\bf Step 10:} In this step, we cut $\\mathscr{H}_2$ to a closed subset $\\mathscr{H}_3$ to ensure that the natural polarization $\\O \\mathscr{H}_3 \\times \\mathbb{P}^N.(1)$ coincides with $\\omega_{\\mathscr{X}_3\/\\mathscr{H}_3}^{[m]} \\otimes \\mathscr{I}_{\\mathscr{D}_3}^{[-\\frac{m(1-\\epsilon)}{r}]}$.\n\n\nBy construction, for every $p$-stable pair $(X;D)$ in our moduli problem, we have $\\omega_{\\mathscr{X}_3\/\\mathscr{H}_3}^{[m]} \\otimes \\mathscr{I}_3^{-[\\frac{m(1-\\epsilon)}{r}]}|_X \\sim \\O X. (m(\\K X. + (1-\\epsilon)D)) \\sim \\O \\mathbb P ^N .(1)|_X$.\nSince $\\omega_{\\mathscr{X}_3\/\\mathscr{H}_3}^{[m]} \\otimes \\mathscr{I}_3^{-[\\frac{m(1-\\epsilon)}{r}]}$ is a line bundle and the natural polarization of $\\mathscr{H}_2$ coming from the original choice of Hilbert scheme restricts to $\\O \\mathbb P ^ N. (1)$ fiberwise, by \\cite{Vie95}*{Lemma 1.19} there is a locally closed subscheme $\\mathscr{H}_3$ where $\\omega_{\\mathscr{X}_3\/\\mathscr{H}_3}^{[m]} \\otimes \\mathscr{I}_3^{-[\\frac{m(1-\\epsilon)}{r}]}$ is linearly equivalent to the natural polarization of $\\mathscr{X}_2 \\rar \\mathscr{H}_2$ over $\\mathscr{H}_3$.\nFurthermore, this subscheme is functorial in nature, it is preserved by the natural action of $\\mathrm{PGL}_{N+1}$.\n\n{\\bf Step 11:} In this step, we show that there is an isomorphism $\\mathscr{F}_{n,p,I}\\cong [\\mathscr{H}_3\/\\operatorname{PGL}_{N+1}] $.\n\nOur argument follows closely \\cite{Alp21}*{Theorem 2.1.11}.\nFirst, observe that from its construction, over $\\mathscr{H}_3$ there is a stable family of $p$-stable pairs.\nThis gives a morphism $\\mathscr{H}_3 \\to \\mathscr{F}_{n,p,I}$, and if we forget the embedding into $\\mathbb{P}^N$, this descends to a morphism $\\Phi^{pre} \\colon [\\mathscr{H}_3\/\\operatorname{PGL}_{N+1}]^{pre} \\to \\mathscr{F}_{n,p,I}$, where the superscript pre stands for prestack (see \\cite{Alp21}*{Definition 1.3.12}).\nThis induces a map $\\Phi \\colon [\\mathscr{H}_3\/\\operatorname{PGL}_{N+1}] \\to \\mathscr{F}_{n,p,I}$, which we now show is an isomorphism.\n\nTo show it is fully faithful, as in \\cite{Alp21}, it suffices to check that $\\Phi^{pre}$ is fully faithful.\nBut $\\Phi^{pre}$ is fully faithful since any isomorphism between two families of $p$-stable\npairs $\\pi \\colon (\\mathscr{Y};\\mathscr{D})\\to B$ and $\\pi \\colon (\\mathscr{Y}',\\mathscr{D}')\\to B$ over $B$ sends $\\mathscr{L} \\coloneqq \\omega_{\\mathscr{Y}\/B}^{[m]} \\otimes \\mathscr{I}_{\\mathscr{Y}}^{-[\\frac{m(1-\\epsilon)}{r}]}$ to $\\mathscr{L}' \\coloneqq\\omega_{\\mathscr{Y}'\/B}^{[m]} \\otimes \\mathscr{I}_{\\mathscr{Y}'}^{-[\\frac{m(1-\\epsilon)}{r}]}$, where we denoted by $ \\mathscr{I}_{\\mathscr{Y}}$ (resp. $\\mathscr{I}_{\\mathscr{Y}'}$) the ideal sheaves of $\\mathscr{D}$ (res. $\\mathscr{D}'$) in $\\mathscr{Y}$ (resp. $\\mathscr{Y}'$). This induces an unique isomorphism $\\mathbb{P}(\\pi_*\\mathscr{L} )\\cong \\mathbb{P}(\\pi'_*\\mathscr{L}')$ which sends $\\mathscr{Y}$ to $\\mathscr{Y}'$.\n\nSince $\\Phi$ is a morphism of stacks, also essential surjectivity can be checked locally on $B$.\nIn particular, it suffices to check that if $\\pi \\colon (\\mathscr{Y};\\mathscr{D})\\to B$ is a family of $p$-stable pairs such that $\\pi_*\\mathscr{L}$ is free, then the morphism $B\\to \\mathscr{F}_{n,p,I}$ lifts to a morphism $B\\to \\mathscr{H}_3$.\nThis follows since if we pick an isomorphism $\\mathbb{P} (\\pi_*\\mathscr{L})\\cong \\mathbb{P}^N \\times B$ then\n$\\mathscr{Y}, \\mathscr{D}\\subseteq \\mathbb{P} (\\pi_*\\mathscr{L}) = \\mathbb{P}^N\\times B$, and then from the functorial properties of $\\mathscr{H}_3$ it induces a morphism $B\\to \\mathscr{H}_3$.\n\\end{proof}\n\\begin{Remark}\nObserve that the stack $\\mathscr{F}_{n,p,I}$ is in fact Deligne--Mumford.\nIndeed, since we are working over a field of characteristic 0, it suffices to show that the automorphisms of the objects over the points are finite. But this follows since an automorphism of a $p$-pair $(X;D)$ induces an automorphism of the stable pair $(X,(1-\\epsilon)D)$, and those are finite from \\cite{KP17}*{Proposition 5.5}.\n\\end{Remark}\n\n\\section{Properness of $\\mathscr{F}_{n,p,I}$}\\label{section_properness}\nThe goal of this section is to prove that $\\mathscr{F}_{n,p,I}$ is proper.\nIn particular, since in the definition of a $p$-stable pair $(X;D)$ there are prescribed conditions on the scheme-theoretic structure of $D$.\nThen, when proving that a moduli functor for $p$-pairs satisfies the valuative criterion for properness, one needs to check that these scheme-theoretic properties are preserved.\nIt is convenient to check that the flat limit of $D^{sc}$ (recall that $D^{sc}$ was introduced in Notation \\ref{notation_Dsc}) is $S_1$.\nThis will be the content of the next proposition.\n\n\\begin{Prop}\\label{prop:flat:limit:is:S1}\nLet $\\operatorname{Spec}(R)$ be the spectrum of a DVR with generic point $\\eta$ and closed point $p$.\nConsider a locally stable family $(X,D)\\to \\operatorname{Spec}(R)$ such that $D$ is $\\mathbb{Q}$-Cartier.\nThen, for every $m\\in \\mathbb{N}$, the ideal sheaf $\\mathcal{O}_X(-mD)$ is $S_3$ on every point $x\\in X_p$.\nIn particular:\n\\begin{enumerate}\n    \\item the restriction $\\mathcal{O}_X(-mD)|_{X_p}$ is $S_2$; and\n    \\item if we denote by $mD$ the closed subscheme of $X$ with ideal sheaf $\\mathcal{O}_X(-mD)$, then $(mD)|_{X_p}$ is $S_1$.\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\nFirst, observe that since $X\\to \\operatorname{Spec}(R)$ is locally stable, $x \\in X_p$ cannot be a log canonical center for $X$ (see \\cite{kol1}*{Proposition 2.13}).\nThe statement is local, so up to shrinking $X$, we can assume that $X$ is affine, and, since $D$ is $\\mathbb{Q}$-Cartier, $\\mathcal{O}_X(m_0D) \\cong \\mathcal{O}_X$ for a certain $m_0 \\in \\mathbb{N}$.\nThen, for every $m \\in \\mathbb{N}$, $mD\\sim_\\bQ0$.\nIn particular, if we apply \\cite{Kol13}*{Theorem 7.20} where, with the notations of \\emph{loc. cit.}, we take $\\Delta'=0$, we conclude that $\\mathcal{O}_X(-mD)$ is $S_3$.\n\nNow, we denote by $\\pi$ the pull-back to $X$ of a uniformizer on $\\operatorname{Spec}(R)$.\nSince $\\mathcal{O}_X(-mD)$ is a subsheaf of $\\mathcal{O}_X$ and $\\pi$ is not a zero divisor on $\\mathcal{O}_X$, it is not a zero divisor on $\\mathcal{O}_X(-mD)$.\nThen, \\begin{center}$(\\mathcal{O}_X(-mD))|_{X_p}$ is $S_2$ by \\cite{KM98}*{Proposition 5.3}.\\end{center}\n\nNotice that $\\mathcal{O}_X$ is $S_3$ since it is a flat and proper family of $S_2$ schemes over a smooth base.\nThen, by \\cite{Kol13}*{Corollary 2.61}, $\\mathcal{O}_{mD}$ is $S_2$.\nIn particular, it is $S_1$, so its generic points are the only associated points.\nHence, if we denote by $mD$ the closed subscheme of $X$ with ideal sheaf $\\mathcal{O}_X(-mD)$, then $mD\\to \\operatorname{Spec}(R)$ is flat.\nThen, if we pull back the exact sequence\n\\[\n0\\to \\mathcal{O}_X(-mD)\\to \\mathcal{O}_X\\to \\mathcal{O}_{mD}\\to 0\n\\]\nto $X_p$, the sequence remains exact and we get\n\\[\n0\\to (\\mathcal{O}_X(-mD))|_{X_p}\\to (\\mathcal{O}_X)|_{X_p}\\to (\\mathcal{O}_{mD})|_{X_p}\\to 0.\n\\]\nThe desired result follows again by \\cite{Kol13}*{Corollary 2.61}.\n\\end{proof}\n\n\\begin{Prop} \\label{properness_lc_case}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that $I$ is closed under sum: that is, if $a,b \\in I$ and $a+b \\leq 1$, then $a+b \\in I$.\nAssume that assumption (A) holds.\nLet $C$ be a smooth affine curve, let $0 \\in C$ be a distinguished closed point, and let $U\\coloneqq C \\setminus \\{ 0 \\}$.\nLet $(\\mathcal{X}_U, \\Delta_U; \\mathcal{D}_U) \\rightarrow U$ be a $p$-stable morphism of relative dimension $n$, polynomial $p(t)$, coefficients in $I$, and constant part $\\Delta_U$, which is reduced.\nFurther, assume that the geometric generic fiber is normal.\nThen, up to a finite base change $B \\rar C$, the family can be filled with a unique $p$-stable pair of dimension $n$, with polynomial $p(t)$ and coefficients in $I$.\n\\end{Prop}\n\n\\begin{proof}\nWe proceed in several steps.\nIn the following, $r$ will denote the index of $I$.\n\n{\\bf Step 1:} In this step, we show that the family of pairs $(\\mathcal{X}_U,\\frac{1}{r}\\mathcal{D}_U + \\Delta_U) \\rar U$ admits a relative ample model $(\\mathcal{Y}_U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U, \\mathcal{Y}})$ over $U$.\n\nBy Proposition \\ref{prop_stable_family_p-pairs} and the fact that $C$ is smooth, it follows that $(\\mathcal{X} _U, \\frac{1}{r}\\mathcal{D}_U + \\Delta_U)$ is a pair.\nFurthermore, by inversion of adjunction, this pair is log canonical.\nWe need to argue that $(\\mathcal{X} _U, \\frac{1}{r}\\mathcal{D}_U + \\Delta_U)$ admits a morphism to its relative ample model $(\\mathcal{Y}_U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U, \\mathcal{Y}})$ over $U$.\nIf we assume case (ii) of assumption (A), this is clear.\nSo, assume that $n \\leq 3$.\nThen, the claim follows by Lemma \\ref{lemma minimal models}.\n\n{\\bf Step 2:} In this step, we show that, up to a base change, $(\\mathcal{Y}_U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U,\\mathcal{Y}}) \\rar U$ can be compactified to a family of stable pairs.\nFurthermore, this model is dominated by a $\\qq$-factorial log canonical model.\n\nBy construction, $(\\mathcal{Y}_U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U,\\mathcal{Y}}) \\rar U$ is a family of stable pairs.\nThen, by \\cite{HX13}*{Corollary 1.5}, up to a finite base change, which we omit in the notation, we may assume that $(\\mathcal{Y} _U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U,\\mathcal{Y}}) \\rightarrow U$ admits a compactification $(\\mathcal{Y}, \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}) \\rightarrow C$ to a stable family.\nFurthermore, arguing as in \\cite{HX13}*{proof of Corollary 1.4}, by semi-stable reduction, we\nmay assume that $(\\mathcal{Y}, \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}+\\mathcal{Y}_0) \\rightarrow C$ admits a $\\mathbb{Q}$-factorial log canonical model $(\\mathcal{X}', \\frac{1}{r}\\mathcal{D}' + \\Delta'+\\mathcal{X}'_0) \\rightarrow (\\mathcal{Y}, \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}+\\mathcal{Y}_0)$, where the special fiber $\\mathcal{X}'_0$ is reduced, and such that $K_{\\mathcal{X}'}+\\frac{1}{r}\\mathcal{D}' + \\Delta'+\\mathcal{X}'_0$ is the pull-back of $K_{\\mathcal{Y}}+ \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}+\\mathcal{Y}_0$, where $\\Delta'$ is the proper transform of $\\Delta_\\mathcal{Y}$.\nFurthermore, by increasing the coefficients of the divisors of log discrepancy 1 on the initial semi-stable model, we may assume that all the divisors extracted in $\\mathcal{X}'$ have log discrepancy in $[0,1)$ with respect to $(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y} + \\Delta_\\mathcal{Y})$.\nIn particular, all the exceptional divisors that are horizontal over $U$ appear in the support of $\\frac{1}{r}\\mathcal{D}'$.\n\n{\\bf Step 3:} In this step, we show that we may choose the $\\qq$-factorial log canonical model $(\\mathcal{X}', \\frac{1}{r}\\mathcal{D}' + \\Delta'+\\mathcal{X}'_0) \\rightarrow (\\mathcal{Y}, \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}+\\mathcal{Y}_0)$ so that $\\mathcal{X}'_U \\dashrightarrow \\mathcal{X}_U$ is a rational contraction over $U$.\n\n\nSince $K_{\\mathcal{X}_U}+\\frac{1-\\epsilon}{r}\\mathcal{D}_U + \\Delta_U$ is ample over $U$ for $0 < \\epsilon \\ll 1$, the divisors contracted by $\\mathcal{X}_U \\rightarrow \\mathcal{Y}_U$ are contained in the support of $\\mathcal{D}$.\nIn particular, they have log discrepancy in $[0,1)$.\nThen, from \\cite{Mor19}*{Theorem 1}, we can produce a morphism $\\mathcal{X}''\\to  \\mathcal{X}'$ extracting all the exceptional divisors of $\\pi \\colon \\mathcal{X}_U \\rightarrow \\mathcal{Y}_U$.\nObserve that $\\mathcal{X}''$ remains $\\mathbb{Q}$-factorial: by \\cite{Mor19}*{Theorem 1}, all the extracted divisors are $\\mathbb{Q}$-Cartier.\nIf a divisor is not exceptional for $\\pi$, we can write it as the difference of its push-pull to $\\mathcal{X}'$ and the exceptional divisors for $\\pi$.\nAll the divisors extracted dominate $U$, as their generic point is disjoint from the special fiber over 0.\nThus, the irreducible components of the special fiber remain unchanged, and, in particular, it is still reduced.\nThe claim then follows by replacing $\\mathcal{X}' $ with $\\mathcal{X}''$.\n\n\nFor clarity, we can represent the morphisms so far constructed with the following diagram\n\\[\n\\begin{tikzcd}\n\t& {\\mathcal{X}'_U} & {\\mathcal{X}'} \\\\\n\t{\\mathcal{X}_U} & {\\mathcal{Y}_U} & {\\mathcal{Y}} \\\\\n\tU && C\n\t\\arrow[hook, from=3-1, to=3-3]\n\t\\arrow[hook, from=2-2, to=2-3]\n\t\\arrow[from=2-2, to=3-1]\n\t\\arrow[from=2-3, to=3-3]\n\t\\arrow[from=1-3, to=2-3]\n\t\\arrow[hook, from=1-2, to=1-3]\n\t\\arrow[from=1-2, to=2-2]\n\t\\arrow[from=2-1, to=3-1]\n\t\\arrow[from=2-1, to=2-2]\n\t\\arrow[dashed, from=1-2, to=2-1]\n\\end{tikzcd}\n\\]\n\n{\\bf Step 4:} In this step, we construct a new model $\\mathcal{X}$, which will be then showed to be the sought compactification of $\\mathcal{X}_U$.\n\nSince $\\mathcal{X}'_U \\dashrightarrow \\mathcal{X}_U$ is a birational contraction and $\\mathcal{D}_U$ is $\\mathbb{Q}$-Cartier, the rational pull-back of $\\mathcal{D}_U$ to $\\mathcal{X}'_U$ is well defined.\nWe denote by $\\mathcal{B}'$ its closure in $\\mathcal{X}'$.\nBy construction, we have that $\\frac{1}{r}\\mathcal{D} ' + \\Delta'\\geq 0$ and\n\\begin{equation} \\label{eq_lin_eq}\nK_{\\mathcal{X}'} + \\frac{1}{r}\\mathcal{D} ' + \\Delta' - \\epsilon \\mathcal{B}' \\sim_{\\mathbb{Q},\\mathcal{Y}} -\\epsilon \\mathcal{B}',\n\\end{equation}\nfor any $\\epsilon \\in \\mathbb{Q}$.\nAs observed at the end of Step 2, $\\operatorname{Supp}(\\frac{1}{r}\\mathcal{D}')$ contains all the divisors that are exceptional for $\\mathcal{X}' \\rar \\mathcal{Y}$.\nThen, as $\\mathcal{X}'_U \\dashrightarrow \\mathcal{X}_U$ is a rational contraction, it follows that $\\operatorname{Supp}(\\mathcal{B}') \\subset \\operatorname{Supp}(\\frac{1}{r}\\mathcal{D}')$.\nThus, for $0< \\epsilon \\ll 1$, the the divisor $ \\frac{1}{r}\\mathcal{D} ' + \\Delta'-\\epsilon \\mathcal{B}'$ is  effective, and hence $({\\mathcal{X}'} , \\frac{1}{r}\\mathcal{D} ' + \\Delta'-\\epsilon \\mathcal{B}')$ is a log canonical pair.\nNow, we want to define $\\mathcal{X}$ as its relative ample model over $\\mathcal{Y}$.\nFor this definition to be well-posed, we need to check that such an ample model exists.\n\n{\\bf Step 5:} In this step, we show that the definition of $\\mathcal{X}$ is well-posed.\n\n\nFirst, observe that since $(\\mathcal{X}', \\frac{1}{r}\\mathcal{D}' + \\Delta'+\\mathcal{X}'_0)$ is log canonical, there are no log canonical centers of $(\\mathcal{X}', \\frac{1}{r}\\mathcal{D}' + \\Delta'-\\epsilon \\mathcal{B}')$ contained in $\\mathcal{X}_0'$.\nTherefore, up to replacing $C$ with an open neighborhood of $0\\in C$, we may assume that all the log canonical centers are horizontal.\nThen, we may apply Lemma \\ref{lemma minimal models}, which guarantees the existence of a relative good minimal model, up to further extracting some log canonical places.\nIn particular, the relative ample model over $\\mathcal{Y}$ exists, and by definition, it coincides with $\\mathcal{X}$.\nWe let $\\mathcal{D}$, $\\Delta$, and $\\mathcal{B}$ be the push-forwards of $\\mathcal{D}'$, $\\Delta'$, and $\\mathcal{B}'$ to $\\mathcal{X}$, respectively. Observe also that from \\eqref{eq_lin_eq}, $\\mathcal{X}$ is independent of $\\epsilon$, as long as $\\epsilon>0$ and $\\epsilon \\in \\mathbb{Q}$, as it is the relative Proj of the same graded algebra, only with shifted degrees. Moreover, $\\operatorname{Supp}(\\frac{1}{r}\\mathcal{D}+\\Delta)$ does not contain any irreducible component of the special fiber, since $(\\mathcal{X},\\frac{1}{r}\\mathcal{D} + \\Delta + \\mathcal{X}_0)$ is log canonical.\n\n{\\bf Step 6:} In this step, we show that $\\mathcal{X} \\times_C U = \\mathcal{X}_U$ and that $\\mathcal{D}\\times_C U  = \\mathcal{D}_U$.\nIn particular, we have $\\mathcal{B} = \\mathcal{D}$.\n\nRecall that the rational map $\\mathcal{X}'_U \\dashrightarrow \\mathcal{X}_U$ is a contraction.\nThus, we can define the rational pull-back of $K_{\\mathcal{X}_U} + \\frac{1-\\epsilon}{r}\\mathcal{D}_U+\\Delta_U$ to $\\mathcal{X}'_U$.\nBy definition of $\\mathcal{B}'$, this pull-back agrees with $K_{\\mathcal{X}'_U}+\\frac{1}{r}\\mathcal{D}'_U+\\Delta'_U -\\epsilon \\mathcal{B}'_U$.\nThus, the relative ample model of $(\\mathcal{X}'_U,\\frac{1}{r}\\mathcal{D}'_U+\\Delta'_U -\\epsilon \\mathcal{B}'_U)$ over $U$ is $(\\mathcal{X}_U,\\frac{1-\\epsilon}{r}\\mathcal{D}_U+\\Delta_U)$.\nThen, since $K_{\\mathcal{X}'}+\\frac{1}{r}\\mathcal{D}'+\\Delta'$ is the pull-back of an ample divisor on $\\mathcal{Y}$, and the model $\\mathcal{X}$ is independent of $\\epsilon>0$, it follows that, if we choose $0<\\epsilon \\ll 1$, the relative ample model of $(\\mathcal{X}'_U,\\frac{1}{r}\\mathcal{D}'_U+\\Delta'_U-\\epsilon \\mathcal{B}'_U)$ over $\\mathcal{Y}$ is a relative ample model over $C$.\nThis shows that $\\mathcal{X} \\times_C U = \\mathcal{X}_U$ and therefore $\\mathcal{D}\\times_C U  = \\mathcal{D}_U$.\n\nSince $\\mathcal{B}'$ is defined as the closure of the rational pull-back of $\\mathcal{D}_U$ to $\\mathcal{X}'_U$, all of its components are horizontal over $U$.\nIn particular, to check the equality $\\mathcal{B}= \\mathcal{D}$, it suffices to show it holds over $U$.\nBut then, this is immediate, by the definition of $\\mathcal{B}'$ and the fact that $\\mathcal{X} \\times_C U = \\mathcal{X}_U$.\n\n{\\bf Step 7:} In this step, we show that $(\\mathcal{X},\\Delta;\\frac{1}{r}\\mathcal{D})$ is a $p$-pair and that the central fiber of $(\\mathcal{X},\\Delta;\\frac{1}{r}\\mathcal{D})$ is a $p$-stable pair with polynomial $p(t)$ and coefficients in $I$.\n\n\nRecall that $\\mathcal{X}$ is independent of $\\epsilon$, as long as $\\epsilon>0$ and $\\epsilon \\in \\mathbb{Q}$.\nIn particular, $\\mathcal{B}$ is $\\mathbb{Q}$-Cartier on $\\mathcal{X}$.\nThen, as $\\K \\mathcal{X}. + \\frac{1}{r}\\mathcal{D} + \\Delta - \\epsilon \\mathcal{B}$ is $\\mathbb{Q}$-Cartier for $0 < \\epsilon \\ll 1$ rational and $\\mathcal{B}=\\mathcal{D}$, it follows that $\\K \\mathcal{X}. + \\frac{1}{r}\\mathcal{D} + \\Delta$ is $\\mathbb{Q}$-Cartier.\nAs $(\\mathcal{X},\\frac{1}{r}\\mathcal{D} + \\Delta)$ is log canonical by construction, it follows that $(\\mathcal{X},\\Delta;\\frac{1}{r}\\mathcal{D})$ is a $p$-pair.\n\n\nBy construction and by adjunction, $(\\mathcal{X}_0,\\frac{1}{r}\\mathcal{D}_0 + \\Delta_0)$ is semi-log canonical.\nFurthermore, $\\K \\mathcal{X}_0. + \\frac{1-\\epsilon}{r}\\mathcal{D}_0 + \\Delta_0$ is ample for $0 < \\epsilon \\ll 1$.\nSince $\\mathcal{D}$ and $K_{\\mathcal{X}} + \\frac{1}{r}\\mathcal{D} +\\Delta$ are $\\mathbb{Q}$-Cartier, the self-intersection $(K_{\\mathcal{X}_c} + \\frac{1}{r}\\mathcal{D}_c +\\Delta_c-\\epsilon \\mathcal{D}_c)^{{\\rm dim}(\\mathcal{X}_0)}$ is well-defined for every $\\epsilon$ and $c\\in C$, and does not depend on $c \\in C$.\nSince the general fiber is $p$-stable with polynomial $p(t)$, we have $(K_{\\mathcal{X}_0} + \\frac{1}{r}\\mathcal{D}_0 +\\Delta_0-\\epsilon \\mathcal{D}_0)^{{\\rm dim}(\\mathcal{X}_0)} = p(1-\\epsilon)$.\nThe coefficients of $\\frac{1}{r}\\mathcal{D}_0$ are still in $I$,  since $I$ is closed under addition.\n\n{\\bf Step 8:} In this step, we show that $(\\mathcal{X}, \\Delta;\\mathcal{D}) \\rar C$ is a $p$-stable morphism with constant part $\\Delta$.\n\nBy construction, the fibers are proper.\nSince the base is a curve and every divisor is horizontal, all the morphisms are flat of the appropriate relative dimension.\nBy Step 7, every fiber is a $p$-stable pair.\nThen, as $\\mathcal{D}$ is $\\qq$-Cartier, every fiber of $(\\mathcal{X},\\Delta) \\rar C$ is semi-log canonical.\nThus, by \\cite{kol1}*{Definition-Theorem 4.45}, the morphism $(\\mathcal{X},\\Delta) \\rar C$ is locally stable.\n\n{\\bf Step 9:} In this step, we show that the limit is unique.\n\nFrom Theorem \\ref{thm_boundedness}, there exists $\\epsilon_0 > 0$ such that, for every $p$-stable pair $(Z,B,\\Gamma)$ with coefficients in $I$ and polynomial $p(t)$, $(Z,(1-\\epsilon_0)B + \\Gamma)$ is a stable pair.\nThen, the claim follows from the separatedness of stable morphisms \\cite{kol1}*{2.45}.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{Thm_properness}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that $I$ is closed under sum: that is, if $a,b \\in I$ and $a+b \\leq 1$, then $a+b \\in I$.\nAssume that assumption (A) holds.\nLet $C$ be a smooth affine curve, let $0 \\in C$ be a distinguished closed point, and let $U\\coloneqq C \\setminus \\{ 0 \\}$.\nLet $(\\mathcal{X}_U, \\Delta_U; \\mathcal{D}_U) \\rightarrow U$ be a $p$-stable morphism of dimension $n$, polynomial $p(t)$, coefficients in $I$, and constant part $\\Delta_U$.\nThen, up to a finite base change $B \\rar C$, the family can be filled with a unique $p$-stable pair of dimension $n$, with polynomial $p(t)$ and coefficients in $I$.\n\\end{theorem}\n\n\n\\begin{proof}\nBy Proposition \\ref{properness_lc_case}, we may assume that the geometric generic fiber is not normal.\nIn the following, $r$ will denote the index of $I$.\n\nLet $(\\overline \\mathcal{X}_U, \\frac{1}{r}\\overline \\mathcal{D}_U + \\overline \\Delta_U)$ denote the normalization of $(\\mathcal{X}_U, \\frac{1}{r}\\mathcal{D}_U + \\Delta_U)$, where $\\overline{\\Delta}_U$ also includes the conductor with coefficient 1.\nBy assumption, $\\mathcal{D}_U$ is $\\mathbb{Q}$-Cartier.\nThen, by \\cite{Kol13}*{Corollary 5.39}, $\\overline{\\mathcal{D}}_U$ is $\\mathbb{Q}$-Cartier.\nThen, since $\\K \\mathcal{X}_U. + \\frac{1-\\epsilon}{r}\\mathcal{D}_U + \\Delta_U$ is ample over $U$, it follows that $(\\overline \\mathcal{X} _U, \\overline \\Delta_U; \\overline \\mathcal{D} _U)$ is $p$-stable morphism with constant part $\\overline \\Delta_U$, where the polynomial on each connected component depends on the original choice of $p$.\n\nThen, as $\\overline{\\mathcal{X}}_U$ has finitely many connected components, by Proposition \\ref{properness_lc_case}, there is a finite base change $B \\rar C$ such that the family can be filled with a unique $p$-stable pair.\nTo simplify the notation, we omit the base change $B \\rar C$, and we assume that the filling is realized over $C$ itself.\nWe denote this family by $(\\overline \\mathcal{X},  \\overline \\Delta; \\overline \\mathcal{D}) \\rar C$.\nThen, we may find $0 < \\epsilon \\ll 1$ such that $(\\overline \\mathcal{X}, \\frac{1-\\epsilon}{r} \\overline \\mathcal{D}  + \\overline \\Delta) \\rar C$ is a stable morphism.\nBy \\cite{kol1}*{Corollary 2.56}, also $(\\mathcal{X} _U, \\frac{1-\\epsilon}{r} \\mathcal{D} _U + \\Delta_U)$ admits a compactification $(\\mathcal{X} , \\frac{1-\\epsilon}{r} \\mathcal{D} + \\Delta)$ over $C$ obtained by gluing $\\overline{\\mathcal{X}}$ along some components of $\\overline{\\Delta}$.\nBy \\cite{Kol13}*{Corollary 5.39}, the divisor $\\mathcal{D}$ is $\\mathbb{Q}$-Cartier.\nThus, we have that $\\mathcal{D}_0$ is $\\mathbb{Q}$-Cartier, as needed.\nSimilarly, the coefficients of $\\frac{1}{r}\\mathcal{D}_0+\\Delta_0$ are in $I$, by construction.\n\nTo conclude, we need to show that $(\\K \\mathcal{X}_0. + \\frac{t}{r}\\mathcal{D}_0 + \\Delta_0)^n=p(t)$.\nThis follows from flatness over the base $C$, as we have\n\\[\n(\\K \\mathcal{X}_0. + \\frac{t}{r}\\mathcal{D}_0 + \\Delta_0)^n = (\\K \\mathcal{X}_c. + \\frac{t}{r}\\mathcal{D}_c + \\Delta_c)^n = p(t),\n\\]\nwhere $c \\in C \\setminus \\{ 0 \\}$.\nThis concludes the proof.\n\\end{proof}\n\n\\begin{Cor}Fix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that $I$ is closed under sum: that is, if $a,b \\in I$ and $a+b \\leq 1$, then $a+b \\in I$.\nAssume that assumption (A) holds.\nThen the algebraic stack $\\mathscr{F}_{n,p,I}$ is proper.\n\\end{Cor}\n\\begin{proof}\nIt suffices to check that it satisfies the valuative criterion for properness. So assume that we have a family of $p$-stable pairs $f_\\eta \\colon (X;D)\\to \\eta$ over the generic point $\\eta$ of the spectrum of a DVR $R$, and we need to fill in the central fiber up to replacing $\\operatorname{Spec}(R)$ with a ramified cover of it. Theorem \\ref{Thm_properness} guarantees the existence and uniqueness of a p-stable morphism $f \\colon (\\mathcal{X};\\mathcal{D})\\to \\operatorname{Spec}(R)$ extending $f_\\eta$, up to a ramified cover of $\\operatorname{Spec}(R)$. We need to check that $f$ satisfies conditions (1) and (2) of Definition \\ref{Def:functor}. Condition (1) follows from Proposition \\ref{prop:flat:limit:is:S1} and \\cite{HK04}*{Corollary 3.8}. Condition (2) follows from \\cite{kol1}*{Proposition 2.76}.\n\\end{proof}\n\n\\section{Morphism from the $p$-moduli to the moduli of stable pairs}\n\\label{section morphism}\nGiven a pair $(X,D)$ with $K_X+D$ semi-ample and big, one can take its ample model. The goal of this subsection is to show that for $p$-pairs, one can take the ample model in families.\n\n\n\\begin{Lemma}\\label{Lemma:slc:canonical:model:for:p:pairs}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nLet $r$ denote the index of $I$.\nConsider a $p$-stable morphism with coefficients in $I$ and polynomial $p(t)$ over a smooth scheme $U$:\n$p \\colon (\\mathcal{X};\\mathcal{D})\\to U.$\nAssume that assumption (A) holds.\nThen, there is a stable family of pairs $(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})\\to U$ such that:\n\\begin{enumerate}\n    \\item there is a contraction $\\pi \\colon \\mathcal{X} \\to \\mathcal{Y}$; and\n    \\item we have $\\pi^* (\\K \\mathcal{Y}\/U. + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) = \\K \\mathcal{X} \/U. + \\frac{1}{r}\\mathcal{D}$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nBy Proposition \\ref{prop_stable_family_p-pairs} and the fact that $U$ is smooth, it follows that $(\\mathcal{X}, \\frac{1}{r}\\mathcal{D})$ is a pair.\nFurthermore, by inversion of adjunction, this pair is semi-log canonical.\n\n\nFirst, we assume that $\\mathcal{X}$ is normal.\nThen, by Lemma \\ref{lemma minimal models} and our assumptions, the canonical model of $(\\mathcal{X},\\frac{1}{r}\\mathcal{D})$ over $U$ exists.\nSince $U$ is smooth, this canonical model gives a family of stable pairs by \\cite{kol1}*{Corollary 4.86}, and the result follows from how canonical models are constructed.\n\nWe now treat the case where $\\mathcal{X}$ is not normal.\nFirst, we normalize $\\mathcal{X}^n\\to \\mathcal{X}$ to get $(\\mathcal{X}^n,\\Delta;\\mathcal{D}') $.\nAs argued in the previous case, we can construct the canonical model of $(\\mathcal{X}^n,\\Delta;\\mathcal{D}') $ over $U$.\nAs above, this gives a family of stable pairs $(\\mathcal{Y}',\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}'+\\Delta_\\mathcal{Y})\\to U$.\nFrom Koll\\'ar's gluing theory (see \\cite{Kol13}*{Chapter 5}), there is an involution $\\tau \\colon \\Delta^n \\to \\Delta^n$ that fixes the different.\nWe first show that this involution descends onto $\\Delta_\\mathcal{Y}^n$.\n\nRecall that, by Lemma \\ref{ample model p-pair}, the map $\\mathcal{X}^n\\to \\mathcal{Y}'$ does not contract any component of $\\Delta$.\nIn particular, for every irreducible component $F\\subseteq \\Delta^n$, there is an irreducible component $F_Y\\subseteq \\Delta_\\mathcal{Y}^n$ birational to it, and we have the following diagram:\n\n$$\\xymatrix{\\Delta^n\\ar[r]^f \\ar[d] & \\Delta^n_\\mathcal{Y} \\ar[d] \\\\ \\mathcal{X} \\ar[r] & \\mathcal{Y}.}$$\nObserve now that\n\\begin{center}\n    {\\bf ($\\ast$)} {\\it a curve $C \\subset \\Delta^n$ gets contracted by $f$ if and only if $(K_{\\Delta^n} + \\operatorname{Diff}_{\\Delta^n}(\\frac{1}{r}\\mathcal{D} ' + \\Delta)).C = 0$.}\n\\end{center}\nIn particular, since the involution $\\tau$ preserves the different, it preserves all the curves that are contracted by $f$.\nHence, the involution descends to an involution $\\tau_\\mathcal{Y}$ on $\\Delta_\\mathcal{Y}^n$.\n\nWe prove that $\\tau_\\mathcal{Y}$ preserves the different.\nIndeed, by Lemma \\ref{ample model p-pair}, the only divisors contracted by $f$ are contained in $\\operatorname{Supp}(\\mathcal{D}')$, so the morphism $f$ is an isomorphism generically around each divisor not contained in $\\operatorname{Supp}(\\mathcal{D})$.\nIn particular, since the computation of the different is local,\n\\[\nf_*(\\operatorname{Diff}_{\\Delta^n}(\\frac{1}{r}\\mathcal{D}' + \\Delta)) = \\operatorname{Diff}_{\\Delta^n_\\mathcal{Y}}(\\frac{1}{r}\\mathcal{D}'_\\mathcal{Y} + \\Delta_\\mathcal{Y}).\n\\]\nSince $\\tau$ preserves the different on $\\Delta^n$, $\\tau_\\mathcal{Y}$ preserves the different on $\\Delta_\\mathcal{Y}^n$.\n\nThen from \\cite{kol1}*{Chapter 5}, we can glue $\\mathcal{Y}'$ to get an semi-log canonical pair $(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})$.\nNow, recall that $\\mathcal{Y}$ is a geometric quotient (see \\cite{Kol13}*{Theorem 5.32} and the proof of \\cite{Kol13}*{Corollary 5.33}), so for any morphism $\\phi \\colon \\mathcal{Y}^n\\to Z$ such that $\\phi|_{\\Delta_\\mathcal{Y}^n} = \\tau \\circ \\phi|_{\\Delta_\\mathcal{Y}^n} $, there is a unique morphism $\\mathcal{Y}\\to Z$ which fits in the obvious commutative diagram.\nTherefore, we have a morphism $\\mathcal{Y}\\to U$, and applying this result to $\\mathcal{X}^n$ and $\\mathcal{X}$, we obtain a morphism $\\mathcal{X}\\to \\mathcal{Y}$.\nNow, by Lemma \\ref{ample model p-pair}, we can apply Lemma \\ref{lemma:canonical:model:of:slc:pairs:has:connected:fibers}, and the desired result follows.\n\\end{proof}\n\n\n\n\n\\begin{theorem}\\label{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nLet $r$ denote the index of $I$.\nAssume that assumption (A) holds.\nLet $(\\mathcal{X};\\mathcal{D})\\to B$ be a $p$-stable morphism of dimension $n$, with coefficients in $I$, and polynomial $p(t)$, over a reduced connected base $B$, and assume that there is an open dense subscheme $U\\subseteq B$ with:\n\\begin{enumerate}\n    \\item a stable family of pairs $(\\mathcal{Y}_U,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},U})\\to U$ of relative dimension $n$; and\n    \\item a contraction $\\pi_U \\colon \\mathcal{X}_U\\to \\mathcal{Y}_U $ such that $\\pi^*(K_{\\mathcal{Y}_U\/U} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},U}) = K_{\\mathcal{X}_U\/U} + \\frac{1}{r}\\mathcal{D}_{U}$.\n\\end{enumerate}\nThen, there there is an $m>0$ such that for every $d$ and every $b\\in B$ we have $$p_*(\\mathcal{O}_{\\mathcal{X}}(m(K_{\\mathcal{X}\/B}+\\frac{1}{r}\\mathcal{D}))^{\\otimes d})\\otimes k(b) = H^0(\\mathcal{X}_b, \\mathcal{O}_{\\mathcal{X}_b}(m(K_{\\mathcal{X}_b}+\\frac{1}{r}\\mathcal{D}_b))^{\\otimes d}) .$$\n\nMoreover, if we define $\\mathcal{Y} \\coloneqq \\operatorname{Proj}(\\bigoplus_d p_*\\mathcal{O}_{\\mathcal{X}}(m(K_{\\mathcal{X}\/B}+\\frac{1}{r}\\mathcal{D}))^{\\otimes d})$, then:\n\\begin{itemize}\n    \\item there is a unique family of divisors $\\mathcal{D}_\\mathcal{Y}$ such that the pair $q: (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\to B $ is a stable family extending $q_U$;\n    \\item there is a contraction $\\pi \\colon \\mathcal{X}\\to \\mathcal{Y}$ over $B$ that extends $\\pi_U$; and\n    \\item $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}) = K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}$.\n\\end{itemize}\n\\end{theorem}\n\\begin{Remark}\nObserve that since $K_{\\mathcal{Y}_U\/U} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},U}$ is $\\mathbb{Q}$-Cartier, we can define its pull-back as a $\\mathbb{Q}$-Cartier divisor. \n\\end{Remark}\n\n\n\\begin{proof}\nWe proceed in several steps.\n\n{\\bf Step 1:} In this step, we make some preliminary considerations and set some notation.\n\nBy Proposition \\ref{prop_stable_family_p-pairs}, we have that $\\K \\mathcal{X}\/B. + \\frac{1}{r}\\mathcal{D}$ and $\\mathcal{D}$ are $\\qq$-Cartier.\nThen, observe that for every $s,t\\in U$, the volumes of the pairs $(\\mathcal{Y}_s,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},s})$ and $(\\mathcal{Y}_t,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},t})$ agree.\nIndeed, since $(\\mathcal{X};\\mathcal{D})\\to B$ is a $p$-stable morphism, $\\K \\mathcal{X}_s.+ \\frac{1}{r}\\mathcal{D}_s$ and $\\K \\mathcal{X}_t.+ \\frac{1}{r}\\mathcal{D}_t$ are nef.\nThus, their volumes are computed by the $n$-fold self-intersection, which is independent of $s,t \\in S$.\nBut from condition (2) the morphisms $\\pi_s \\colon \\mathcal{X}_s\\to \\mathcal{Y}_s$ and $\\pi_t \\colon \\mathcal{X}_t\\to \\mathcal{Y}_t$ have connected\nfibers, and we have $\\pi_s^*(\\K \\mathcal{Y}_s. + \\frac{1}{r}\\mathcal{D} \\subs \\mathcal{Y},s.)=\\K \\mathcal{X}_s. +\\frac{1}{r} \\mathcal{D}_s$ and $\\pi_t^*(\\K \\mathcal{Y}_t. +\\frac{1}{r} \\mathcal{D} \\subs \\mathcal{Y},t.)=\\K \\mathcal{X}_t. +\\frac{1}{r} \\mathcal{D}_t$.\nThen, by Remark \\ref{remark:cohom:connected:implies:one:can:take:global:sections:after:pull:back}, the volumes of $\\K \\mathcal{Y}_s. + \\frac{1}{r}\\mathcal{D} \\subs \\mathcal{Y},s.$ (resp. $\\K \\mathcal{Y}_t. + \\frac{1}{r}\\mathcal{D} \\subs \\mathcal{Y},t.$) and $\\K \\mathcal{X}_s. +\\frac{1}{r} \\mathcal{D}_s$ (resp. $\\K \\mathcal{X}_t. + \\frac{1}{r}\\mathcal{D}_t$) agree.\nLet then $v$ be the volume of any fiber of $q_U$ and let $k$ be a natural number such that $r$ divides $k$ and, for every stable pair $(Y,D)$ of dimension $n$, volume $v$ and coefficients in $I$, the line bundle $\\mathcal{O}_Y(k(K_Y + D))$ is very ample and the higher cohomologies of all of its natural multiples vanish.\nNotice that $k$ exists by \\cite{HMX18}*{Theorem 1.2.2}.\nThen, we set $\\mathcal{L} \\coloneqq  \\O \\mathcal{X}. (k(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}))$.\nUp to replacing $k$ with a multiple, we may further assume that $\\mathcal{L}$ is Cartier.\n\n{\\bf Step 2:} In this step, we show that the theorem holds if $B$ is a smooth curve.\n\nFrom Lemma \\ref{Lemma:slc:canonical:model:for:p:pairs}, we can construct a family of stable pairs $(\\mathcal{Z},\\frac{1}{r}\\mathcal{D}_\\mathcal{Z})\\to B$ with a contraction $\\phi \\colon \\mathcal{X}\\to \\mathcal{Z}$.\nFirst, observe that $(\\mathcal{Z}_U,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Z},U})\\cong (\\mathcal{Y}_U,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},U})$ over $U$.\nIndeed, consider the reflexive sheaves $\\mathcal{L} _\\mathcal{Z} \\coloneqq \\O \\mathcal{Z}. (k(\\K \\mathcal{Z}. + \\frac{1}{r}\\mathcal{D} _\\mathcal{Z}))$ and $\\mathcal{L} _{\\mathcal{Y},U} \\coloneqq \\O \\mathcal{Y}_U. (k(\\K \\mathcal{Y}_U. + \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y},U}))$.\nBy construction, the fibers of $(\\mathcal{Z},\\frac{1}{r}\\mathcal{D}_\\mathcal{Z}) \\rar B$ and $(\\mathcal{Y}_U,\\frac{1}{r}\\mathcal{D} \\subs \\mathcal{Y},U.) \\rar U$ belong to the moduli problem of stable pairs with volume $v$ and coefficients in the finite set $I$.\nThus, by \\cite{kol1}*{Theorem 5.7.(4)} and the choice of $k$, $\\mathcal{L} _\\mathcal{Z}$ and $\\mathcal{L} _{\\mathcal{Y},U}$ are line bundles.\nIn order to apply \\cite{kol1}*{Theorem 5.7.(4)}, notice that we know that $\\mathcal{L} _\\mathcal{Z}$ and $\\mathcal{L} _{\\mathcal{Y},U}$ are line bundles away from the exceptional locus of $\\phi$ and $\\pi_U$, which are big open subsets restricting to big open subsets fiberwise.\n\nSince $\\pi_U$ and $\\phi_U$ have connected fibers, we have\n\\[\n\\pi_U^*\\mathcal{L}_{\\mathcal{Y},U} \\cong \\mathcal{L}_U \\cong \\phi^*_U\\mathcal{L}_{\\mathcal{Z},U}.\n\\]\nBut both $\\pi_U$ and $\\phi_U$ have connected fibers so, by the projection formula, for every $m \\geq 1$, we have\n\\[\nH^0(\\mathcal{Y}_U,\\mathcal{L}_{\\mathcal{Y},U}^{\\otimes m}) = H^0(\\mathcal{X}_U,\\pi^*_U\\mathcal{L}_{\\mathcal{Y},U}^{\\otimes m}) =H^0(\\mathcal{X}_U,\\mathcal{L}_{U}^{\\otimes m}) = H^0(\\mathcal{X}_U,\\phi^*_U\\mathcal{L}_{\\mathcal{Z},U}^{\\otimes m})  =  H^0(\\mathcal{Z}_U,\\mathcal{L}_{\\mathcal{Z},U}^{\\otimes m}).\n\\]\nBut $\\mathcal{L}_{\\mathcal{Z},U}$ and $\\mathcal{L}_{\\mathcal{Y},U}$ are ample over $U$, so $\\mathcal{Z}_U$ and $\\mathcal{Y}_U$ are isomorphic as $U$-schemes, as they are the relative Proj of the same sheaf of graded algebras.\nIn particular, the three final claims of the theorem hold if we consider the family $(\\mathcal{Z},\\frac{1}{r}\\mathcal{D}_\\mathcal{Z})$.\n\nThus, we are left with proving that for every $d$ and every $b\\in B$, we have $p_*(\\mathcal{L}^{\\otimes d})\\otimes k(b) = H^0(\\mathcal{X}_b, \\mathcal{L}^{\\otimes d}_b) $.\nBut since $\\phi^*\\mathcal{L}_\\mathcal{Z} = \\mathcal{L}$, we have $\\phi_b^*(\\mathcal{L}_{\\mathcal{Z},b}) = \\mathcal{L}_{b}$.\nThus, by Remark \\ref{remark:cohom:connected:implies:one:can:take:global:sections:after:pull:back}, we have $h^0(\\mathcal{X}_b,\\mathcal{L} _b^{\\otimes m}) = h^0(\\mathcal{Z}_b,\\mathcal{L} \\subs \\mathcal{Z},b.^{\\otimes m})$ for all $m \\geq 1$.\nThen, the latter is locally constant from the assumptions on $k$, as by the vanishing of the higher cohomologies we have $h^0(\\mathcal{Z}_b,\\mathcal{L} \\subs \\mathcal{Z},b.^{\\otimes m})= \\chi(\\mathcal{Z}_b,\\mathcal{L} \\subs \\mathcal{Z},b.^{\\otimes m})$, and the Euler characteristic of $\\mathcal{L} \\subs \\mathcal{Z},b.^{\\otimes m}$ is independent of $b \\in B$. Now the desired statement follows from \\cite{Mum74}*{Corollary 2, page 50}.\n\n{\\bf Step 3:} In this step, we return to the general case and we show that for every $m$, the morphism $B\\ni b\\mapsto h^0(\\mathcal{X}_b,\\mathcal{L}|_{\\mathcal{X}_b}^{\\otimes m})$ is constant and the algebra $\\bigoplus_m H^0(\\mathcal{X}_b,\\mathcal{L}|_{\\mathcal{X}_b}^{\\otimes m})$ is finitely generated.\n\n\nObserve that the claim holds for every $p\\in U$. Indeed, the following diagram commutes:\n$$\\xymatrix{\\mathcal{X}_b \\ar[r] \\ar[d]_{\\pi_b} & \\mathcal{X}_U \\ar[d] \\\\ \\mathcal{Y}_b \\ar[r] & \\mathcal{Y}_U}$$\nwhich from point (2) guarantees $\\pi_b^*(\\mathcal{L}_{\\mathcal{Y},b}) = \\mathcal{L}_{b}$.\nThen, in this case, we can conclude as at the end of Step 2.\nFurthermore, the finite generation follows from the fact that $\\mathcal{L}_{b}$ is very ample and $\\mathcal{Y}_b$ is the Proj of its associated graded ring.\n\nNow, we treat the case when $b \\not \\in U$.\nConsider a smooth curve $C$ with a map $C\\to B$.\nAssume that the generic point of $C$ maps into $U$.\nNotice that any point $b \\in B$ is contained in the image of such a curve.\nThen, for any point $s\\in C$, we have the following diagram:\n$$\\xymatrix{\\mathcal{X}_s\\ar[r] \\ar[d] & \\mathcal{X}_C \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ \\{s\\}\\ar[r] & C \\ar[r] & B.}$$\nSince both squares are fibered squares, the big rectangle is a fibered square.\nIn particular, since we want to prove that $B\\ni b\\mapsto h^0(\\mathcal{X}_b,\\mathcal{L}|_{\\mathcal{X}_b}^{\\otimes m})$\nis constant, and since we know it is constant as long as $b\\in U$, it suffices to check that for every such $C$ the functions $C\\ni s\\mapsto h^0(\\mathcal{X}_s,\\mathcal{L}|_{\\mathcal{X}_s}^{\\otimes m})$ are constant.\nNow, this follows by Step 2.\nSimilarly, the finite generation of $\\bigoplus_m H^0(\\mathcal{X}_s,\\mathcal{L}|_{\\mathcal{X}_s}^{\\otimes m})$ follows from Step 2.\n\n\n{\\bf Step 4:} In this step, we construct the model $\\mathcal{Y}$ and the morphism $\\pi$.\n\nBy Step 3 and cohomology and base change (see \\cite{Mum74}), the sheaves $p_*(\\mathcal{L}^{\\otimes m})$ commute with the restriction to points. In particular, the algebra $\\mathcal{A}  \\coloneqq  \\bigoplus_{m} p_*(\\mathcal{L}^{\\otimes m})$ is finitely generated since it is finitely generated when restricted to every point $b\\in B$.\nSo we can consider $\\mathcal{Y} \\coloneqq \\operatorname{Proj}_B(\\mathcal{A})$.\n\nAs observed in Step 3, the pluri-sections of $\\mathcal{L}|_{\\mathcal{X} b}$ are deformation invariant.\nAs $\\mathcal{L}|_{\\mathcal{X} b}$ is semi-ample for every $b \\in B$ by our assumptions, it follows that $\\mathcal{L}$ is relatively semi-ample.\nIn particular, we have a morphism $\\mathcal{X} \\rar \\mathcal{Y}$.\nFurthermore, this implies the equality $\\mathcal{Y}_b=\\mathrm{Proj}(\\bigoplus_m H^0(\\mathcal{X}_b,\\mathcal{L}_b^{\\otimes m}))$ for every $b \\in B$.\n\nAs already discussed, this construction commutes with base change.\nIn particular, for every $b\\in B$, for checking properties of $\\mathcal{Y}_b$ we can consider a smooth curve $C\\to B$ which sends the generic point to $U$ and the special one to $b$, and first pull back $\\mathcal{Y}$ to $C$ and then restrict it to $b$:\n$$\\xymatrix{\\mathcal{Y}_b\\ar[r] \\ar[d] & \\mathcal{Y}_C \\ar[r] \\ar[d] & \\mathcal{Y} \\ar[d] \\\\ \\{b\\}\\ar[r] & C \\ar[r] & B.}$$\nThe advantage is that now we can apply the results of Step 2 to $\\mathcal{Y}_C$.\n\nSince we have $\\mathcal{L}_U \\cong \\pi_U^* \\mathcal{L}_{\\mathcal{Y},U}$ and $\\mathcal{L}_{\\mathcal{Y},U}$ is relatively ample over $U$, it follows that $\\mathcal{Y} \\times_B U = \\mathcal{Y}_U$ and that $\\pi$ extends $\\pi_U$.\nFurthermore, since $B$ is reduced, the construction commutes with base change, and each fiber is reduced, it follows that $\\mathcal{Y}$ is reduced.\n\n{\\bf Step 5:} In this step, we show that $\\pi$ is a contraction and we construct the divisor $\\mathcal{D}_\\mathcal{Y}$.\n\nWe first prove that $\\pi$ is a contraction. We denote by $V\\subseteq \\mathcal{Y}$ the locus where $\\pi^{-1}(V)\\to V$ is an isomorphism. Then\nit follows from Lemma \\ref{ample model p-pair}:\n\\begin{center}\n    {\\bf ($\\ast \\ast$)} {\\it for every fiber $\\mathcal{Y}_b$, the complement of $V_b \\coloneqq V \\cap \\mathcal{Y}_b$ has codimension at least 2 in $\\mathcal{Y}_b$ and it does not contain any irreducible component of the conductor of $\\mathcal{Y}_b$.}\n\\end{center}\nConsider now the inclusion $i:\\pi^{-1}(V)\\to \\mathcal{X}$, which induces the injective map $0\\to \\mathcal{O}_\\mathcal{X} \\to i_*\\mathcal{O}_{\\pi^{-1}(V)}$. We can push this sequence forward via $\\pi$ and we obtain $0\\to \\pi_*\\mathcal{O}_\\mathcal{X} \\to \\pi_*i_*\\mathcal{O}_{\\pi^{-1}(V)}$. But $\\pi\\circ i :\\pi^{-1}(V)\\to \\mathcal{Y}$ is the inclusion $j: V\\hookrightarrow Y$, so $\\pi_*i_*\\mathcal{O}_{\\pi^{-1}(V)} = j_*j^*\\mathcal{O}_\\mathcal{Y}$ is reflexive from \\cite{HK04}*{Corollary 3.7}, and it is isomorphic to $\\mathcal{O}_{\\mathcal{Y}}$ from \\cite{HK04}*{Proposition 3.6.2}. In particular, this gives an injective map $\\pi_*\\mathcal{O}_\\mathcal{X}\\to \\mathcal{O}_\\mathcal{Y}$. One can check that this is the inverse of the canonical morphism $\\mathcal{O}_\\mathcal{Y}\\to \\pi_*\\mathcal{O}_\\mathcal{X}$, so in particular the latter is an isomorphism.\n\nConsider the ideal sheaf $\\mathcal{I}$ of $\\mathcal{D}$, and consider the inclusion\n\\[\n0\\to \\mathcal{I} \\to \\mathcal{O}_\\mathcal{X}.\n\\]\nThen, as $\\pi$ is a contraction, if we push it forward via $\\pi$, we get\n\\[0\\to \\pi_*\\mathcal{I} \\to \\mathcal{O}_\\mathcal{Y}.\n\\]\nIn particular, $\\pi_* \\mathcal{I}$ is an ideal sheaf on $\\mathcal{Y}$.\nWe denote by $\\mathcal{S}\\subseteq \\mathcal{Y}$ the closed subscheme with ideal sheaf $\\pi_*\\mathcal{I}$.\n\nA priori, $\\mathcal{S}$ may not be pure dimensional, however, consider the intersection between $\\pi^{-1}(V)$ and the locus in $\\mathcal{X}$ where $\\mathcal{D}$ is $\\mathbb{Q}$-Cartier: we denote this locus with $W$.\nObserve that, since $\\pi^{-1}(V)\\to V$ is an isomorphism, we can identify $W$ with a subset of $\\mathcal{Y}$.\nMoreover, since $\\mathcal{D}$ is Cartier on codimension one point of $\\mathcal{X}$ and $V$ is a big open subset, the locus $W$ contains all the codimension one points of $\\mathcal{Y}_b$ for every $b$.\nThen, we consider $\\mathcal{S} \\cap W$, and we define $\\mathcal{D}_\\mathcal{Y}$ to be the closure of $\\mathcal{S} \\cap W$ in $\\mathcal{Y}$.\n\n\n{\\bf Step 6:} In this step, we show that $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\rar B$ is a well defined family of pairs.\n\nBy construction, it is immediate that $\\mathcal{D}_\\mathcal{Y}$ is a relative Mumford divisor in the sense of \\cite{kol19s}*{Definition 1}. Indeed, the three conditions of \\cite{kol19s}*{Definition 1} are now clear, since they hold on $\\mathcal{X}$.\n\nThus, we just need to check that $q \\colon \\mathcal{Y}\\to B$ is flat.\nBy pulling back $\\mathcal{Y}\\to B$ along a smooth curve through $U$, it follows from Step 1 that all the fibers of $q$ are reduced and equidimensional.\nThus, $q$ is an equidimensional morphism with reduced fibers over a reduced base, so \\cite{kol1}*{Lemma 10.48} applies.\n\n{\\bf Step 7:} In this step, we show that $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\rar B$ is a stable family of pairs and $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}) = K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}$.\n\nSince $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\rar B$ is a well defined family of pairs and its fibers belong to a prescribed moduli problem for stable pairs, we can argue as in the proof of Proposition \\ref{prop_stable_family_p-pairs} to conclude that $K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}$ is $\\mathbb Q$-Cartier.\nIn particular, $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\rar B$ is a stable family of pairs.\n\nBy construction, we have that $K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}} =\\pi_*( K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D})$.\nFurthermore, we have that the equality $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}) = K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}$ holds over $U$.\nThen, by construction, all the exceptional divisors of $\\pi$ dominate $B$, as they are contained in the support of $\\mathcal{D}$.\nThus, as $K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}$ is $\\mathbb Q$-Cartier and so $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}})$ is well defined, it follows that $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}) = K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}$.\n\\end{proof}\n\n\\begin{Lemma}\\label{lemma:finitedness:paper:zsolt}\nWith the notation and assumptions of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}, assume that $B$ is an affine curve and that there is a stable pair $(Y,D_Y)$ such that $(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})\\cong (Y\\times B,D_Y\\times B)$.\nThen there are finitely many isomorphism classes of $p$-stable pairs in the fibers of $(\\mathcal{X};\\mathcal{D})\\to B$.\n\\end{Lemma}\n\\begin{proof}\nThe proof is analogous to the proof of \\cite{ABIP}*{Claim 6.2}, we summarize here the most salient steps of the argument.\n\n\\begin{bf}Step 1:\\end{bf} Using Koll\\'ar's gluing theory and the fact that stable pairs have finitely many isomorphisms, up to normalizing and disregarding finitely many points on $B$, we can assume that $\\mathcal{X}$ (and therefore also $\\mathcal{Y}$) is normal.\nThis is achieved in \\cite{ABIP}*{Lemma 6.5}.\nIn particular, $(Y\\times B,D_Y\\times B)$ is the canonical model of $(\\mathcal{X},\\frac{1}{r}\\mathcal{D})$.\n\n\\begin{bf}Step 2:\\end{bf} We observe that the divisors contracted by $\\mathcal{X}\\to Y\\times B$ have negative discrepancies and can be extracted by a log resolution of the form $Z\\times B\\to Y\\times B$, where $Z\\to Y$ is a log resolution.\nThis is achieved in \\cite{ABIP}*{Proposition 6.13}.\n\n\n\\begin{bf}Step 3:\\end{bf} To conclude, we observe that all the fibers of $(\\mathcal{X};\\mathcal{D})\\to B$ are isomorphic in codimension 2.\n\nBut two \\emph{stable} pairs $(X_1, D_1)$ and $(X_2, D_2)$ which are isomorphic in codimension 2 must be isomorphic. Indeed, if $U$ is the open subset where they agree, and $L_1$ (resp. $L_2$) is the log-canonical divisor $K_{X_1} + D_1$ (resp. $K_{X_2} + D_2$) then $$H^0(X_1, L_1^{[m]}) = H^0(U, L_1^{[m]}) = H^0(U, L_2^{[m]}) =  H^0(X_2, L_2^{[m]}).$$\nTherefore $X_1$ and $X_2$ are $\\mathrm{Proj}$ of the same graded algebra, so they are all isomorphic.\n\\end{proof}\n\n\\section{Projectivity of the moduli of stable pairs}\n\\label{section projectivity ksba}\nThe goal of this section is to provide a different proof of the projectivity of the moduli of stable pairs, established in \\cite{KP17}, using $p$-pairs.\n\\begin{Lemma}\\label{Lemma_there_is_a_section_vanishing_on_the_divisor}\nWith the notation of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}, we will denote by $p\\colon (\\mathcal{X};\\mathcal{D})\\to B$ the $p$-stable family, and by $q:(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})\\to B$ the resulting stable family of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}.\nAssume this $p$-stable morphism is a $p$-stable family, and let $m_0$ be the smallest positive integer such that $m_0\\frac{1}{r}\\mathcal{D}$ is Cartier.\nThen, there is $k_0$ dividing $m_0$ such that, for every $k=\\ell k_0$ positive multiple of $k_0$, the sheaf  $\\mathcal{L}_{\\mathcal{Y}} \\coloneqq \\mathcal{O}_\\mathcal{Y}(k(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$ satisfies the following properties:\n\\begin{enumerate}[(a)]\n    \\item $\\mathcal{L}_\\mathcal{Y}$ is Cartier;\n    \\item $R^iq_*(\\mathcal{L}_\\mathcal{Y}^{\\otimes j}) = 0$ for every $i>0$ and $j>0$;\n    \\item $\\mathcal{L}_\\mathcal{Y}$ gives an embedding $\\mathcal{Y}\\hookrightarrow \\mathbb{P}(q_*\\mathcal{L}_\\mathcal{Y})$;\n    \\item $\\mathcal{O}_\\mathcal{X}(k(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) - \\frac{m_0}{r}\\mathcal{D})$ is relatively ample, and\n    \\item for every $b\\in B$, there is a section $s\\in H^0(\\mathcal{Y}_b, \\mathcal{L}_{\\mathcal{Y},b})$ such that $V(s)$ has codimension 1 in each irreducible component of $\\mathcal{Y}_b$ and the scheme-theoretic image of $m_0\\mathcal{D}\\to \\mathcal{Y}$ restricted to $\\mathcal{Y}_b$ is contained in $V(s)$.\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{Remark}\nObserve that the definition of $m_0\\frac{1}{r}\\mathcal{D}$ is given in Notation \\ref{Notation_mD_for_pstable_fam}.\nIn our case, we can still define $m_0\\frac{1}{r}\\mathcal{D}$ even if $(\\mathcal{X};\\mathcal{D})\\to B$ was a $p$-stable morphism instead, since the base $B$ is reduced, and Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor} would go through verbatim.\nSince we will use Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor} only in the case in which $(\\mathcal{X};\\mathcal{D})\\to B$ is a $p$-stable family, for simplicity we stick with the family case.\n\\end{Remark}\n\n\\begin{proof}\nBy construction, $K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}$ is relatively ample.\nThus, for a sufficiently divisible $k_0$, properties (a) to (d) are satisfied for $k=k_0$.\n\nThen, we can achieve (a) to (d) since if $\\mathcal{O}_\\mathcal{Y}(k_0(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$ satisfies any condition between (a) to (e), then for every $m>0$ also $\\mathcal{O}_\\mathcal{Y}(mk_0(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$ satisfies the same condition.\nWe only need to check that up to choosing $k_0$ divisible enough, also (e) holds.\n\nLet $\\mathcal{G} \\coloneqq \\mathcal{O}_\\mathcal{Y}(k_0(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$ for a $k_0$ which satisfies (a)-(d), and we need to show that\n$\\mathcal{G}^{\\otimes m}$ satisfies (a)-(e) for some $m\\gg 0$.\nFirst observe that from cohomology and base change and point (b), for every $b\\in B$ we have $q_*(\\mathcal{G}^{\\otimes m})\\otimes k(b) = H^0(\\mathcal{Y}_b,\\mathcal{G}_{|\\mathcal{Y}_b}^{\\otimes m})$.\n\nLet $\\mathcal{Z}\\subseteq \\mathcal{Y}$ be the scheme-theoretic image of $\\pi_{|\\frac{m_0}{r}\\mathcal{D}}$.\nUp to replacing $B$ with a locally closed stratification $B'\\to B$, we can assume that $\\mathcal{Z}' \\coloneqq \\mathcal{Z}\\times_B B'\\to B'$ is flat, and $q_*\\mathcal{G}$ is free.\nThen, we define $\\mathcal{Y}' \\coloneqq \\mathcal{Y}\\times_B B'$, the first projection $\\pi_1 \\colon \\mathcal{Y}'\\to \\mathcal{Y}$, the second projection $q' \\colon \\mathcal{Y}'\\to B'$, and $\\mathcal{G}' \\coloneqq \\pi_1^*\\mathcal{G}$.\nThen, consider the exact sequence \n\\[\n0\\to \\mathcal{I}\\to \\mathcal{O}_{\\mathcal{Y}'}\\to \\mathcal{O}_{\\mathcal{Z}'} \\to 0.\n\\]\nWe twist the exact sequence above by $\\mathcal{G}'^{\\otimes m}$ and we obtain \n\\[\n0\\to \\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes m} \\to  \\mathcal{G}'^{\\otimes m}\\to \\mathcal{O}_{\\mathcal{Z}'}\\otimes \\mathcal{G}'^{\\otimes m} \\to 0.\n\\]\nSince $\\mathcal{Z}'\\to B'$ is flat, also $\\mathcal{I}$ is flat over $B'$. Therefore since $\\mathcal{G}'$ is relatively ample, up to choosing $m$ big enough, the following is an exact sequence on $B'$:\n\n\\[\n0\\to q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes m})\\to  q'_*(\\mathcal{G}'^{\\otimes m})\\to q'_*(\\mathcal{O}_{\\mathcal{Z}'}\\otimes \\mathcal{G}'^{\\otimes m}) \\to 0.\n\\]\nBy \\cite{ACH11}*{Corollary 4.5 (iii)}, there is $n_0$ such that for every $n\\ge n_0$, the multiplication map\n\\[\n\\operatorname{Sym}^a(q'_*\\mathcal{G}')\\otimes q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes m}) \\to q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes m+a})\n\\]\nis surjective. Then for every $b\\in B'$, also \n$$\\operatorname{Sym}^a(q'_*\\mathcal{G}')\\otimes q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n}) \\otimes k(b) \\cong( \\operatorname{Sym}^a(q'_*\\mathcal{G}')\\otimes k(b) )\\otimes_{k(b)}( q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n}) \\otimes k(b)) \\to q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n+a})\\otimes k(b)$$\nis surjective.\nFrom \\cite{ACH11}*{Corollary 4.5 (vi)} (or cohomology and base change) we have:\n\\[\n\\operatorname{Sym}^a(q'_*\\mathcal{G}')\\otimes k(b) \\cong H^0(\\mathcal{O}_{\\mathbb{P}^N}(a)),\n\\]\n\\[ q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n}) \\otimes k(b) \\cong H^0(\\mathcal{Y}_b,(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n})_{|\\mathcal{Y}_b}) = H^0(\\mathcal{Y}_b,\\mathcal{I}_{|\\mathcal{Y}_b }(n))\\text{ } \\text{ }\\text{ and }\n\\]\n\\[\nq'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n+s}) \\otimes k(b) \\cong H^0(\\mathcal{Y}_b,(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n+s})_{|\\mathcal{Y}_b}) = H^0(\\mathcal{Y}_b,\\mathcal{I}_{|\\mathcal{Y}_b }(n+s)).\n\\]\nIn other terms, for every $b\\in B$ the ideal $\\mathcal{I}_{\\mathcal{Y}_b}$ is generated in degree $n_0$. Then for every $b\\in B$ there is a section $s\\in H^0(\\mathcal{Y}_b, \\mathcal{L}_{\\mathcal{Y},b})$ (and therefore also a section $s^{\\otimes m}\\in H^0(\\mathcal{Y}_b, \\mathcal{L}_{\\mathcal{Y},b}^{\\otimes m})$) such that $V(s)$ does not contain any irreducible component of $\\mathcal{Y}_b$ and the scheme-theoretic image of $\\frac{m_0}{r}\\mathcal{D}\\to \\mathcal{Y}$ restricted to $\\mathcal{Y}_b$ is contained in $V(s)$.\n\\end{proof}\n\n\\begin{Cor}\\label{corollary_the_section_that_vanishes_on_the_scheme_theoretic_image_of_m0\/rD_can_be_lifted_on_X}\nWith the notation and assumptions of Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}, for every $b\\in B$ and every $a>0$, there is a global section $t$ of $\\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a}$ that is not zero on the generic points of $\\mathcal{X}_b$ but its maps to 0 via the restriction map $H^0(\\mathcal{X}_b, \\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a})\\to H^0(\\frac{am_0}{r}\\mathcal{D}_b, \\mathcal{L}^{\\otimes a}_{|\\frac{am_0}{r}\\mathcal{D}_b})$.\n\\end{Cor}\n\n\\begin{proof}\nFrom Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}, for every $b \\in B$, there is a section $s$ that does not vanish on the generic points of $\\mathcal{Y}_b$ but vanishes on the restriction to $\\mathcal{Y}_b$ of the scheme theoretic image of $\\frac{m_0}{r}\\mathcal{D}$.\nFrom point (b) of  Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}, $q_*\\mathcal{L}_\\mathcal{Y}$ is a vector bundle on $B$, so there is an open subset $b\\in U\\subseteq B$ such that $(q_*\\mathcal{L}_\\mathcal{Y})_{|U}$ is free.\nThen, we can extend the section $s$ to a global section $s'\\in H^0(U,q_*(\\mathcal{L}_\\mathcal{Y})_{|U}) = H^0(U,(q_U)_*(\\mathcal{L}_{\\mathcal{Y},U}))$.\nBut $\\pi$ is a contraction, so $\\pi_{U} \\colon \\mathcal{X}_U\\to \\mathcal{Y}_U$ is still a contraction, and $\\pi_{U}^*\\mathcal{L}_{\\mathcal{Y},U} = \\mathcal{L}_U$.\nIn particular, consider the section $t\\in H^0(\\mathcal{X}_U,\\mathcal{L}_U)$ that is the pull-back of $s'$.\nThen, $t$ does not vanish on the generic points of $\\mathcal{X}_b$, as by Lemma \\ref{ample model p-pair} those are in bijection with the generic points of $\\mathcal{Y}_b$.\nMoreover, it vanishes along $\\frac{m_0}{r}\\mathcal{D}_b$: this is a local computation. Indeed, if we replace $\\mathcal{Y}$ and $\\mathcal{X}$ with appropriate open subsets $\\operatorname{Spec}(A)$, $\\operatorname{Spec}(B)$; and $f$ is the generator for the ideal sheaf of $\\frac{m_0}{r}\\mathcal{D}$, we have the following commutative diagram:\n$$\\xymatrix{B\\otimes k(b)\/f\\otimes 1 & B\/f \\ar[l] \\\\ B\\otimes k(b) \\ar[u] & t\\in B  \\ar[u] \\ar[l] \\\\ A\\otimes k(b) \\ar[u] & s\\in A. \\ar[u] \\ar[l]}$$\nSince the image of $s$ vanishes in $B\\otimes k(b)\/f\\otimes 1 $, so does the image of $t$. \n\nTherefore for every $a>0$, the section $t^{\\otimes a}\\in H^0(\\mathcal{X}_U,\\mathcal{L}_U^{\\otimes a})$ will vanish along $\\frac{am_0}{r}\\mathcal{D}_b$.\nIn other terms, if we look at the exact sequence\n\\[\n0 \\to H^0(\\mathcal{X}_b, \\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a}\\otimes \\mathcal{I}_{\\mathcal{D}_b}^{[\\frac{am_0}{r}]})\\to H^0(\\mathcal{X}_b, \\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a})\\xrightarrow{\\psi} H^0(\\frac{am_0}{r}\\mathcal{D}_b, \\mathcal{L}^{\\otimes a}_{|\\frac{am_0}{r}\\mathcal{D}_b}),\n\\]\nwe have an element of $H^0(\\mathcal{X}_b, \\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a})$ (namely, the restriction of $t^{\\otimes a}$ to $\\mathcal{X}_b$) that does not vanish along the generic points of $\\mathcal{X}_b$ but maps to 0 via $\\psi$.\n\\end{proof}\n\n\n\\begin{Lemma}\\label{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}\nWith the notation and assumptions of Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}, we will denote by $p\\colon (\\mathcal{X};\\mathcal{D})\\to B$ the $p$-stable family, and with $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})\\to B$ the resulting stable family of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}.\nLet $k$ be a multiple of $k_0$, where $k_0$ is as in Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}.\nAlso, set $\\mathcal{L} \\coloneqq \\mathcal{O}_\\mathcal{X}(k(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}))$.\n\nThen, there is an $a_0$ such that, for every $a\\ge a_0$, the sheaf $p_*(\\mathcal{L}_{|\\frac{am_0}{r}\\mathcal{D}}^{\\otimes a})$ is a vector bundle, and fits in an exact sequence as follows: \n\\[\n0\\to p_*(\\mathcal{O}_\\mathcal{X}(ka(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}) - \\frac{am_0}{r}\\mathcal{D})) \\to p_*( \\mathcal{L}^{\\otimes a}) \\to p_*(\\mathcal{L}_{|\\frac{am_0}{r}\\mathcal{D}}^{\\otimes a})\\to 0.\n\\]\n\\end{Lemma}\n\n\n\n\n\\begin{proof}\nFirst, from relative Serre's vanishing and Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor} (b) and (d), there is an integer $a_0$ such that for every $b\\in B$ and $a \\geq a_0$, we have\n    \\begin{itemize}\n        \\item $H^i(\\mathcal{O}_{\\mathcal{X}_b}(ak(K_{\\mathcal{X}_b} + \\frac{1}{r}\\mathcal{D}_b) - \\frac{am_0}{r}\\mathcal{D}_b)=0$ for $i>0$; and\n        \\item $H^i(\\mathcal{O}_{\\mathcal{Y}_b}(ak(K_{\\mathcal{Y}_b} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},b})))=0$ for $i>0$.\n    \\end{itemize}\n\n\n\nFix a positive integer $a>a_0$.\nWe begin by considering the following exact sequence, where $\\mathcal{J}$ is the locally free ideal sheaf $\\mathcal{O}_\\mathcal{X}\\left(-\\frac{m_0}{r}\\mathcal{D}\\right) $:\n$$0\\to \\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a} \\to \\mathcal{L}^{\\otimes a} \\to \\mathcal{L}_{|\\frac{am_0}{r}\\mathcal{D}}^{\\otimes a}\\to 0.$$\nWe push it forward via $p$, and from the first bullet point and from cohomology and base change,\n$R^1p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a}) = 0$, so we have:\n\\begin{equation} \\label{eq_ses}\n    0\\to p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a} ) \\to p_*( \\mathcal{L}^{\\otimes a}) \\to p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\to 0.\n\\end{equation}\n\nRecall that there is a contraction $\\pi \\colon \\mathcal{X}\\to \\mathcal{Y}$ such that, if we denote $\\mathcal{L}_\\mathcal{Y} \\coloneqq \\mathcal{O}_{\\mathcal{Y}}(k(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$, then  $\\pi^*(\\mathcal{L}_\\mathcal{Y}) = \\mathcal{L}$.\nIn particular\n\\begin{equation} \\label{eq_push_fwd}\n    p_*(\\mathcal{L}^{\\otimes a}) = q_*\\pi_* (\\mathcal{L}^{\\otimes a}) = q_*(\\mathcal{L}_\\mathcal{Y}^{\\otimes a}).\n\\end{equation}\n\nNow, pick $b\\in B$. From the first bullet point and from cohomology and base-change, we have \\begin{equation} \\label{eq_13}\n    p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a} )\\otimes k(b)\\cong H^0(\\mathcal{X}_b, (\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})_{|\\mathcal{X}_b})  \\cong H^0(\\mathcal{X}_b,\\mathcal{O}_{\\mathcal{X}_b}(ak(K_{\\mathcal{X}_b} + \\frac{1}{r}\\mathcal{D}_b) - \\frac{am_0}{r}\\mathcal{D}_b))\n\\end{equation}\nand from the second bullet point, equation \\eqref{eq_push_fwd} and the fact that $\\pi_b\\colon \\mathcal{X}_b\\to \\mathcal{Y}_b$ is a contraction,\n\n\\begin{equation} \\label{eq_14}\n    p_*(\\mathcal{L}^{\\otimes a})\\otimes k(b)\\cong H^0(\\mathcal{Y}_b, \\mathcal{L}^{\\otimes a}_{\\mathcal{Y}, b}) \\cong  H^0(\\mathcal{Y}_b,\\mathcal{O}_{\\mathcal{Y}_b}(ak(K_{\\mathcal{Y}_b} + \\frac{1}{r}\\mathcal{D}_b)))  \\cong H^0(\\mathcal{X}_b, \\mathcal{L}^{\\otimes a}_{|\\mathcal{X}_b}).\n\\end{equation}\n\n\nNow, we tensor the sequence \\eqref{eq_ses} by $k(b)$, and we obtain \n\\[\n     p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})\\otimes k(b) \\to p_*( \\mathcal{L}^{\\otimes a})\\otimes k(b) \\to p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\otimes k(b)\\to 0.\n\\]\nFrom the natural adjunction between pull-back and push-forward, we have the following commutative diagram:\n\\[\n\\xymatrix{ & p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})\\otimes k(b) \\ar[d]_{\\cong \\text{ from \\eqref{eq_13}}}\\ar[r]^-\\phi & p_*( \\mathcal{L}^{\\otimes a})\\otimes k(b) \\ar[r] \\ar[d]^{\\cong \\text{ from \\eqref{eq_14}}} &p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\otimes k(b)\\ar[r] \\ar[d] & 0\n\\\\ 0 \\ar[r] & H^0( \\mathcal{X}_b, (\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})_{|\\mathcal{X}_b})\\ar[r] & H^0(\\mathcal{X}_b, \\mathcal{L}^{\\otimes a}_{|\\mathcal{X}_b}) \\ar[r] & H^0(\\frac{m_0a}{r}\\mathcal{D}, (\\mathcal{L}^{\\otimes a})_{|\\frac{m_0a}{r}\\mathcal{D}_b}) \\ar[r]^-{(\\ast)}& 0}\n\\]\nObserve that the exactness of the map $(\\ast)$ follows from the first bullet point.\nThen, from diagram chasing, also $\\phi$ is injective.\nIn particular,\n\\[\n\\dim_{k(b)}(p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\otimes k(b)) = \\dim_{k(b)}(H^0(\\mathcal{X}_b, \\mathcal{L}^{\\otimes a}_{|\\mathcal{X}_b})) - \\dim_{k(b)}(H^0( \\mathcal{X}_b, (\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})_{|\\mathcal{X}_b})).\n\\]\nHowever the right-hand side does not depend on $b\\in B$. Therefore also the left-hand side does not depend on $b\\in B$, so $p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})$ is a vector bundle.\n\\end{proof}\n\n\\begin{Cor}\\label{cor_determinant}\nWith the notation and assumptions of Lemma \\ref{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}, there is an isomorphism\n\\[\n\\det p_*(\\mathcal{L}^{\\otimes a}) \\cong \\det (p_*(\\mathcal{O}_{\\mathcal{X}}(ak(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}) - \\frac{am_0}{r}\\mathcal{D})))\\otimes \\det( p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})).\n\\]\n\\end{Cor}\n\n\\begin{theorem}\\label{thm:main:step:proj}\nWith the notation and assumptions of Lemma \\ref{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}, further assume that the map $B\\to \\mathscr{F}_{n,p,I}$ given by the $p$-stable family $(\\mathcal{X};\\mathcal{D})\\to B$ is finite.\nThen for $a$ divisible enough, the line bundle $\\det p_*(\\mathcal{L}^{\\otimes a})$ is ample.\n\\end{theorem}\n\n\\begin{proof}\nWe plan on using Koll\\'ar's Ampleness Lemma, see \\cite{Kol90}.\nBy Corollary \\ref{cor_determinant}, it suffices to show that $\\det p_*(\\mathcal{L}^{\\otimes a}) \\otimes \\det (p_*(\\mathcal{O}_\\mathcal{X}(ak(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D}))))\\otimes \\det( p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a}))$ is ample.\n\nConsider the vector bundles\n\\[\n\\mathcal{Q}_a \\coloneqq p_*(\\mathcal{L}^{\\otimes a}) \\oplus p_*(\\mathcal{O}_\\mathcal{X}(ak(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})))) \\oplus p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\text{ }\\text{ }\\text{ }\\text{ and }\n\\]\n\\[\n\\mathcal{W}_{b,c} \\coloneqq \\operatorname{Sym}^b( p_*(\\mathcal{L}^{\\otimes c})) \\oplus \\operatorname{Sym}^b(\\mathcal{O}_\\mathcal{X}(ak(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})) \\oplus \\operatorname{Sym}^b( p_*(\\mathcal{L}^{\\otimes c}))\n\\]\nfor appropriate choices of $a$, $b$, and $c$.\n\nWhen $a = bc$, there is a morphism $\\mathcal{W}_{b,c}\\to \\mathcal{Q}_{bc}$ given by the sum of the multiplication morphisms\n\\[\n\\operatorname{Sym}^b( p_*(\\mathcal{L}^{\\otimes c})) \\xrightarrow{\\alpha} p_*(\\mathcal{L}^{\\otimes bc}),\\quad \\operatorname{Sym}^b( p_*(\\mathcal{O}_\\mathcal{X}(ck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D}))) \\xrightarrow{\\beta} p_*(\\mathcal{O}_\\mathcal{X}(bck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})),\n\\]\nand the composition of $\\alpha$ with the surjection in the exact sequence of Lemma \\ref{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}, denoted by\n\\[\n\\operatorname{Sym}^b( p_*(\\mathcal{L}^{\\otimes c})) \\xrightarrow{\\gamma} p_*(\\mathcal{L}_{|\\frac{m_0bc}{r}\\mathcal{D}}^{\\otimes bc}).\n\\]\n\nFirst, observe that for $c$ divisible enough, the vector bundles $p_*(\\mathcal{L}^{\\otimes c})$ and $  p_*(\\mathcal{O}_\\mathcal{X}(ck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})))$ are nef. Indeed, from the assumptions of Lemma \\ref{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}, the formation of $p_*(\\mathcal{L}^{\\otimes c})$ and $  p_*(\\mathcal{O}_\\mathcal{X}(ck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})))$ commutes with base change, so we can assume that the base $B$ is a smooth curve.\nThen the statement follows from \\cite{Fuj18}*{Theorem 1.11}.\nTherefore, their symmetric powers and sum are nef, so $\\mathcal{W}_{b,c}$ is nef for $c$ divisible enough.\n\nSince $\\mathcal{L}_{\\mathcal{Y}}$ is relatively ample and $q_*(\\mathcal{L}_{\\mathcal{Y}}^{\\otimes m}) \\cong p_*(\\mathcal{L}^{\\otimes m})$, there is a $b_0$ such that, for every $b\\ge b_0$, the map $\\alpha$ is surjective, and for the same reason also $\\beta$ is surjective (see  \\cite{ACH11}*{Corollary 4.5}).\nThen, also $\\gamma$ is surjective since it is the composition of surjective morphisms.\nTherefore we have a surjection $\\Phi:\\mathcal{W}_{b,c}\\to \\mathcal{Q}_{bc}$.\nWe denote by $G$ the structure group of $\\mathcal{W}_{b,c}$.\nThis gives a map of sets\n\\[\n\\Psi \\colon |B|\\to |[\\operatorname{Gr}(w,q)\/G]|,\n\\]\nwhere $w$ (resp. $q$) is the rank of $\\mathcal{W}_{b,c}$ (resp. $\\mathcal{Q}_{bc}$), and where for a stack $\\mathcal{Z}$ we denote by $|\\mathcal{Z}|$ its associated topological space.\nIf we show that $\\Psi$ has finite fibers, then the theorem follows from \\cite{Kol90}*{Ampleness Lemma, 3.9}.\n\nConsider two points $x_1$, $x_2$ that map to the same point via $\\Psi$.\nOver the point $x_1$ we have the surjection $\\Phi_{x_1} \\colon \\mathcal{W}_{b,c}\\otimes k(x_1)\\to \\mathcal{Q}_{bc}\\otimes k(x_1)$, and similarly we have one denoted by $\\Phi_{x_2}$ over $x_2$.\nChoose two isomorphisms\n\\[\n\\tau_1 \\colon H^0(\\mathbb{P}^{N}, \\mathcal{O}_{\\mathbb{P}^{N}}(1)) \\to p_*(\\mathcal{L}^{\\otimes c})\\otimes k(x_1), \\text{ and }\n\\]\n\\[\n\\tau_1' \\colon H^0(\\mathbb{P}^{M}, \\mathcal{O}_{\\mathbb{P}^{M}}(1)) \\to  p_*(\\mathcal{O}_\\mathcal{X}(ck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{M}{k})\\mathcal{D})))  \\otimes k(x_1).\n\\]\nSimilarly we define  $\\tau_2$ and $\\tau_2'$ for the same isomorphisms over $x_2$.\nThis gives an isomorphism\n\\[\nH^0(\\mathbb{P}^{N}, \\mathcal{O}_{\\mathbb{P}^{N}}(b)) \\oplus H^0(\\mathbb{P}^{M}, \\mathcal{O}_{\\mathbb{P}^{M}}(b)) \\oplus H^0(\\mathbb{P}^{N}, \\mathcal{O}_{\\mathbb{P}^{N}}(b)) \\xrightarrow{\\operatorname{Sym}^b(\\tau_1) \\oplus \\operatorname{Sym}^b(\\tau_1') \\oplus \\operatorname{Sym}^b(\\tau_1)} \\mathcal{W}_{b,c}\\otimes k(x_1).\n\\]\nSince $x_1$ and $x_2$ map to the same point via $\\Psi$, using the identifications above, there is an element $g\\in G$ such that $g\\operatorname{Ker}( \\Phi_{x_1}) =\\operatorname{Ker}(\\Phi_{x_2}) $.\n\nIn particular, we can choose a basis for $\\mathcal{W}_{b,c}\\otimes k(x_1)$ by first choosing a basis for each summand of the left-hand side, choosing the same basis for the first and third summand.\nThen, since $\\mathcal{W}_{b,c}$ is a direct sum of vector bundles, in this basis the linear transformation $g$ is block diagonal:\n\\[\ng = \\begin{bmatrix}A & 0 & 0 \\\\ 0 & B & 0 \\\\ 0 & 0 & A\\end{bmatrix}.\n\\]\nIn particular, $g$ will send $\\operatorname{Ker}(\\alpha_{x_1})$ (resp. $\\operatorname{Ker}(\\gamma_{x_1})$) to $\\operatorname{Ker}(\\alpha_{x_2})$ (resp. $\\operatorname{Ker}(\\gamma_{x_2})$).\nBut the kernel of $\\alpha_{x_1}$ corresponds to the symmetric functions of degree $b$ that vanish on a subvariety of $\\mathbb{P}^{n_1}$ isomorphic to $\\mathcal{Y}_{x_1}$, which we will still denote by $\\mathcal{Y}_{x_1}$.\nFrom \\cite{ACH11}*{Corollary 4.5}, up to choosing $b_0$ big enough, this kernel generates a graded ideal that corresponds to a subvariety isomorphic to $\\mathcal{Y}_{x_1}$.\nThe same conclusion holds for $\\operatorname{Ker}(\\alpha_{x_2})$, since there is a linear transformation (given by a block of the matrix $A$) that induces an isomorphism $\\operatorname{Ker}(\\alpha_{x_1})\\to \\operatorname{Ker}(\\alpha_{x_2})$.\nSo, in particular, this linear transformation induces a map of projective spaces, that gives an isomorphism $h \\colon \\mathcal{Y}_{x_1}\\to \\mathcal{Y}_{x_2}$.\n\nBy Corollary \\ref{corollary_the_section_that_vanishes_on_the_scheme_theoretic_image_of_m0\/rD_can_be_lifted_on_X}, $\\operatorname{Ker}(\\gamma_{x_1})$ contains a function that does not vanish along the generic points of the irreducible components of $\\mathcal{Y}_{x_1}$ but, if pulled back via $\\pi_{x_1} \\colon \\mathcal{X}_{x_1}\\to \\mathcal{Y}_{x_1}$, it vanishes along $\\frac{m_0a}{r}\\mathcal{D}_{x_1}$.\nTherefore, the zero locus of the polynomials in $\\operatorname{Ker}(\\gamma_{x_1})$ has codimension 1 in $\\mathcal{Y}_{x_1}$ and the union of its irreducible components of codimension 1, which we denote by $\\Gamma_{x_1}$, contains $\\operatorname{Supp}(\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},x_1})$.\nSince $\\Gamma_{x_1}$ has finitely many irreducible components of codimension one and the coefficients of $\\mathcal{D}_{\\mathcal{Y},x_1}$ are in the finite set $rI$, the divisor $\\mathcal{D}_{\\mathcal{Y},x_1}$ is determined up to finitely many possible choices of prime divisors and coefficients. \n\nA similar conclusion holds by replacing $x_1$ with $x_2$. Since in the description of $g$ the block at position (1,1) is the same as the block at position (3,3), we also know that $h(\\Gamma_{x_1}) = \\Gamma_{x_2}$.\n\nTherefore the fiber of $\\Psi(x_1)$ corresponds to stable pairs $(Y,D)$ in our moduli problem where $Y\\cong \\mathcal{Y}_{x_1}$, $\\Supp(D)\\subseteq \\Supp(\\Gamma_{x_1})$.\nBut there are finitely many such subvarieties, and since $B\\to \\mathscr{F}_{n,p,I}$ is finite by our assumptions, the fiber $\\Psi(x_1)$ has to be finite.\n\\end{proof}\n\n\\begin{Cor} \\label{cor proj moduli}\nConsider a proper DM stack $ \\mathcal{K}_{n,v,I}$ which satisfies the following two conditions:\n\\begin{enumerate}\n    \\item for every normal scheme $S$, the data of a morphism $f \\colon S\\to \\mathcal{K}_{n,v,I}$ is equivalent to a stable family of pairs $q \\colon (\\mathcal{Y},\\mathcal{D})\\to B$ with fibers of dimension $n$, volume $v$ and coefficients in $I$; and\n    \\item there is $m_0\\in \\mathbb{N}$ such that, for every $k \\in \\mathbb{N}$, there is a line bundle $\\mathcal{L}_k$ on $\\mathcal{K}_{n,v,I}$ such that, for every morphism $f$ as above, $f^*\\mathcal{L}_k \\cong \\det(q_*(\\omega_{\\mathcal{Y}\/B}^{[km_0]}(km_0\\mathcal{D})))$.\n\\end{enumerate}\nThen, for $m_0$ divisible enough, $\\mathcal{L}_k$ descends to an ample line bundle on the coarse moduli space of $\\mathcal{K}_{n,v,I}$ for every $k\\ge 1$. In particular, the coarse moduli space of $\\mathcal{K}_{n,v,I}$ is projective. \n\\end{Cor}\n\n\\begin{Remark}\nAs a concrete example of Corollary \\ref{cor proj moduli}, one can consider $\\mathcal{K}_{n,v,I}$ to be any moduli space of KSB-stable pairs $\\mathscr{K}$, such that, if we denote by $\\pi \\colon (\\mathscr{Y},\\mathscr{D})\\to \\mathscr{K}$ its universal family, then $\\omega_{\\mathscr{Y}\/\\mathscr{K}}^{[m_0]}(m_0\\mathscr{D})$ is $\\pi$-ample for $m_0$ divisible enough.\nIndeed, in this case, by cohomology and base change $\\pi_*\\omega_{\\mathscr{Y}\/\\mathscr{K}}^{[m_0]}(m_0\\mathscr{D})$ is a vector bundle for $m_0$ divisible enough, its formation commutes with base change from cohomology and base change, and the formation of the determinant commutes with base change as well.\n\\end{Remark}\n\n\\begin{proof}\nThe argument is divided into two steps. We denote by $K_{n,v,I}$ the coarse moduli space of $\\mathcal{K}_{n,v,I}$, and let $X$ be an irreducible component of it.\n\n\\begin{bf}Step 1.\\end{bf} There is a $p$-stable morphism $(\\mathcal{X};\\mathcal{D})\\to B$ which satisfies the following conditions:\n\\begin{enumerate}\n    \\item $B$ is normal and projective,\n    \\item The map $B\\to \\mathscr{F}_{n,p,I}$ is finite,\n    \\item There is a dense open subset $U\\subseteq B$ and a stable family $(\\mathcal{Y}_U,\\mathcal{D}_{\\mathcal{Y}, U})$ which satisfies the assumptions of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}, and\n    \\item The family $q:(\\mathcal{Y}_U,\\mathcal{D}_{\\mathcal{Y}, U})\\to U$ induces a map $U\\to \\mathcal{K}_{n,v,I} \\to K_{n,v,I}$ which dominates $X$.\n\\end{enumerate}\n\nConsider the generic point of $X$, and consider the stable pair $(\\mathcal{Z}_\\eta, \\mathcal{D}_{\\mathcal{Z},\\eta})$ it corresponds to. By Lemma \\ref{lift stable pair slc}, there is the spectrum of a field $\\operatorname{Spec}(\\mathbb{F})$ and a $p$-stable family over it $f:(\\mathcal{X}_\\eta; \\mathcal{D}_\\eta) \\to \\operatorname{Spec}(\\mathbb{F})$ such that the\nample model of $(\\mathcal{X}_\\eta; \\mathcal{D}_\\eta)$ is $(\\mathcal{Z}_\\eta, \\mathcal{D}_{\\mathcal{Z},\\eta})$. The family $f$ induces a morphism $\\operatorname{Spec}(\\mathbb{F}) \\to \\mathscr{F}_{n,p,I}$, and we can take $\\mathcal{F}$ to be the closure of its image. This is still a proper DM stack, it admits a finite and surjective cover by a scheme $B'\\to \\mathcal{F}$ with $B'$ a normal and \\emph{projective} scheme using \\cite{LMB18}*{Theorem 16.6}, Chow's lemma and potentially normalizing. Up to replacing $\\mathbb{F}$ with a finite cover of it, we can lift $\\operatorname{Spec}(\\mathbb{F})\\to \\mathcal{F}$ to $\\operatorname{Spec}(\\mathbb{F})\\to B'$, and consider $B$ an irreducible component of $B$ containing the image of $\\operatorname{Spec}(\\mathbb{F})\\to B'$.\n\nThe morphism $B\\to \\mathscr{F}_{n,p,I}$ induces a $p$-stable family $(\\mathcal{X};\\mathcal{D})\\to B$, and by construction its generic fiber admits an ample model.\nThen such an ample model can be spread out: there is an open subset $U\\subseteq B$ and a family $(\\mathcal{Y}_U,\\mathcal{D}_{\\mathcal{Y},U})$ as in Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}, and the generic fiber $(\\mathcal{Y}_U,\\mathcal{D}_{\\mathcal{Y},U})\\to U$ is isomorphic to $(\\mathcal{Z}_\\eta, \\mathcal{D}_{\\mathcal{Z},\\eta})$. Therefore the image of the corresponding map $U\\to \\mathcal{K}_{n,v,I}\\to K_{n,v,I}$ contains the generic point of $X$, and since both $U$ and $X$ are irreducible, $U\\to X$ is dominant.\n\n\\begin{bf}Step 2.\\end{bf} The map $U\\to \\mathcal{K}_{n,v,I}$ extends to a finite map $\\Phi:B\\to \\mathcal{K}_{n,v,I}$.\n\nThe extension follows from Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}. To prove that $\\Phi$ is finite, we can use Lemma \\ref{lemma:finitedness:paper:zsolt}. Indeed, if it was not finite, there would be a curve contained in a fiber of $\\Phi$. But then, up to replacing $C$ with an open subset of it, there would be a stable pair $(Y,D_Y)$ and a $p$-stable family $g:(\\mathcal{X};\\mathcal{D})\\to C$ as in Lemma \\ref{lemma:finitedness:paper:zsolt}. Therefore there would be finitely many isomorphism types of $p$-stable pairs in the fibers of $g$: this contradicts point (2) above.\n\n\\begin{bf}End of the proof.\\end{bf} From Theorem \\ref{thm:main:step:proj}, up to replacing $m_0$ with a multiple (which depends only on $X$), the line bundle $\\Phi^*\\mathcal{L}_k$ is ample. But a multiple of $\\mathcal{L}_k$ descends to a line bundle on $K_{n,v,I}$. In other terms, there is a line bundle $G$ on $K_{n,v,I}$ whose pull-back is $\\mathcal{L}_k^{\\otimes c}$ for a certain $c>0$. Therefore $G_{|X}$ is ample since it is ample once pulled-back via the \\emph{finite} map $B\\to X$. But if a line bundle is ample when restricted to each irreducible component, it is ample.\n\\end{proof}\n\\begin{Remark}\nWe remark that for the proof of Corollary \\ref{cor proj moduli} we do not need assumption (A).\nIndeed, with the notation of the proof of Corollary \\ref{cor proj moduli}, we just need assumption (A) to hold for the pairs parametrized by $\\mathscr{F}$.\nBut for the generic points of $\\mathscr{F}$, assumption (A) holds by construction, whereas for the other points of $\\mathscr{F}$ it holds from \\cite{HX13}*{Section 7}.\n\\end{Remark}\n\n\\begin{Cor}\\label{cor_projectivity_our_moduli} The stack $\\mathscr{F}_{n,p,I}$ admits a projective coarse moduli space.\n\\end{Cor}\n\\begin{proof}\nWe will denote with $F_{n,p,I}$ the coarse moduli space of $\\mathscr{F}_{n,p,I}$, which exists from Keel-Mori's theorem, and with $F_{n,p,I}^n$ the one of the normalization $\\mathscr{F}_{n,p,I}^n$ of $\\mathscr{F}_{n,p,I}$.\n\nConsider $\\epsilon_0$ such that $K_X+(1-\\epsilon_0)D$ is ample for every $p$-pair $(X;D)$ parametrized by $\\mathscr{F}_{n,p,I}$. Recall that such an\n$\\epsilon_0$ exists from Lemma \\ref{lemma_bound_for_epsilon} (or from boundedness). In particular, if we denote with $(\\mathscr{X};\\mathscr{D})\\to \\mathscr{F}_{n,p,I}$ the universal family, for $m$ divisibie enough the formation of $\\mathscr{G}:= \\det(p_*(\\mathcal{O}_{\\mathscr{X}}(m(K_{\\mathscr{X}\/\\mathscr{F}_{n,p,I}} + \\frac{1-\\epsilon}{r}\\mathscr{D}))))$ commutes with base change and an high enough power of $\\mathscr{G}$ descends to a line bundle $G$ on $F_{n,p,I}$. Therefore to prove that $G$ is ample, it suffices to replace $F_{n,p,I}$ (resp. $G$) with $F_{n,p,I}^n$ (resp. the pull-back $G^n$ of $G$ via $F_{n,p,I}^n \\to F_{n,p,I}$).\n\nConsider a proper DM stack $\\mathcal{K}_{n,p(1-\\epsilon_0),I}$ as in Corollary \\ref{cor proj moduli}. When $D$ has coefficient $1-\\epsilon_0$ the pairs parametrized by $\\mathscr{F}_{n,p,I}^n$ are stable of volume $p(1-\\epsilon_0)$, so we have a map $\\mathscr{F}_{n,p,I}^n\\to \\mathcal{K}_{n,p(1-\\epsilon_0),I}$ which is finite, as different points of $\\mathscr{F}_{n,p,I}$ parametrize different $p$-pairs, and the normalization is a finite morphism. From Corollary \\ref{cor proj moduli}, the formation of $\\mathcal{L}_k$ commutes with base change, so $\\mathcal{L}_k$ pulls back to $G^n$. Then $G^n$ is ample as it is the pull-back of an ample line bundle via a finite morphism.\\end{proof}\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Sec:Intro}\n\nLet $S_n$ be the set of permutations of $[n]=\\{1,\\ldots,n\\}$, which we write as words in one-line notation. Given $\\tau=\\tau_1\\cdots\\tau_m\\in S_m$, we say a permutation $\\sigma=\\sigma_1\\cdots\\sigma_n$ \\emph{contains the pattern} $\\tau$ if there exist indices $i_1<\\cdots<i_m$ in $[n]$ such that for all $j,k\\in[m]$, we have $\\sigma_{i_j}<\\sigma_{i_k}$ if and only if $\\tau_j<\\tau_k$. We say $\\sigma$ \\emph{avoids} $\\tau$ if it does not contain $\\tau$. Let $\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)$ be the set of permutations avoiding the patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$ (this list of patterns could be finite or infinite). \n\nThe study of pattern avoidance in permutations, which began with Knuth's analysis of permutations that are sortable via a stack \\cite{Knuth}, has grown into a large, thriving area of research in combinatorics \\cite{Bona,Kitaev,Linton}. Recently, there has been a great amount of interest in pattern avoidance in other combinatorial objects such as words, inversion sequences, set partitions, and trees \\cite{Albert, Anderson, Baxter, Bevan, Branden, Burstein2, Corteel, Dairyko, Daly, DefantEnumeration, DefantFertilityWilf, DefantPoset, DefantClass, DefantKravitz, DefantSupertrees, Godbole, Levin, Mansour1, Mansour2, Martinez, Rowland, Yakoubov}. The papers \\cite{Anderson2, Bevan, DefantPoset, Levin, Yakoubov} consider the following general type of problem. Let $\\mathcal P$ be an $n$-element poset, and suppose we are given some canonical total ordering of the elements of $\\mathcal P$. We can view each linear extension of $\\mathcal P$ as a bijective labeling of the elements of $\\mathcal P$ with the elements of $[n]$. If we read these labels in the canonical order, we obtain a permutation. We can then ask how many linear extensions give rise to permutations that avoid certain patterns. An alternative, yet essentially equivalent, formulation of this problem is to consider a canonical \\emph{labeling} of $\\mathcal P$ and then view each linear extension of $\\mathcal P$ as an ordering. Again, each linear extension gives rise to a permutation, so we can ask the same enumerative questions about pattern avoidance. \n\nAs a first example, we consider the problems introduced in \\cite{Levin}. A \\emph{complete $k$-ary tree} is a rooted tree in which each vertex that is not in the penultimate or the last level has exactly $k$ children and all vertices in the last level are as far left as possible. We make the convention that trees grow up from their roots (as in real life). A \\emph{$k$-ary heap} is a complete $k$-ary tree whose vertices are bijectively labeled with the elements of $[n]$ so that each vertex is given a label that is smaller than the labels of its children. Associating complete $k$-ary trees with posets in the natural way, one can view a $k$-ary heap as a linear extension of the underlying complete $k$-ary tree. We endow each complete $k$-ary tree with the breadth-first ordering (traversing the levels from bottom to top with each level traversed from left to right). Reading the labels in a $k$-ary heap in this order yields a permutation in $S_n$, which we call the permutation \\emph{associated to} the $k$-ary heap. For example, the left image in Figure \\ref{Fig1} shows the unique complete binary tree with $12$ vertices. The right image shows a binary heap obtained by labeling this tree. The permutation associated to this binary heap is $1\\,3\\,2\\,4\\,6\\,9\\,7\\,12\\,5\\,11\\,8\\,10$. In Section \\ref{Sec:Heaps}, we prove a conjecture from \\cite{Levin} concerning binary heaps whose associated permutations avoid the pattern $321$. In fact, we will prove a much more general result. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=.8\\linewidth]{LinearExtensionsPIC1}\n\\end{center}  \n\\caption{The complete binary tree with $12$ vertices (left) and a $12$-vertex binary heap (right). Reading the labels of this heap in the breadth-first order yields the associated permutation $1\\,3\\,2\\,4\\,6\\,9\\,7\\,12\\,5\\,11\\,8\\,10$.}\n\\end{figure}\\label{Fig1}\n\nThe second type of problem we consider deals with the rectangular posets $\\EN_{s,t}$ studied in \\cite{Anderson2}. The poset $\\EN_{s,t}$ has $st$ elements that are labeled in a canonical fashion. The easiest way to define these posets is via examples. The Hasse diagrams of $\\EN_{3,2}$, $\\EN_{3,5}$, and $\\EN_{4,3}$, along with their canonical labelings, are shown in Figure \\ref{Fig2}. Each linear extension of $\\EN_{s,t}$ can be viewed as an ordering of the labels, which is a permutation in $S_{st}$. For example, the linear extensions of $\\EN_{3,2}$ correspond to the permutations $531642,536142,536412,563142$, and $563412$. In Section \\ref{Sec:Rectangular}, we prove several of the conjectures that the authors of \\cite{Anderson2} posed about pattern-avoiding linear extensions of these posets. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=.65\\linewidth]{LinearExtensionsPIC2}\n\\end{center}  \n\\caption{The Hasse diagrams of three rectangular posets. The numbers show the canonical labelings.}\n\\end{figure}\\label{Fig2}\n\n\\section{Pattern-Avoiding $k$-ary Heaps}\\label{Sec:Heaps}\n\nLet $\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)$ denote the set of $n$-vertex $k$-ary heaps whose associated permutations avoid \\linebreak the patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$. The authors of \\cite{Levin} found formulas for $|\\mathcal H_n^k(\\tau^{(1)},\\ldots,\\tau^{(r)})|$ for every set $\\{\\tau^{(1)},\\ldots,\\tau^{(r)}\\}\\subseteq S_3$ except the singleton set $\\{321\\}$. In fact, they were unable to explicitly enumerate binary heaps avoiding $321$. They did, however, compute $|\\mathcal H_n^2(321)|$ for $1\\leq n\\leq 31$ and prove that \\[2^{n-1}<|\\mathcal H_n^2(321)|<4^n.\\] Their data led them to conjecture that\\footnote{The actual conjecture was written as ``$|\\mathcal H_n^2(321)|\\sim c^n$ for some $c\\in(3.66,4)$,\" but it is clear from context that the authors really meant what is written in \\eqref{Eq1}.} \n\\begin{equation}\\label{Eq1}\n\\lim_{n\\to\\infty}|\\mathcal H_n^2(321)|^{1\/n}\\text{ exists and is in the interval }(3.66,4].\n\\end{equation} We will see that this conjecture follows as a special consequence of the main theorem of this section. To state this theorem, we need one additional piece of terminology. \n\nGiven $\\lambda=\\lambda_1\\cdots\\lambda_\\ell\\in S_\\ell$ and $\\mu=\\mu_1\\cdots\\mu_m\\in S_m$, the \\emph{direct sum} of $\\lambda$ and $\\mu$, denoted $\\lambda\\oplus\\mu$, is the permutation in $S_{\\ell+m}$ obtained by ``placing $\\mu$ above and to the right of $\\lambda$.\" More formally, the $i^\\text{th}$ entry of $\\lambda\\oplus\\mu$ is \\[(\\lambda\\oplus\\mu)_i=\\begin{cases} \\lambda_i & \\mbox{if } 1\\leq i\\leq \\ell; \\\\ \\mu_{i-\\ell}+\\ell & \\mbox{if } \\ell+1\\leq i\\leq \\ell+m. \\end{cases}\\] A permutation is called \\emph{sum indecomposable} if it cannot be written as the direct sum of two shorter permutations. \n\n\\begin{theorem}\\label{Thm1}\nFix an integer $k\\geq 2$ and a nonempty sequence of permutation patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$. If the permutations $\\tau^{(1)},\\tau^{(2)},\\ldots$ are all sum indecomposable, then \\[\\lim_{n\\to\\infty}|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}=\\lim_{n\\to\\infty}|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}.\\]\n\\end{theorem}\n\n\\begin{remark}\\label{Rem1}\nHow do we know that the limits in Theorem \\ref{Thm1} actually exist? Because the permutations $\\tau^{(1)},\\tau^{(2)},\\ldots$ are sum indecomposable, we have an injective map $S_\\ell\\times S_m\\hookrightarrow S_{\\ell+m}$ given by $(\\lambda,\\mu)\\mapsto\\lambda\\oplus\\mu$ for all $\\ell,m\\geq 1$. This shows that the sequence $(|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|)_{n\\geq 1}$ is supermultiplicative, so the limit on the left-hand side exists by Fekete's lemma. We also have an injective map $\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)\\hookrightarrow\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)$ sending each $k$-ary heap to its associated permutation. This shows that $\\limsup\\limits_{n\\to\\infty}|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}\\leq\\lim\\limits_{n\\to\\infty}|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}$. We will see in the proof of Theorem \\ref{Thm1} that $\\lim\\limits_{n\\to\\infty}|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}\\leq\\liminf\\limits_{n\\to\\infty}|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}$, so the limit on the right-hand side exists as well. It is interesting to note that this right-hand limit does not depend on $k$. \\hfill$\\lozenge$ \n\\end{remark}\n\n\nLet $C_n=\\frac{1}{n+1}{2n\\choose n}$ denote the $n^\\text{th}$ Catalan number. It is well known that $|\\operatorname{Av}_n(321)|=C_n$. This implies that $\\lim\\limits_{n\\to\\infty}|\\operatorname{Av}_n(321)|^{1\/n}=4$. Consequently, we can specialize Theorem \\ref{Thm1} to obtain the following result. In the special case in which $k=2$, this settles the conjecture from \\cite{Levin} that is stated in \\eqref{Eq1}. \n\n\\begin{corollary}\\label{Cor2}\nIf $k\\geq 2$, then $\\displaystyle\\lim_{n\\to\\infty}|\\mathcal H_n^k(321)|^{1\/n}=4$.\n\\end{corollary}\n\n\\begin{proof}[of Theorem \\ref{Thm1}]\nFix an integer $k\\geq 2$ and a sequence of sum indecomposable permutation patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$. As mentioned in Remark \\ref{Rem1}, we must prove that \\[\\lim_{n\\to\\infty}|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}\\leq\\liminf_{n\\to\\infty}|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}.\\] To ease \nnotation, let \\[a(n)=|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n},\\quad b(n)=|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n},\\] \\[\\quad L_a=\\lim_{n\\to\\infty}a(n),\\quad L_b=\\liminf_{n\\to\\infty}b(n).\\] The proof is trivial if \none of the patterns $\\tau^{(i)}$ is the permutation $1\\in S_1$, so we may assume otherwise. Because the patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$ are sum indecomposable, the identity permutation $123\\cdots n$ is in $\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)$ and is the permutation associated to an element of $\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)$. Thus, $L_a,L_b\\geq 1$. Fix $\\varepsilon\\in(0,1)$, and let $N\\geq 1$ be such that $a(n-\\left\\lfloor(n-1)\/k\\right\\rfloor)>L_a-\\varepsilon$ and $b(\\left\\lfloor(n-1)\/k\\right\\rfloor)\\!>L_b-\\varepsilon$ for every $n\\geq N$. Now choose $n\\geq N$ such that $b(n)<L_b+\\varepsilon$, and put $m=\\left\\lfloor(n-1)\/k\\right\\rfloor$. \n\n\\noindent {\\bf Claim:} If $\\lambda\\in S_m$ is a permutation associated to a $k$-ary heap in $\\mathcal H_m^k(\\tau^{(1)},\\tau^{(2)},\\ldots)$ and \\linebreak $\\mu\\in\\operatorname{Av}_{n-m}(\\tau^{(1)},\\tau^{(2)},\\ldots)$, then $\\lambda\\oplus\\mu$ is a permutation associated to a $k$-ary heap in $\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)$. \n\nLet us see how this claim implies Theorem \\ref{Thm1}. Suppose by way of contradiction that $L_b<L_a$. Assuming the claim, we have \\[(L_b+\\varepsilon)^n>b(n)^n=|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|\\geq|\\mathcal H_m^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|\\cdot|\\operatorname{Av}_{n-m}(\\tau^{(1)},\\tau^{(2)},\\ldots)|\\] \\[=b(m)^ma(n-m)^{n-m}>(L_b-\\varepsilon)^m(L_a-\\varepsilon)^{n-m}.\\] Since $m<n\/2$ and $L_b<L_a$, we have $(L_b-\\varepsilon)^m(L_a-\\varepsilon)^{n-m}>(L_b-\\varepsilon)^{n\/2}(L_a-\\varepsilon)^{n\/2}$. Hence, $(L_b+\\varepsilon)^2>(L_b-\\varepsilon)(L_a-\\varepsilon)$. Letting $\\varepsilon$ tend to $0$ shows that $L_b\\geq L_a$, contradicting our assumption that $L_b<L_a$. This completes the proof of Theorem \\ref{Thm1} assuming the claim. \n\nNow, let $\\lambda$ and $\\mu$ be as in the claim. Let $T$ be the complete $k$-ary tree with $n$ vertices, and let $v_i$ be the vertex of $T$ that appears $i^\\text{th}$ in the\nbreadth-first ordering. It is straightforward to check that the vertices $v_{m+1},\\ldots,v_n$ are incomparable (meaning none is a descendant of another). It follows that $\\lambda\\oplus\\mu$ is the permutation associated to some $k$-ary heap in $\\mathcal H_n^k$. Because $\\lambda$ and $\\mu$ avoid the sum indecomposable permutations $\\tau^{(1)},\\tau^{(2)},\\ldots$, their direct sum $\\lambda\\oplus\\mu$ must also avoid these permutations. This completes the proof of the claim. \n\\end{proof}\n\n\\section{Rectangular Posets}\\label{Sec:Rectangular}\n\nAn \\emph{inversion} of a permutation $\\pi=\\pi_1\\cdots\\pi_n\\in S_n$ is a pair $(i,j)$ such that $i<j$ and $\\pi_i>\\pi_j$. Let $\\inv(\\pi)$ denote the number of inversions of $\\pi$. Following \\cite{Anderson2}, we let $\\EN_{s,t}(\\tau)(q)=\\sum_{\\pi\\in\\EN_{s,t}(\\tau)}q^{\\inv(\\pi)}$, where $\\EN_{s,t}(\\tau)$ is the set of linear extensions of $\\EN_{s,t}$ (viewed as permutations of the labels of $\\EN_{s,t}$) that avoid the pattern $\\tau$. The following three theorems were stated as conjectures in \\cite{Anderson2}\\footnote{Technically speaking, Theorem \\ref{Thm2} was stated incorrectly in that article.}.\n  \n\\begin{theorem}\\label{Thm2}\nFor all $t\\geq 1$, we have \\[\\EN_{3,t}(1243)(q)=\\frac{q^{3(t^2-t+1)}(1-q^{2t-1}-2q^{2t}+q^{3t-1}+q^{3t})}{(1-q)(1-q^2)}.\\]\n\\end{theorem}  \n\n\\begin{theorem}\\label{Thm3}\nFor all $s\\geq 1$, we have \\[\\EN_{s,2}(2143)(q)=q^{(2s-1)(s-1)}(1+q)^{s-1}.\\]\n\\end{theorem}\n  \n\\begin{theorem}\\label{Thm4}\nFor all $s\\geq 1$, we have \\[\\EN_{s,3}(2143)(q)=q^{9{s\\choose 2}}F_s(1\/q),\\] where $F_s(r)$ is defined by $F_0(r)=F_1(r)=1$ and $F_s(r)=(1+r+2r^2)F_{s-1}(r)+r^3F_{s-2}(r)$ for $s\\geq 2$. \n\\end{theorem}\n\n\\begin{proof}[of Theorem \\ref{Thm2}]\nIt is easy to verify this theorem when $t\\in\\{1,2\\}$, so assume $t\\geq 3$. Let $\\pi=\\pi_1\\cdots\\pi_n$ be a linear extension of $\\EN_{3,t}(1243)$ (viewed as a permutation of the labels). Note that $2t$ appears before $2$ in $\\pi$ because, otherwise, the entries $1,2,2t,t$ would form a $1243$ pattern. Similarly, $3t$ must appear before $t+2$, lest the entries $t+1,t+2,3t,2t$ form a $1243$ pattern. It follows that if we remove the entries $1$ and $t+1$ from $\\pi$, then we will be left with the permutation \\[(2t+1)(2t+2)\\cdots(3t)(t+2)(t+3)\\cdots(2t)23\\cdots t.\\] Let $i,j$ be such that $\\pi_{i+1}=t+1$ and $\\pi_{j+1}=1$. We can easily check that the possibilities for $i$ and $j$ are given by $i\\in\\{1,\\ldots,t\\}$ and $j\\in\\{i+1,\\ldots,2t\\}$. We find that \\[\\EN_{3,t}(1243)(q)=\\sum_{i=1}^t\\sum_{j=i+1}^{2t}q^{3t^2-3t+i+j}.\\] This can easily be rewritten as $\\dfrac{q^{3(t^2-t+1)}(1-q^{2t-1}-2q^{2t}+q^{3t-1}+q^{3t})}{(1-q)(1-q^2)}$.  \n\\end{proof}\n\n\\begin{proof}[of Theorem \\ref{Thm3}]\nLet $S$ be the collection of subsets of $\\{1,3,5,\\ldots,2s-3\\}$, and define $\\eta\\colon\\EN_{s,2}(2143)\\to S$ by \\[\\eta(\\pi)=\\{i\\in\\{1,3,5,\\ldots,2s-3\\}\\colon i+3\\text{ appears before }i\\text{ in }\\pi\\}.\\] The map $\\eta$ is a bijection. It is now easy to check that \\[\\EN_{s,2}(2143)(q)=\\sum_{X\\subseteq\\{1,3,5,\\ldots,2s-3\\}}q^{(2s-1)(s-1)+|X|}=q^{(2s-1)(s-1)}(1+q)^{s-1}. \\qedhere\\] \n\\end{proof}\n\n\\begin{proof}[of Theorem \\ref{Thm4}]\nLet $H_\\ell$ be the set of $\\pi=\\pi_1\\cdots\\pi_{3\\ell}\\in\\EN_{\\ell,3}(2143)$ such that $\\pi_2=3\\ell-1$. Let $H_\\ell(q)=\\sum_{\\pi\\in H_\\ell}q^{\\inv(\\pi)}$. Fix $s\\geq 2$, and let $J(r_1,\\ldots,r_k)$ be the set of permutations $\\pi=\\pi_1\\cdots\\pi_{3s}\\in\\EN_{s,3}(2143)$ such that $\\pi_i=r_i$ for all $i\\in\\{1,\\ldots,k\\}$. One can check that the sets \\[J(3s-2,3s-1,3s),\\hspace{.5cm}J(3s-2,3s-1,3s-5,3s),\\hspace{.5cm}J(3s-2,3s-5,3s-1,3s),\\] \\[J(3s-2,3s-5,3s-1,3s-4,3s),\\hspace{.5cm}\\text{and}\\hspace{.5cm}J(3s-2,3s-1,3s-5,3s-4,3s)\\] partition $\\EN_{s,3}(2143)$. Call these sets $J_1,J_2,J_3,J_4$, and $J_5$, respectively. \n\nThe operation that consists of removing the entries $3s-2,3s-1$, and $3s$ from a permutation establishes bijections $J_1\\to\\EN_{s-1,3}(2143)$, $J_2\\to\\EN_{s-1,3}(2143)$, $J_3\\to\\EN_{s-1,3}(2143)$, $J_4\\to H_{s-1}$, and $J_5\\to H_{s-1}$. After taking into account the number of inversions that are removed by each of these bijections, we obtain the identities \\[\\sum_{\\pi\\in J_1}q^{\\text{inv}(\\pi)}=q^{9(s-1)}\\EN_{s-1,3}(2143)(q),\\quad\\sum_{\\pi\\in J_2}q^{\\text{inv}(\\pi)}=q^{9(s-1)-1}\\EN_{s-1,3}(2143)(q),\\] \n\\[\\sum_{\\pi\\in J_3}q^{\\text{inv}(\\pi)}=q^{9(s-1)-2}\\EN_{s-1,3}(2143)(q),\\quad\\sum_{\\pi\\in J_4}q^{\\text{inv}(\\pi)}=q^{9(s-1)-3}H_{s-1}(q),\\] and \n\\[\\sum_{\\pi\\in J_5}q^{\\text{inv}(\\pi)}=q^{9(s-1)-2}H_{s-1}(q).\\] This yields \n\\begin{equation}\\label{Eq2}\n\\EN_{s,3}(2143)(q)=q^{9(s-1)-2}(1+q+q^2)\\EN_{s-1,3}(2143)(q)+q^{9(s-1)-3}(1+q)H_{s-1}(q).\n\\end{equation}\nThe sets $J_1,J_2,J_5$ partition $H_s$, so \n\\begin{equation}\\label{Eq3}\nH_s(q)=q^{9(s-1)-1}(1+q)\\EN_{s-1,3}(2143)(q)+q^{9(s-1)-2}H_{s-1}(q).\n\\end{equation}\nSolving the recurrence relations \\eqref{Eq2} and \\eqref{Eq3} subject to the initial conditions $\\EN_{1,3}(2143)(q)=H_1(q)=1$, we obtain the desired identity $\\EN_{s,3}(2143)(q)=q^{9{s\\choose 2}}F_s(1\/q)$. \n\\end{proof}\n\nThe article \\cite{Anderson2} also poses several conjectures concerning the polynomials $F_s(q)$ and various OEIS sequences \\cite{OEIS}. We settle many of these conjectures\\footnote{We also correct some typos made in the original statements of these conjectures.} in the following theorem. Since our focus is on the combinatorics of pattern-avoiding linear extensions and not these specific polynomials, we omit some details from the proof. In what follows, let $[q^r]G(q)$ denote the coefficient of $q^r$ in the Laurent series $G(q)$.\n\n\\begin{theorem}\\label{Thm5}\nDefine the polynomials $F_s(r)$ by $F_0(r)=F_1(r)=1$ and $F_s(r)=(1+r+2r^2)F_{s-1}(r)+r^3F_{s-2}(r)$ for $s\\geq 2$. For $s\\geq 2$, \n\\begin{itemize}\n\\item the values of $[q^3]F_s(q)$ are given by OEIS sequence A134465;\n\\item the values of $[q^{2s-2}]F_s(q)$ are given by OEIS sequence A098156; \n\\item the values of $[q^{s-1}]F_s(q)$ are given by OEIS sequence A116914;  \n\\item the values of $[q^s]F_s(q)$ are given by OEIS sequence A072547;\n\\item the values of $[q^{s+1}]F_s(q)$ are given by OEIS sequence A002054;  \n\\item the values of $[q^{s+2}]F_s(q)$ are given by OEIS sequence A127531.\n\\end{itemize}\n\\end{theorem} \n\n\\begin{proof}\nLet $A(x,q)=\\sum_{s\\geq 0}F_s(q)x^s$. The recurrence for $F_s(q)$ translates into the identity \n\\begin{equation}\\label{Eq4}\nA(x,q)=\\frac{1-(q+2q^2)x}{1-(1+q+2q^2)x-q^3x^2}.\n\\end{equation}\nComputing $\\dfrac{1}{6}\\dfrac{\\partial^3}{\\partial q^3}A(x,q)$ proves the first bullet point. The remainder of the proof makes use of the method of diagonals, which is discussed in Section 6.3 of \\cite{Stanley}.\n\nIf we view $qA(x\/q^2,q)$ as a function of the complex variable $q$, then \\[\\sum_{s\\geq 0}([q^{2s-2}]F_s(q))x^s=[q^{-1}](qA(x\/q^2,q))=\\frac{1}{2\\pi i}\\int_{|q|=\\rho}qA(x\/q^2,q)\\,dq,\\] where $\\rho>0$ is sufficiently small and the integral is taken over the circle of radius $\\rho$ centered at the origin. By the Residue Theorem, this is \\[\\sum_{j=1}^r\\Res_{q=u_j(x)}(qA(x\/q^2,q)),\\] where $u_1(x),\\ldots,u_r(x)$ are the singularities of $qA(x\/q^2,q)$ (viewed as functions of $x$) that tend to $0$ as $x\\to 0$. We can explicitly compute that $r=2$ and that \\[u_1(x)=\\frac{x(1+x)+(1-x)\\sqrt{x(4+x)}}{2(1-2x)}\\quad\\text{and}\\quad u_2(x)=\\frac{x(1+x)-(1-x)\\sqrt{x(4+x)}}{2(1-2x)}.\\] Let $U(x,q)=q^2(x-q(1-2x))$ and $V(x,q)=x+qx(1+x)-q^2(1-2x)$ so that $qA(x\/q^2,q)=U(x,q)\/V(x,q)$. Let $V_q(x,q)=\\dfrac{\\partial}{\\partial q}V(x,q)$. Since $u_1(x)$ and $u_2(x)$ are simple poles of $qA(x\/q^2,q)$, we find that \\[\\Res_{q=u_j(x)}(qA(x\/q^2,q))=\\frac{U(x,u_j(x))}{V_q(x,u_j(x))}\\quad\\text{for }j\\in\\{1,2\\}.\\] We have \\[\\sum_{j=1}^2\\Res_{q=u_j(x)}(qA(x\/q^2,q))=\\sum_{j=1}^2\\frac{U(x,u_j(x))}{V_q(x,u_j(x))}=\\frac{x(1-2x+x^2+x^3)}{(1-2x)^2},\\] and this proves the second bullet point. \n\nTo prove the third, fourth, fifth, and sixth bullet points, we choose an integer $\\ell\\leq 2$ and view \\linebreak $q^{-\\ell-1}A(x\/q,q)$ as a complex function of the variable $q$. As above, we have \\[\\sum_{s\\geq 0}([q^{s+\\ell}]F_s(q))x^s=[q^{-1}](q^{-\\ell-1}A(x\/q,q))=\\sum_{j=1}^t\\Res_{q=v_j(x)}(q^{-\\ell-1}A(x\/q,q)),\\] where $v_1(x),\\ldots,v_t(x)$ are the singularities of $q^{-\\ell-1}A(x\/q,q)$ that tend to $0$ as $x\\to 0$. Let $Y(x,q)=-1+x+2qx$ and $Z(x,q)=x-(1-x)q+x(2+x)q^2$ so that $q^{-\\ell-1}A(x\/q,q)=q^{-\\ell}Y(x,q)\/Z(x,q)$. If $\\ell\\leq 0$, then the only singularity of $q^{-\\ell-1}A(x\/q,q)$ that tends to $0$ as $x\\to 0$ is \\[v_1(x)=\\dfrac{1-x-\\sqrt{1-2x-7x^2-4x^3}}{2x(2+x)}.\\] If $\\ell\\in\\{1,2\\}$, then there is one other singularity, which is $v_2(x)=0$. One can check that \\[\\Res_{q=0}(q^{-2}A(x\/q,q))=1-x^{-1}\\quad\\text{\\and}\\quad\\Res_{q=0}(q^{-3}A(x\/q,q))=1+2x^{-1}-x^{-2}.\\] Note that in each of these expressions, the coefficient of $x^s$ is $0$ for every $s\\geq 2$. It follows that for all integers $\\ell\\leq 2$ and $s\\geq 2$, the coefficient of $x^s$ in $\\sum_{s\\geq 0}([q^{s+\\ell}]F_s(q))x^s$ agrees with the coefficient of $x^s$ in $\\Res_{q=v_1(x)}(q^{-\\ell-1}A(x\/q,q))$. We have \\[\\Res_{q=v_1(x)}(q^{-\\ell-1}A(x\/q,q))=\\frac{v_1(x)^{-\\ell}Y(x,v_1(x))}{Z_q(x,v_1(x))},\\] where $Z_q(x,q)=\\dfrac{\\partial}{\\partial q}Z(x,q)$. When $\\ell\\in\\{-1,0,1,2\\}$, we can explicitly compute and simplify these expressions in order to obtain proofs of the last four bullet points. \n\\end{proof}\n\n\\section{Acknowledgments}\nThe author thanks the anonymous referees for helpful comments. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}