diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhapx" "b/data_all_eng_slimpj/shuffled/split2/finalzzhapx"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhapx"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nA recurrent theme in classical projective geometry is the study of\nspecial subvarieties of some given family of varieties, e.g. how\nmany isolated singular points can a surface of degree $d$ in\n$\\mathbb{P}^3$ have? when is it true that the members of a certain family\nof varieties contain rational curves? contain a linear space of\nsome positive dimension?  Other examples of similar questions can be\neasily provided by the reader.\n\nThe study of the special case of complete intersection\nsubvarieties of hypersurfaces in $\\mathbb{P}^n$ has been the subject of a great deal of research. It\nwas known to Severi \\cite{Severi} that for $n \\geq 4$ the only\ncomplete intersections, of codimension one, on a general hypersurface are obtained by\nintersecting that hypersurface with another.\n\nThis observation was extended to $\\mathbb{P}^3$ by Noether (and\nLefschetz)  \\cite{Lefschetz, GH85} for general hypersurfaces of\ndegree $\\geq 4$.  These ideas were further generalized by\nGrothendieck \\cite{SGA2}.\n\nIn \\cite{CaChGe}, we proposed a new approach to the problem of\nstudying complete intersection subvarieties of hypersurfaces. This\napproach used a mix of projective geometry and commutative algebra\nand is more elementary and direct than, for example, the\napproach of Grothendieck.  With our approach we were able to give a complete\ndescription of the situation for complete intersections of\ncodimension $r$ in $\\mathbb{P}^n$ which lie on a general hypersurface of\ndegree $d$ whenever $2r \\leq n+2$. The main result of\n\\cite{CaChGe} is the following:\n\n\\begin{thm}\\label{nostro}\nLet $X\\subset\\mathbb{P}^n$ be a generic degree $d$ hypersurface, with\n$n,d>1$. Then $X$ contains a complete intersection of type\n$(a_1,\\ldots,a_r)$, with $2r\\leq n+2$, and the $a_i$ all less than\n$d$, in the following (and only in the following) instances:\n\\medskip\n\n\\begin{itemize}\n\\item $n=2$: then $r=2$, $d$ arbitrary and $a_1$ and $a_2$ can\nassume any value less than $d$;\n\n\\item $n=3$, $r=2 $:  for $d \\leq 3$ we have that $a_1$ and $a_2$\ncan assume any value less than $d$;\n\n\\item $n=4$, $r = 3$: for $d \\leq 5$ we have that $a_1, a_2$ and\n$a_3$ can assume any value less than $d$;\n\n\\item $n=6,r = 4$ or $n=8,r = 5$: for $d \\leq 3$ we have that\n$a_1, \\ldots ,a_r$ can assume any value less than $d$;\n\n\\item $n=5,7$ or $n>8$, $2r=n+1$ or $2r = n+2$:  we have only\nlinear spaces on quadrics, i.e. $d = 2$ and $a_1 = \\ldots =a_r =\n1$.\n\n\n\\end{itemize}\n\n\\end{thm}\n\nIn this paper, we are interested in the first case not covered by\nTheorem \\ref{nostro}. Namely, the case  $n=3,r=3$, i.e.\ncomplete intersection points on surfaces of $\\mathbb{P}^3$. Although this\na very natural question, we are not aware of any reference to the\nsubject in the literature. Using the methods of \\cite{CaChGe} we prove the\nfollowing:\n\n\\begin{thm}\\label{mainTHM} For non-negative integers $a,b,c,d$, such that $a\\leq b\\leq c<\nd$ we have the following:\n\n\\begin{itemize}\n\n\\item if $a\\leq 4$, then the generic degree $d$ surface of $\\mathbb{P}^3$ contains a\n$CI(a,b,c)$;\n\n\\item if $a=5,b\\leq 11$, then the generic degree $d$\nsurface of $\\mathbb{P}^3$ contains a $CI(5,b,c)$; if $a=5,b=12$ and\n$c=12$ then the generic degree $d$ surface of $\\mathbb{P}^3$ contains a\n$CI(5,12,12)$; if $a=5,b=12$ and $c\\geq 13$ then the generic\ndegree $d\\geq 2c+15$ surface does not contain a $CI(5,12,c)$; if\n$a=5$ and $b\\geq 13$, then the generic degree $d\\geq b+c+2$\nsurface does not contain a $CI(5,b,c)$;\n\n\\item if $a=6,b\\leq 7$, then the generic degree $d$\nsurface of $\\mathbb{P}^3$ contains a $CI(6,b,c)$; if $a=6,b=8$ and\n$c=8,9$ then the generic degree $d$ surface of $\\mathbb{P}^3$ contains a\n$CI(6,8,c)$; if $a=6,\\ b=8$ and $c\\geq 10$ then the generic degree\n$d\\geq 2c+12$ surface does not contain a $CI(6,8,c)$; if $a=6$ and\n$b\\geq 9$, then the generic degree $d\\geq b+c+3$ surface does not\ncontain a $CI(6,b,c)$;\n\n\\item if $a\\geq 7$, then the generic degree $d\\geq a+b+c-3$ surface of $\\mathbb{P}^3$\ndoes not contain a $CI(a,b,c)$.\n\\end{itemize}\n\\end{thm}\n\nNotice that Theorem \\ref{mainTHM} gives a complete asymptotic solution to the existence problem for $CI(a,b,c)$ on a general surface of degree $d$ in $\\mathbb{P}^3$.  More precisely,\n\n\\begin{cor} Let $a \\leq b \\leq c <d$ be integers.  Then for $d\\gg 0$ the generic degree $d$ surface contains a $CI(a,b,c)$ if:\n\\begin{itemize}\n  \\item $a \\leq 4$;\n  \\item $a = 5,\\ b\\leq 11$;\n  \\item $a = 5,\\ b=12,\\ c=12$;\n  \\item $a=6,\\ b\\leq 7$;\n  \\item $a=6,\\ b=8,\\ c=8,9$.\n \\end{itemize}\nand does not contain a $CI(a,b,c)$ in all other cases.\n\\end{cor}\n\n\\begin{rem}\nWe also notice that the kind of asymptotic problem we solved above can only be considered for points.  More precisely, if we choose a family $\\mathcal{F}$ of subschemes of $\\mathbb{P}^n$ we can ask the following: is it true that for $d\\gg 0$ the generic degree $d$ hypersurface of $\\mathbb{P}^n$ contains an object of the family $\\mathcal{F}$?\n\nUsing a standard incidence correspondence argument, it is easy to see that a positive answer can be given only if\n$$\n\\dim {\\mathcal{F}} + 1 - h_{\\mathcal{F}}(d) \\geq 0,\n$$\nwhere $h_{\\mathcal{F}}(d)$ is the Hilbert polynomial of the objects in $\\mathcal{F}$.  Clearly this can be the case only if $h_{\\mathcal{F}}(d)$ is bounded and hence constant.  This implies that $\\mathcal{F}$ is a family parameterizing $0$-dimensional schemes.\n\\end{rem}\n\nThe paper is structured as follows: in Section \\ref{questionSEC},\nwe formalize the question we want to study and we treat the first\nsimple instances; in Section \\ref{technicalFACTSsec}, we recall\nthe results we need from \\cite{CaChGe}; in Sections\n\\ref{aleq4SEC}, \\ref{ageq5nonexistSEC} and \\ref{ageq5existSEC} we\napply our method to produce the intermediate results necessary to\nprove Theorem \\ref{mainTHM}. Finally, in Section \\ref{finalSEC},\nwe prove Theorem \\ref{mainTHM} and we state a conjecture for the\nexpected behavior in the cases which still remain open.\n\nIn the proof of Theorem \\ref{ageq5THM} we used the\ncomputer algebra system CoCoA \\cite{cocoa} for which we thank the\ndevelopers of the software.\n\nThe first author wishes to thank Queen's University for its kind\nhospitality during the writing of this paper, the research group\nGNSAGA of INDAM and the special fund ``Fondo giovani ricercatori\"\nof the Politecnico di Torino for financial support.  The first and third author also wish to thank NSERC (Canada) for its financial support during the writing of this paper.\n\n\n\n\\section{The question}\\label{questionSEC}\nIn this paper we study complete intersection points in projective\nthree space. We say that $\\mathbb{X}\\subset\\mathbb{P}^3$ is a complete\nintersection 0-dimensional scheme if its ideal\n$I_\\mathbb{X}=(F,G,H)$ where the forms $F,G$ and $H$ are a regular\nsequence in the ring $R=\\mathbb{C}[x_0,\\ldots,x_3]$. Moreover, if\n$\\deg F=a,\\deg G=b$ and $\\deg H=c$ we say that $\\mathbb{X}$ is a\n{\\it complete intersection of type} $(a,b,c)$. We will always\nassume $a\\leq b\\leq  c$ and we will write $CI(a,b,c)$\nto describe a complete intersection of type $(a,b,c)$.\n\nOur basic question is: {\\em for which integers $a,b,c$ and $d$\ndoes the general degree $d$ surface of $\\mathbb{P}^3$ contain a $CI(a,b,c)$}?\n\nThere are cases where the answer is straightforward. If $d=c$, the\nanswer is clearly affirmative as we are cutting a complete\nintersection curve of type $(a,b)$ with a surface of degree $d$\n(similarly for $d=a$ or $d=b$). If $d<a$, the answer is negative\nas no form of degree less than $a$ belongs to the ideal of a\n$CI(a,b,c)$, and similarly for $a<d<b$ as a generic form is\nirreducible. If $b<d<c$, then we are really looking for a complete\nintersection of type $(a,b)$ on the generic degree $d$ surface,\nand this is dealt with in Theorem \\ref{nostro}. Hence, it is\nenough to focus on the following refinement of our question: {\\em\nfor which integers $a,b,c$ and $d$, $a\\leq b\\leq c<d$, does the\ngeneral degree $d$ surface of $\\mathbb{P}^3$ contain a $CI(a,b,c)$}?\n\n\n\n\n\n\\section{Technical facts}\\label{technicalFACTSsec}\n\nWe will treat this question using the method introduced in\n\\cite{CaChGe}. Our method proceeds as follows: translate the problem of finding a $CI(a,b,c)$ on a general surface of degree $d$, say $M = 0$, as the problem of writing $M$ as\n$$\nM = F F^\\prime + G G^\\prime + H H^\\prime\n$$\nwhere $F,\\ G,\\ H$ and $F^\\prime,\\ G^\\prime, \\ H^\\prime$ are forms of degree $a,\\ b,\\ c$ and $d-a,\\ d-b,\\ d-c$ respectively.  As $M$ is generic, this decomposition problem is actually a problem about joins of varieties of splitting forms.  Then we use Terracini's lemma to translate the computation of the dimension of the join, into a Hilbert function computation.  Namely, as first observed in \\cite{Mamma}, the tangent space to the variety of splitting forms at the point $[FF^\\prime]$ corresponds to the degree $d$ homogeneous piece of the ideal $ (F, F^\\prime )$.  Thus, the tangent space at $M$ to the join corresponds to the degree $d$ homogeneous piece of the ideal spanned by $F,F^\\prime, G, G^\\prime, H, H^\\prime$.  For more details we refer the reader to \\cite{CaChGe}.\n\nIn particular we will need the following (see \\cite[Lemma 4.3]{CaChGe}):\n\n\\begin{lem}\\label{equivalence}\nFor given integers $a,b,c$ and $d$, such that $a\\leq b\\leq c<\nd$, the following are equivalent facts:\n\\begin{enumerate}\n\\item  The general degree $d$ surface of $\\mathbb{P}^3$ contains a\n$CI(a,b,c)$; \\item\\label{algebraicQUESTION} For a generic choice\nof forms $F,G,H,H',G',F'\\in R$ of degrees $a,b,c,d-c,d-b,d-a$ one\nhas that\n    \\[H\\left({R\\over (F,G,H,H',G',F')},d\\right)=0\\]\nwhere $H(\\cdot,d)$ denotes the Hilbert function in degree $d$.\n\\end{enumerate}\n\\end{lem}\n\nUsing Lemma \\ref{equivalence} we translate our geometric question\ninto a purely algebraic one. In particular, we can take advantage of\nresults about the Lefschetz property \\cite{Stanley,Anick} to deal\nwith our question.\n\nAs $F,G,H$ and $H'$ are a\nregular sequence in $R$ we have a good understanding of the ring\n\\[W={R\\over (F,G,H,H')}\\]\nand we will use this to study the Hilbert function of the ring\n\\[{R\\over(F,G,H,H',G',F')}\\simeq {W\\over ([F'],[G'])},\\]\nwhere $[\\cdot]$ denotes the class in $W$.\n\nVia the Koszul complex we compute the minimal free resolution of $W$:\n\\begin{equation}\\label{Wres}\n0\\leftarrow W\\leftarrow M_0 \\leftarrow M_1\\leftarrow M_2\\leftarrow\nM_3\\leftarrow M_4\\leftarrow 0\n\\end{equation}\nwhere\n\\[M_0=R,\\]\n\\[M_1=R(-a)\\oplus R(-b)\\oplus R(-c)\\oplus R(-d+c),\\]\n\\[M_2=R(-a-b)\\oplus R(-a-c)\\oplus R(-a-d+c)\\oplus R(-b-c)\\oplus R(-b-d+c)\\oplus R(-d),\\]\n\\[M_3=R(-a-b-c)\\oplus R(-a-b-d+c)\\oplus R(-a-d)\\oplus R(-b-d)\\]\n\\[M_4=R(-a-b-d).\\]\n\nWe also notice that (see  \\cite[Lemma 4.1 and Remark 4.2]{CaChGe}):\n\n\\begin{lem}\\label{switchLEM}\nThe following are equivalent:\n\\begin{itemize}\n\\item for integers $a\\leq b\\leq c<d$ a $CI(a,b,c)$ exists on the generic degree $d$ surface of $\\mathbb{P}^3$;\n\\item for integers $a'\\leq b'\\leq c'\\leq {d}$ a $CI(a',b',c')$ exists on the generic degree $d$ surface of $\\mathbb{P}^3$, where $a=a'$ or $a+a'=d$, and $b=b'$ or $b+b'=d$, and $c=c'$ or $c+c'=d$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{rem}\\label{dover2LEM}\nUsing Lemma \\ref{switchLEM} we can study our question for integers $a\\leq b\\leq c<d\/2$ and produce a complete answer for the general case. In fact, either $a\\leq d\/2$ or $a'\\leq d\/2$.\n\\end{rem}\n\n\n\\section{The $a\\leq 4$ case}\\label{aleq4SEC}\nHere we use Stanley's result \\cite{Stanley} showing that the\nquotient of $R=\\mathbb{C}[x_0,\\ldots,x_3]$ by four generic forms\nhas the Strong Lefschetz Property. More precisely, given generic\nforms $F,G,H,F',G'\\in R$, of degrees $a,b,c,d-c,d-b$, we consider\n$W={R\/ (F,G,H,H')}$. Then the multiplication by the class of $G'$\nhas maximal rank. Hence the sequence\n\\[W(-d+b){\\rightarrow} W\\rightarrow {W\\over ( [G'])}\\rightarrow 0\\]\nproduces $H(W\/([G'])),d)=\\max\\{H(W,d)-H(W,b),0\\}$.\n\n\\begin{prop}\\label{asymptoticamenodi5}\nFor any choice of $a,b,c$ and $d$ positive integers such that\n$a\\leq 4\\leq b\\leq c$ and $d\\geq a+b+c-3$, the general degree $d$ surface in $\\mathbb{P}^3$\ncontains a $CI(a,b,c)$.\n\n\\end{prop}\n\\begin{proof} Using Lemma \\ref{dover2LEM} it is enough to consider the case when $a\\leq b\\leq c\\leq {d\\over 2}$. Using Proposition \\ref{equivalence} part \\eqref{algebraicQUESTION} we only have to show that\n\\[H(W\/([G']),d)=\\max\\{H(W,d)-H(W,b),0\\}=0.\\]\nBy the resolution of $W$ given in (\\ref{Wres}) we immediately get:\n\\begin{itemize}\n\\item if $b<c$, then\n\\begin{eqnarray*}\nH(W,b) & = & {b+3\\choose 3}-{b-a+3\\choose 3}-1 \\\\\n       & = & 1\/6 a^{3}-1\/2a^{2}b + 1\/2ab^{2} - a^{2} + 2ab + 11\/6a-1;\n\\end{eqnarray*}\n\\item if $b=c$, then\n\\begin{eqnarray*}\nH(W,b) & = & {b+3\\choose 3}-{b-a+3\\choose 3}-2 \\\\\n       & = & 1\/6 a^{3}-1\/2a^{2}b + 1\/2ab^{2} - a^{2} + 2ab + 11\/6a-2.\n\\end{eqnarray*}\n\\end{itemize}\n\nLed by the resolution of $W$, we also consider the following polynomial\n\n\\[h(W,d)=\\]\n\\[={d+3\\choose 3}-\\left[{d-a+3\\choose 3}+{d-b+3\\choose\n3}+{d-c+3\\choose 3}+\\right. \\left. {c+3\\choose 3}\\right]+\\]\n\\[+{d-a-b+3\\choose\n3}+{d-a-c+3\\choose 3}+{c-a+3\\choose 3}+{d-b-c+3\\choose 3}+\\]\n\n\\[+{c-b+3\\choose3}+1-\\left[{d-a-b-c+3\\choose 3}\\right. \\left. +{c-a-b+3\\choose 3}\\right],\\]\nwhere ${x \\choose 3}$ is the polynomial ${1\\over 6}x(x-1)(x-2)$. Making the computation we get\n\\[h(W,d)=1\/2 a^{2}b + 1\/2ab^{2}-2ab + 1.\\]\n\nNotice that, for given $a,b$ and $c$ such that $c-a-b\\geq -3$ and\n$d\\geq a+b+c-3$, the evaluation of $h(W,d)$ coincides with the\nHilbert function of $W$ in degree $d$, i.e. $h(W,d)=H(W,d)$.\nMoreover, the inequalities\n\\[a\\leq 4 \\mbox{ and } c-a-b\\leq -4\\]\nonly hold when $a=4$ and $b=c$ (recall that $a\\leq b\\leq c$) and in this case\n\\[H(W,d)=h(W,d)+{c-a-b+3\\choose 3}=h(W,d)-1.\\]\n\nFinally we compute $H(W,d)-H(W,b)$ distinguishing two cases.\n\nThe $a< 4$ or $b<c$ case. If $b<c$ we use the value of $H(W,b)$\nand the equality $H(W,d)=h(W,d)$  previously determined to  get\n\\[H(W,d)-H(W,b)=-1\/6 a^{3} + a^{2}b + a^{2}-4ab-11\/6a + 2.\\]\nThis polynomial is linear in $b$ and it does not involve $d$ and\nit is easy to see that for $a\\leq 4$\n\\[H(W,d)-H(W,b)\\leq 0.\\]\nWhen $a < 4$ and $b=c$ a completely analogous argument can be applied.\n\nThe $a=4$ and  $b=c$ case. Mutatis mutandis, we compute again and we get\n\\[H(W,d)-H(W,b)=-1\/6 a^{3} + a^{2}b + a^{2}-4ab-11\/6a + 2,\\]\nhence the same polynomial of the previous case and this finishes the proof.\n\\end{proof}\n\nProposition \\ref{asymptoticamenodi5} gives an asymptotic result\nyielding that, when one of the degree of the $CI$ is at most 4, then\nfor $d$ big enough a complete intersection of the given type\nexists on a generic surface of degree $d$. With a slightly more\ncareful analysis this can be improved and the condition on $d$ can\nbe dropped.\n\n\\begin{thm}\\label{aleq4THM}\nLet $a,b,c$ and $d$ be integers such that $a\\leq  b\\leq c<d$.\nIf $a\\leq 4$, then a $CI(a,b,c)$ exists on the generic degree $d$\nsurface in $\\mathbb{P}^3$.\n\\end{thm}\n\\begin{proof}\nUsing Proposition \\ref{asymptoticamenodi5} we have only to check\nvalues of $d$ in the range $c<d\\leq a+b+c-4$. The idea is to\nuse Lemma \\ref{switchLEM} to reduce the degree of the complete\nintersection not changing $d$ so that Proposition\n\\ref{asymptoticamenodi5} can be applied.\n\nFor $a=2$, we consider the existence of a $CI(2,b,c)$ on the\ngeneric degree $d$ surface for $c<d\\leq b+c-2$. Such a\ncomplete intersection exists if the same happen for a\n$CI(2,d-c,d-b)$. But, by Proposition \\ref{asymptoticamenodi5},\nthis is the case as soon as\n\\[d\\geq 2+(d-c)+(d-b)-3\\]\nand this equivalent to $d\\leq b+c-1$ which is actually the case. Similarly for $a=3$.\n\nThe case $a=4$ is treated in analogy with the previous ones,\nexcept for $d=b+c$. In this situation, applying Lemma\n\\ref{switchLEM}, we have to study $CI(4,b,b)$'s on a generic\nsurface of degree $d\\geq b$. Repeating the same argument above we\nhave to treat values of $d$ in the range $b\\leq d\\leq 2b$.\nProposition \\ref{existdbigPROP} gives the existence for all $d$,\nbut for $d=2b$. By Proposition $\\ref{equivalence}$ we have to\nconsider the coordinate ring $W$ of a complete intersection of\ntype $(b,b,b,b)$ and its Hilbert function $H(\\cdot)$. Using the\nfact that multiplication by one form  has maximal rank in $W$, we\nhave only to compare $H(2b)$ and $H(2b-4)$, but these values are\nthe same being $W$ a Gorenstein ring with socle degree $4b-4$, and\nthis finishes the proof.\n\\end{proof}\n\n\n\\section{The case $a> 4$: non-existence results}\\label{ageq5nonexistSEC}\n\nIn this section we will prove asymptotic non-existence results\nwhen $a> 4$. For non-negative integers $a,b,c$ and $d$, $a\\leq\nb\\leq c<d$, we consider generic forms $F,G,H,H',G',F'\\in R$ of\ndegrees $a,b,c,d-c,d-b$ and $d-a$. Consider the ring $W={R\/\n(F,G,H,H')}$ and notice that, by a straightforward dimensional\nargument, if\n\\[H(W,a)+H(W,b)-H(W,d)<0\\]\nthen\n\\[H\\left({W\\over ([G'],[F'])},d\\right)\\neq 0.\\]\nHence, by Lemma \\ref{equivalence} \\eqref{algebraicQUESTION},  if\n$H(W,d)-H(W,a)-H(W,b)<0$, then the generic degree $d$ surface of\n$\\mathbb{P}^3$ does not contain a\n$CI(a,b,c)$. Using this idea we prove the following:\n\n\\begin{thm}\\label{nonEXISTa5}\nLet $a\\leq b\\leq c$ and $d$  be non-negative integers such that\n\\[a=5 \\mbox{ and } b\\geq 13\\]\nor\n\\[a=6 \\mbox{ and } b\\geq 9\\]\nor\n\\[a \\geq 7.\\]\n\nThen, for $d\\geq a+b+c-3$ the generic degree $d$ surface of\n$\\mathbb{P}^3$ does not contain a\n$CI(a,b,c)$.\n\\end{thm}\n\nIn order to prove this theorem we need the following technical\nfact:\n\n\\begin{lem}\\label{inequalityLEMMA}\nLet $a,b,c$ be non-negative integers, such that $4<a$ .\nAssume that, for integers $c_0$ and $d$ such that\n\\[c_0\\geq b \\mbox{ and } d>a+b+c_0-4,\\]\none has the Hilbert function inequality\n\\[H(W,a)+H(W,b)-H(W,d)<0,\\]\nwhere $W$ is the ring\n\\[\nW={R\\over (F,G,H,H')}\n\\]\nand the forms $F,G,H$ and $H'$ are generic and have degrees\n$a,b,c_0$ and $d-c_0$.\n\nThen, if $A,\\ B,\\ C$ and $D$ are forms of degrees $a,\\ b,\\ c\\geq c_0$ and $d-c$ and $$W^\\prime = {R\\over (A,B,C,D)}$$ then the following inequality holds:\n\\[H(W^\\prime,a)+H(W^\\prime,b)-H(W^\\prime,d)<0,\\]\nfor $d>a+b+c-4$.\n\\end{lem}\n\\begin{proof}\nThe key observation is that\n\\[H(W,d)=H(W,a+b-4).\\]\nIn fact, being $W$ a Gorenstein ring,  its Hilbert function is symmetric and $H(W,x)=H(W,y)$ if $x+y=d+a+b-4$.\nThen, we compute\n\\[H(W,a)+H(W,b)-H(W,a+b-4)\\]\nusing the formulae in the proof of Proposition\n\\ref{asymptoticamenodi5}, for which we need the assumption on $d$.\nOne sees that the final expression does not involve neither $c$ or\n$d$ and the proof follows. For example, in the case $a<b$ one gets\n$H(W,a)+H(W,b)-H(W,a+b-4)<0$ if and only if\n\\[b\\geq \\frac{{1\\over 2}a^3-a^2+{11\\over 2}a -4}{(a^2-4a)}.\\]\n\\end{proof}\n\nWe can now prove Theorem \\ref{nonEXISTa5}.\n\n\\begin{proof}[Proof of Theorem \\ref{nonEXISTa5}]\nWe let $b=c$ and we show that for the required values of $a$ and\n$b$ we have the inequality $H(W,a)+H(W,b)-H(W,d)<0$. Then we apply\nLemma \\ref{inequalityLEMMA} to get the result when $c\\geq b$.\n\nAgain, we notice that $H(W,d)=H(W,a+b-4)$.\n\nWe divide the proof in two cases depending on whether $a=b$ or\n$a<b$.\n\n\\noindent{\\bf Case $a<b$.}\n\nUsing the resolution of the ring $W$, the\ninequality\n\\[H(W,a)+H(W,b)-H(W,a+b-4)\\geq 0\\]\nis readily seen to be equivalent to\n\\[b\\leq \\frac{{1\\over 2}a^3-a^2+{11\\over 2}a -4}{(a^2-4a)}.\\]\nRecalling that $a<b$ we get\n\\[H(W,a)+H(W,b)-H(W,d)\\geq 0\\]\nonly if\n\\[-{1\\over 2}a^3+3a^2+{11\\over 2}a-4\\geq 0\\]\nand this inequality holds if and only if\n\\[a=5 \\mbox{ or } a=6.\\]\nHence\n\\[H(W,a)+H(W,b)-H(W,d)\\geq 0\\]\nimplies\n\\[a=5,b\\leq 12\\]\nor\n\\[a=6,b\\leq 8.\\]\n\n{\\bf Case $a=b$.}\n\n Computing we get\n\\[H(W,a)+H(W,b)-H(W,d)=-{2\\over 3}a^3+4a^2+{11\\over 3}a-3\\geq 0\\]\nonly if $a< 7$ and this finishes the proof.\n\\end{proof}\n\nTo prove some more non-existence results, we need the following:\n\n\\begin{prop}\\label{nonexistEVENTUALLY}\n\nLet $a\\leq b\\leq c< {d}$ and $d> 2c+b+a-3$. If no $CI(a,b,c)$\nexists on the generic degree $d$ hypersurface, then it does not\nexist on the generic hypersurface of degree $d'>d$ either.\n\\end{prop}\n\\begin{proof}\nIt is enough to treat the case $d'=d+1$.  For generic forms $F,\\ G,\\ H$ of degrees $a,\\ b$ and $c$ let\n$$\nA = {\\mathbb{C}[x_0,\\cdots,x_3]\\over(F,G,H)}.\n$$\nBy hypothesis, for the\ngeneric choice of $F',G'$ and $H'$ of degrees $d-a,d-b$ and $d-c$\nin $A$, we know that the degree $d$ part of\n\\[{A\\over (F',G',H')}\\]\nis not zero. Now, consider elements $F'',G''$ and $H''$ of degrees\n$d+1-a,d+1-b$ and $d+1-c$. Notice that\n\n\\[d-a\\geq d-b\\geq d-c >c+b+a-3\\]\nand recall that $A_i\\simeq A_j$ as $\\mathbb{C}$ vector spaces if $i$ and $j$ are $>2c + b + a - 3$. Thus for a general linear form\n$L$ we have\n\\[F''=LF^*, G''=LG^*\\mbox{ and } H''=LH^*\\]\nand the forms $F^*,G^*$ and $H^*$ have degrees $d-a,d-b$ and\n$d-c$. Hence we have a isomorphism\n\\[(F'',G'',H'')_{d+1}\\simeq (F^*,G^*,H^*)_d\\]\nand this is enough to conclude that the degree $d+1$ part of\n\\[{A\\over (F'',G'',H'')}\\]\nis not zero and the result follows.\n\\end{proof}\n\n\n\\begin{lem}\\label{nonexistb8and12}\nIf $c\\geq 13$, then the generic degree $d\\geq 2c+15$ surface of\n$\\mathbb{P}^3$ does not contain a $CI(5,12,c)$.\n\nIf $c\\geq 10$, then the generic degree $d\\geq 2c+12$ surface of\n$\\mathbb{P}^3$ does not contain a $CI(6,8,c)$.\n\\end{lem}\n\n\\begin{proof}\nWe begin with the study of $CI(5,12,c)$. Let $c=13+x$,\n$d=2c+a+b-2=41+2x$ and consider the ring\n\\[\nW={R\\over (F,G,H,H')}\n\\]\nwhere the forms $F,G,H$ and $H'$ have degrees $5,12,13+x$ and\n$28+x$. The generic degree $d$ surface does not contain a\n$CI(5,12,c)$ if $H(W,5)+H(W,12)-H(W,d)<0$, where\n$H(W,d) = H(W,41+2x)=H(W,14)$. Now we compute\n\\[H(5)={8\\choose 3}-1=55,\\]\n\\[H(12)={15\\choose 3}-{10\\choose 3}-1=334,\\]\n\\[H(14)=\\left\\lbrace\n\\begin{array}{lr}\n{17\\choose 3}-{12\\choose 3}-{5\\choose 3}=450 & \\mbox{ if } x>1 \\\\ \\\\\n449 & \\mbox{ if } x=1 \\\\ \\\\\n446 & \\mbox{ if } x=0\n\\end{array}\n\\right. .\\] Hence, $H(W,5)+H(W,12)-H(W,d)<0$, and by Proposition\n\\ref{nonexistEVENTUALLY} we conclude that the generic degree $d'$\nsurface does not contain a $CI(5,12,c)$ for $d'\\geq\nd=41+2x=15+2c$.\n\nThe case of $CI(6,8,c)$ is solved by completely analogous computations.\n\\end{proof}\n\n\\section{The case $a> 4$: existence results}\\label{ageq5existSEC}\n\nTheorem \\ref{nonEXISTa5} does not cover small values of $a$ and $b$.\nIn this Section we derive a result analogous to Theorem\n\\ref{aleq4THM} in these cases.\n\nWe begin with proving two technical facts.\n\n\\begin{prop}\\label{existdbigPROP}\nLet $a\\leq b\\leq c\\leq {d}$ and $d\\geq a+b+c-3$. If a $CI(a,b,c)$\nexists on the generic degree $d$ surface, then it also exists on\nthe generic surface of degree $d'>d$.\n\\end{prop}\n\\begin{proof}\nLet $d'=d+1$ and notice that it is enough to treat this case. The\nhypothesis reads as follows: the degree $d$ part of the ring\n\\[{A\\over (F',G',H')}\\]\nis zero for generic forms $F',G'$ and $H'$ of degrees $d-a,d-b$\nand $d-c$ where $A=R\/(F,G,H)$. If $L\\in A$ is a generic linear\nform, by \\cite{Stanley}, we know that multiplication by $L$ is\nan isomorphism in degree bigger than or equal to $a+b+c-4$. Hence,\nthe degree $d+1$ piece of\n\\[{A\\over (LF',LG',LH')}\\]\nis zero and this is enough to complete the proof since, if three special\nforms, namely $LF',LG'$ and $LH'$, have maximal span then the same property holds\nfor a generic choice.\n\\end{proof}\n\n\n\\begin{lem}\\label{existdbigLEM}\nLet  $a,b$ and $d$ be non-negative integers such that $4<a\\leq b$\nand $d=2a+2b-6$. If the generic degree $d$ surface in $\\mathbb{P}^3$\ncontains a $CI(a,b,a+b-3)$, then the generic degree $d'$ surface\ncontains a $CI(a,b,c)$ for any $d'\\geq a+b+c-3$ and any $c\\geq\na+b-3$.\n\\end{lem}\n\\begin{proof}\nNotice that, by Proposition \\ref{existdbigPROP}, the generic\ndegree $d+\\epsilon$ surface in $\\mathbb{P}^3$ contains a $CI(a,b,a+b-3)$\nfor all $\\epsilon\\geq 0$. Hence, by Lemma \\ref{switchLEM}, the\nsame holds for $CI(a,b,a+b-3+\\epsilon)$ and surfaces of degree\n$d+\\epsilon$. Making $\\epsilon$ vary and again applying\nProposition \\ref{existdbigPROP} the result follows.\n\\end{proof}\n\n\\begin{thm}\\label{ageq5THM}\nLet  $a,b,c$ and $d$ be non-negative integers such that $a\\leq\nb\\leq c<d$. If $a=5$ and $b\\leq 11$, or $a=6$ and $b\\leq 7$,\nthen a $CI(a,b,c)$ exists on the generic degree $d$\nsurface of $\\mathbb{P}^3$. If $a=5,b=12$ and $c=12$, or $a=6,b=8$ and $c=8,9$,\nthen a $CI(a,b,c)$ exists on the generic degree $d$\nsurface of $\\mathbb{P}^3$.\n\\end{thm}\n\n\\begin{proof}\nTo prove the thesis we combine all the previous results and\ntechnical facts. Crucial ingredients are also some explicit\ncomputations that we performed using the computer algebra system\nCoCoA \\cite{cocoa}.\n\nTo determine whether a $CI(a,b,c)$ exists on the generic surface of degree $d$ in $\\mathbb{P}^3$, we proceed as follows:\n\\begin{itemize}\n\\item if $c\\leq a+b-3$, we make explicit computations for all\n$d\\leq a+b+c-3$; a positive answer for $d=a+b+c-3$ solves the\ncases for bigger $d$'s by Proposition \\ref{existdbigPROP};\n\\item if $c=a+b-3$ and $d\\geq 2a+2b-6$, we verify each statement\nwith an explicit computation for $d=2a+2b-6$; if the answer is\npositive we conclude the same for $c\\geq a+b-3$ and $d\\geq\na+b+c-3$ by Lemma \\ref{existdbigLEM}.\n\\end{itemize}\n\nWe sketch this procedure for $a=6$, the case $a=5$ is completely\nanalogous but lengthier. We need to perform explicit computations in the following cases:\n\\begin{itemize}\n\\item $CI(6,6,c_1)$, for $c_1\\leq 9$ and $d_1\\leq 9+c_1$;\n\\item $CI(6,7,c_2)$, for $c_2\\leq 10$ and $d_3\\leq 10+c_2$;\n\\end{itemize}\nThe computations (see Example \\ref{computationEX}) show that the  complete intersections exist on\nthe generic surfaces of the required degrees. Hence we conclude\nthat the generic surface of degree $d$ contains a $CI(6,b,c)$ for\nall $b\\leq 7$ and any $c,d$ such that $d> c$. We conclude the proof for $a=6$ by verifying existence in the cases: $CI(6,8,8)$ for $d=19$, and $CI(6,8,9)$ for $d=20$.\n\n\\end{proof}\n\n\\begin{ex}\\label{computationEX}\nWe begin with verifying that the generic surface of degree $7\\leq d\\leq 15$ contains a\n$CI(6,6,6)$. Using Proposition \\ref{equivalence} it is enough to\nshow that the ring\n\\[S={\\mathbb{C}[x_0,\\ldots,x_3]\\over(F,G,H,H',G',F')}\\]\nis zero in degree $d$, where the forms $F,G,H,H',G'$ and $F'$ are\ngeneric and have degrees $6,6,6,d-6,d-6$ and $d-6$. Hence, for each $d$, we choose\nrandom forms with rational coefficients of the  required degrees.\nThen we ask CoCoA \\cite{cocoa} to compute the Hilbert function of\n$S$ in degree $d$. Since for all $d$'s we get $H(S,d)=0$, we conclude (by semicontinuity)\nthat this is the case for a generic choice of forms of the\nappropriate degrees. In particular, as $15=6+6+6-3$ and $H(S,15)=0$, Proposition \\ref{existdbigPROP} yields that a $CI(6,6,6)$ exists on the generic degree $d\\geq 15$ surface of $\\mathbb{P}^3$.\n\nThe same argument works in complete analogy for $c\\leq 8$. For $c=9$ we make an explicit computation for $d=18$ and using Lemma \\ref{existdbigLEM} we show existence of a $CI(6,6,c)$ on the generic degree $d$ surface for $c\\geq 9$ and $d\\geq c+9$. The cases for $c<d<c+9$ are solved using Lemma \\ref{equivalence} and the results for $c\\leq 8$ and $a\\leq 4$.\n\\end{ex}\n\n\\section{Main theorem and final remarks}\\label{finalSEC}\n\nWe can now prove our main theorem:\n\\begin{proof}[Proof of Theorem \\ref{mainTHM}]\nThe existence part for the case $a\\leq 4$ is Theorem\n\\ref{aleq4THM} while existence for the remaining cases is Theorem\n\\ref{ageq5THM}. The asymptotic non-existences are given by Lemma\n\\ref{nonexistb8and12} and Theorem \\ref{nonEXISTa5}.\n\\end{proof}\n\nTheorem \\ref{mainTHM} produces a complete asymptotic answer to our\noriginal question. We also get many existence and non existence\nresults for small value of $d$. However, there are still\ninfinitely many cases which we have not solved, e.g. $a=7$ any\n$b,c$ and $d$ such that $7\\leq b\\leq c$ and $c+5\\leq d\\leq\na+b+c-4$.\n\nWe state a conjecture completing Theorem \\ref{mainTHM}:\n\\begin{quote}{\\it{ \\textbf{Conjecture}:}\ngiven non-negative integers $a,b,c$ and $d$ such that $ a\\leq\nb\\leq c<d$, there exists a function $d(a,b,c)$, possibly assuming\nthe value $+\\infty$, such that the generic degree $d$ surface in\n$\\mathbb{P}^3$ contains a $CI(a,b,c)$ if and only if $d<d(a,b,c)$.\n}\\end{quote}\n\nAs support for this conjecture, notice that it fits with the\nasymptotic statement and with the other results of Theorem\n\\ref{mainTHM}. For example, $d(a,b,c)=+\\infty$ for $a\\leq 4$ and\n$d(a,b,c)<a+b+c-3$ for $7\\leq a$.\n\n\n\n\n\n\n\\bibliographystyle{alpha}\n\n\\def$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn arithmetically Cohen-Macaulay (for short, aCM) sheaf on a projective scheme $X$ is a coherent sheaf supporting $X$, which has trivial intermediate cohomology and the stalk at each point whose depth equals the dimension of $X$. ACM vector bundles correspond to maximal Cohen-Macaulay modules over the associated graded ring and they reflect the properties of the graded ring. It is believed that the category generated by aCM sheaves on $X$ measures the complexity of $X$. Indeed, a classification of aCM varieties was proposed as {\\it finite, tame or wild} representation type according to the complexity of this category in \\cite{DG} and there are several contributions to this trichotomy such as \\cite{EH,BGS,CMP,FM}. It is only recent when such a representation type is determined for each aCM variety that is not a cone; see \\cite{FP}. \n\nIn this article, we pay our attention to the representation type of surfaces in three-dimensional projective space. Since the aCM vector bundles on smooth surfaces of degree at most two are completely classified due to the work by Horrocks and \\cite{Kapranov, Knorrer}, we may focus on surfaces of degree at least three. The case of cubic surfaces is dealt in \\cite{CH, faenzi} and the case of quartic surfaces is from \\cite{madonna}. Our main result is the following, which implies that the surfaces in Theorem \\ref{thth} are of wild representation type. \n\n\\begin{theorem}\\label{thth}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree at least four with $X_{\\mathrm{reg}} \\ne \\emptyset$ and assume either $\\mathrm{Pic}(X) =\\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}}_X(1)\\rangle$ or that $X$ is integral. For every even and positive integer $r$, there exists a family $\\{\\enm{\\cal{E}} _\\lambda\\}_{\\lambda \\in \\Lambda}$ of indecomposable aCM vector bundles of rank $r$ such that $\\Lambda$ is an integral quasi-projective variety with $\\dim \\Lambda=r$ and $\\enm{\\cal{E}} _{\\lambda}\\ncong \\enm{\\cal{E}} _{\\lambda'}$ for all $\\lambda \\ne \\lambda'$ in $\\Lambda$.\n\\end{theorem}\n\nIt has to be noticed that although the result in \\cite{FP} is more general than the implication of Theorem \\ref{thth} regarding the wildness of the representation type, Theorem \\ref{thth} provides a concrete way of constructing families of indecomposable aCM `vector bundles' with prescribed rank, even on singular surfaces. \n\nOn the other hand, every non-integral aCM projective schemes of arbitrary dimension at least two is of `very wild' representation type, in a sense that there exist arbitrarily large dimensional families of pairwise non-isomorphic aCM non-locally free sheaves of rank one; see Proposition \\ref{zz3}. \n\nHere we summarize the structure of this article. In Section \\ref{prem} we collect several definitions and basic results that are used throughout the article. In Section \\ref{sec3} we state the main result in Theorem \\ref{i1}, which would automatically imply Theorem \\ref{thth}. We also give a proof of Theorem \\ref{i1} in special case and suggest a number of its variation to construct aCM vector bundles. Then we spend the whole Section \\ref{sec4} for the proof of Theorem \\ref{i1}; basically we use induction on rank and the main ingredient for the proof is Lemma \\ref{x1} and the use of monodromy argument. Then we show in Section \\ref{sec5} the wildness of any aCM projective scheme of dimension at least two by investigating non-locally free ideal sheaves. \n\n\n\\section{Preliminary}\\label{prem}\nThroughout the article our base field $\\mathbf{k}$ is algebraically closed of characteristic $0$. We always assume that our projective schemes $X\\subset \\enm{\\mathbb{P}}^N$ are arithmetically Cohen-Macaulay, namely, $h^1(\\enm{\\cal{I}} _{X, \\enm{\\mathbb{P}}^N}(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$ and $h^i(\\enm{\\cal{O}} _X(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$ and all $i=1,\\dots ,\\dim X-1$, of pure dimension at least two. Then by \\cite[Th\\'eor\\`eme 1 in page 268]{serre} all local rings $\\enm{\\cal{O}} _{X,x}$ are Cohen-Macaulay of dimension $\\dim X$. From $h^1(\\enm{\\cal{I}} _{X, \\enm{\\mathbb{P}}^N})=0$ we see that $X_{\\mathrm{red}}$ is connected. Since in all our results we have $N =\\dim X+1=3$, the reader may just assume that $X$ is a surface in $\\enm{\\mathbb{P}}^3$. For a vector bundle $\\enm{\\cal{E}}$ of rank $r\\in \\enm{\\mathbb{Z}}$ on $X$, we say that $\\enm{\\cal{E}}$ {\\it splits} if all its indecomposable factors are $\\enm{\\cal{O}}_X(t)$ for some $t\\in \\enm{\\mathbb{Z}}$; $\\enm{\\cal{E}} \\cong \\oplus_{i=1}^r \\enm{\\cal{O}}_X(t_i)$ for some $t_i\\in \\enm{\\mathbb{Z}}$ with $i=1,\\ldots, r$.\n\nWe always fix the embedding $X\\subset \\enm{\\mathbb{P}}^N$ and the associated polarization $\\enm{\\cal{O}}_X(1)$. For a coherent sheaf $\\enm{\\cal{E}}$ on a closed subscheme $X$ of a fixed projective space, we denote $\\enm{\\cal{E}} \\otimes \\enm{\\cal{O}}_X(t)$ by $\\enm{\\cal{E}}(t)$ for $t\\in \\enm{\\mathbb{Z}}$. For another coherent sheaf $\\enm{\\cal{G}}$, we denote by $\\mathrm{hom}_X(\\enm{\\cal{F}}, \\enm{\\cal{G}})$ the dimension of $\\op{Hom}_X(\\enm{\\cal{F}}, \\enm{\\cal{G}})$, and by $\\mathrm{ext}_X^i(\\enm{\\cal{F}}, \\enm{\\cal{G}})$ the dimension of $\\mathrm{Ext}_X^i (\\enm{\\cal{F}}, \\enm{\\cal{G}})$. Finally we denote the canonical sheaf of $X$ by $\\omega_X$. \n\n\\begin{definition}\\label{deff}\nA coherent sheaf $\\enm{\\cal{E}}$ on $X$ is called {\\it arithmetically Cohen-Macaulay} (for short, aCM) if the following hold:\n\\begin{itemize}\n\\item [(i)] $\\enm{\\cal{E}}$ is locally Cohen-Macaulay, that is, the stalk $\\enm{\\cal{E}}_x$ has depth equal to $\\dim \\enm{\\cal{O}}_{X,x}$ for any point $x$ on $X$, and\n\\item [(ii)] $H^i(\\enm{\\cal{E}}(t))=0$ for all $t\\in \\enm{\\mathbb{Z}}$ and $i=1, \\ldots, \\dim (X)-1$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{remark}\\label{rremm}\nIn the condition (i) of Definition \\ref{deff}, we may only require that the stalk $\\enm{\\cal{E}}_x$ has positive depth for any point $x\\in X$; see \\cite[Remark 2.2]{BHMP} and \\cite[Th\\'eor\\`eme 1 in page 268]{serre}. \n\\end{remark}\n\nIf $\\enm{\\cal{E}}$ is a coherent sheaf on a closed subscheme $X$ of a fixed projective space, then we may consider its Hilbert polynomial $\\mathrm{P}_{\\enm{\\cal{E}}}(t)\\in \\enm{\\mathbb{Q}} [t]$ with the leading coefficient $\\mu(\\enm{\\cal{E}})\/d!$, where $d$ is the dimension of $\\mathrm{Supp}(\\enm{\\cal{E}})$ and $\\mu=\\mu(\\enm{\\cal{E}})$ is called the {\\it multiplicity} of $\\enm{\\cal{E}}$. The {\\it normalized} Hilbert polynomial $p_{\\enm{\\cal{E}}}(t)$ of $\\enm{\\cal{E}}$ is defined to be the Hilbert polynomial of $\\enm{\\cal{E}}$ divided by $\\mu (\\enm{\\cal{E}})$.\n\n\\begin{definition}\\label{def}\nIf $\\dim \\mathrm{Supp}(\\enm{\\cal{E}})=\\dim (X)$, then the {\\it rank} of $\\enm{\\cal{E}}$ is defined to be\n$$\\mathrm{rank}(\\enm{\\cal{E}})=\\frac{\\mu(\\enm{\\cal{E}})}{\\mu(\\enm{\\cal{O}}_X)}.$$\nOtherwise it is defined to be zero.\n\\end{definition}\nFor an integral scheme $X$, the rank of $\\enm{\\cal{E}}$ is the dimension of the stalk $\\enm{\\cal{E}}_x$ at the generic point $x\\in X$. But in general $\\mathrm{rank}(\\enm{\\cal{E}})$ needs not be integer.\n\n\\begin{lemma}\\label{z1}\nLet $(X, \\enm{\\cal{O}}_X(1))$ be an aCM projective scheme of dimension $n\\ge 2$. For a fixed coherent sheaf $\\enm{\\cal{G}}$ with pure depth $n$ on $X$, assume the existence of $t_0\\in \\enm{\\mathbb{Z}}$ such that $s:= h^1(\\enm{\\cal{G}} (t_0))> 0$. Then the vector space $W:= H^1(\\enm{\\cal{G}} (t_0))$ induces the following unique extension up to isomorphisms\n\\begin{equation}\\label{eqz1}\n0 \\to \\enm{\\cal{G}} \\to \\enm{\\cal{E}} \\to \\enm{\\cal{O}} _X(-t_0)\\otimes W^\\vee \\to 0\n\\end{equation}\nand the sheaf $\\enm{\\cal{E}}$ in the middle satisfies the following:\n\\begin{itemize}\n\\item [(i)] $h^1(\\enm{\\cal{E}} (t)) = h^1(\\enm{\\cal{G}} (t))$ for all $t\\ne t_0$, and $h^1(\\enm{\\cal{E}} (t_0)) =0$;\n\\item [(ii)] $h^i(\\enm{\\cal{E}} (t)) =h^i(\\enm{\\cal{G}} (t))$ for all $t\\in \\enm{\\mathbb{Z}}$ and all $i$ with $2\\le i \\le n-1$.\n\\end{itemize}\nIf $\\enm{\\cal{G}}$ is locally free, then $\\enm{\\cal{E}}$ is locally free.\n\\end{lemma}\n\n\\begin{proof}\nAll statements, except the one concerning $h^1(\\enm{\\cal{E}} (t_0))$, are true for any sheaf $\\enm{\\cal{E}}$ fitting into (\\ref{eqz1}). The vanishing\nof $H^1(\\enm{\\cal{E}} (t_0))$ is equivalent to the bijectivity of the coboundary map $\\delta : H^0(\\enm{\\cal{O}} _X)\\otimes W^\\vee \\rightarrow H^1(\\enm{\\cal{G}} (t_0))$ associated to the twist by $\\enm{\\cal{O}} _X(t_0)$ of (\\ref{eqz1}). The bijectivity of $\\delta$ is a standard result on the extension functor.\n\\end{proof}\n\n\\begin{theorem}\\label{e2}\nLet $X\\subset \\enm{\\mathbb{P}}^N$ be a projective Gorenstein scheme with pure dimension two and pure depth two, satisfying that \n\\begin{itemize}\n\\item $h^1(\\enm{\\cal{O}}_X(t))=0$ for all $t\\in \\enm{\\mathbb{Z}}$ and $h^1(\\enm{\\cal{I}}_{X,\\enm{\\mathbb{P}}^N})=0$; \n\\item $X_{\\mathrm{reg}} \\ne \\emptyset$ and $\\deg (\\omega _X)+\\deg (X) \\ge 0$.\n\\end{itemize}\nThen there exists a two-dimensional family of pairwise non-isomorphic aCM vector bundles of rank two on $X$ whose very general member is indecomposable; here ``very general\" means outside countably many proper subvarieties.\n\\end{theorem}\n\n\\begin{proposition}\\label{e1}\nLet $X\\subset \\enm{\\mathbb{P}}^N$ be as in Theorem \\ref{e2}. Assume $X_{\\mathrm{reg}} \\ne \\emptyset$ and fix $p\\in X_{\\mathrm{reg}}$. Then there exists an aCM vector bundle $\\enm{\\cal{E}} _p$ of rank two on $X$ fitting into the exact sequence\n\\begin{equation}\\label{eqe1}\n0 \\to \\omega _X(1)\\to \\enm{\\cal{E}} _p\\to \\enm{\\cal{I}} _{p,X}\\to 0.\n\\end{equation}\nMoreover, if $\\deg (\\omega _X)+\\deg (X) \\ge 0$ and $p,q\\in X_{\\mathrm{reg}}$ with $p\\ne q$, then we have $\\enm{\\cal{E}} _p\\ncong \\enm{\\cal{E}}  _q$.\n\\end{proposition}\n\n\\begin{proof}\nSince $X$ is Gorenstein, $\\omega _X(1)$ is a line bundle and we get\n$$\\op{Ext}_X^1(\\enm{\\cal{I}} _{p,X},\\omega _X(1)) \\cong H^1(\\enm{\\cal{I}} _{p,X}(-1))^\\vee \\cong \\mathbf{k}.$$ \nSo up to isomorphism there exists a unique sheaf $\\enm{\\cal{E}}_p$ fitting into an extension (\\ref{eqe1}) with a nonzero extension class. Since $h^0(\\enm{\\cal{O}} _X(-1)) =0$ and $p\\in X_{\\mathrm{reg}}$, the Cayley-Bacharach condition is satisfied for (\\ref{eqe1}) and so $\\enm{\\cal{E}}_p$ is locally free; see \\cite{cat}. Note that the restriction map \n$$H^0(\\enm{\\cal{O}} _X(t)) \\to H^0(\\enm{\\cal{O}} _X(t)_{|\\{p\\}})$$\nis surjective for any $t\\ge 0$. This implies that $h^1(\\enm{\\cal{I}}_{p,X}(t))=0$ for any $t\\ge 0$, because we have $h^1(\\enm{\\cal{O}}_X(t))=0$. Then we see from (\\ref{eqe1}) that $h^1(\\enm{\\cal{E}}_p(t))=0$ for any $t\\ge 0$. On the other hand, from $\\det (\\enm{\\cal{E}}_p) \\cong \\omega_X(1)$, we get that $h^1(\\enm{\\cal{E}}_p(t))=h^1(\\enm{\\cal{E}}_p^\\vee\\otimes \\omega_X(-t))=h^1(\\enm{\\cal{E}}_p(-t-1))=0$ for $t<0$ by Serre's duality. Thus $\\enm{\\cal{E}}_p$ is aCM. \n\nFor the second assertion, assume $\\enm{\\cal{E}} _p\\cong \\enm{\\cal{E}} _q$. From the assumption $\\deg (\\omega _X(1)) \\ge 0$, we get $h^0(\\omega _X^\\vee (-1)) \\le 1$ with equality if and only if $\\omega _X \\cong \\enm{\\cal{O}} _X(-1)$. In particular, we have $h^0(\\enm{\\cal{I}} _{p,X}\\otimes \\omega _X^\\vee (-1)) =0$. Then from the assumption $h^1(\\enm{\\cal{O}} _X)=0$ and (\\ref{eqe1}), we get $h^0(\\enm{\\cal{E}} _p\\otimes \\omega _X^\\vee (-1)) =1$ and that $p$ is the only zero of a nonzero section of $H^0(\\enm{\\cal{E}} _p\\otimes \\omega _X^\\vee (-1))$. Thus we get $p=q$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{e2}:]\nBy assumption $X_{\\mathrm{reg}}$ is a two-dimensional quasi-projective smooth variety. By Proposition \\ref{e1} there is a flat family of aCM vector bundles $\\{\\enm{\\cal{E}} _p\\}_{p\\in X_{\\mathrm{reg}}}$ of rank two such that if $p, q\\in X_{\\mathrm{reg}}$ and $p\\ne q$, then $\\enm{\\cal{E}} _p\\ncong \\enm{\\cal{E}} _q$. Now assume that $\\enm{\\cal{E}}_p$ is decomposable for some $p\\in X_{\\mathrm{reg}}$, say $\\enm{\\cal{E}} _p\\cong \\enm{\\cal{A}}_1 \\oplus \\enm{\\cal{A}}_2$ with each $\\enm{\\cal{A}}_i$ a line bundle on $X$. Since $\\det (\\enm{\\cal{E}} _p)\\cong \\omega _X(1)$, we have $\\enm{\\cal{A}}_2 \\cong \\enm{\\cal{A}}_1^\\vee\\otimes \\omega _X(1)$. Now from the assumption that $h^1(\\enm{\\cal{O}} _X)=0$, we see that $\\mathrm{Pic}(X)$ is discrete and countable. This implies that there can exist only countably many decomposable vector bundles in the family. Since the base field $\\mathbf{k}$ is algebraically closed and so uncountable, there exists some indecomposable vector bundle in the family $\\{\\enm{\\cal{E}} _p\\}_{p\\in X_{\\mathrm{reg}}}$ and for a very general point $o$ on any connected component of $X_{\\mathrm{reg}}$ the vector bundle $\\enm{\\cal{E}} _o$ is indecomposable.\n\\end{proof}\n\nThroughout the article, as in Proposition \\ref{e1}, our construction of aCM sheaf of rank two on $X$ is in terms of the following extension\n\\begin{equation}\\label{popo}\n0\\to \\omega_X \\to \\enm{\\cal{E}} \\to \\enm{\\cal{I}}_{Z,X}(a) \\to 0\n\\end{equation}\nwith $Z$ a locally complete intersection of codimension two in $X$ and $a\\in \\enm{\\mathbb{Z}}$. Such extensions are parametrized by $\\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{Z,X}(a), \\omega_X)$. In case when $X$ is a surface, the coboundary map associated to (\\ref{popo}) is\n$$\\delta_1 : H^1(\\enm{\\cal{I}}_{Z,X}(a)) \\to H^2(\\omega_X)\\cong \\mathbf{k}$$ \nand by Serre's duality in \\cite[Theorem 3.12]{H} its dual is\n$$\\mathbf{k}\\cong \\mathrm{Hom}_X(\\omega_X, \\omega_X) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{Z,X}(a), \\omega_X),$$\nwhich is obtained by applying the functor $\\mathrm{Hom}_X(-, \\omega_X)$ to (\\ref{popo}). Thus the coboundary map $\\delta_1$ is surjective if and only if (\\ref{popo}) is a non-trivial extension. Since we assume $h^1(\\enm{\\cal{O}}_X)=h^1(\\omega_X)=0$, this implies that $h^1(\\enm{\\cal{E}})=h^1(\\enm{\\cal{I}}_{Z,X}(a))-1$. \n\n\n\n\n\\section{aCM vector bundle on surfaces in $\\enm{\\mathbb{P}}^3$}\\label{sec3}\nWe always assume that $X\\subset \\enm{\\mathbb{P}}^3$ is a surface of degree $m$, not necessarily smooth. In particular, its dualizing sheaf is $\\omega_X \\cong \\enm{\\cal{O}}_X(m-4)$ and we get $h^2(\\enm{\\cal{O}}_X)=\\binom{m-1}{3}$. We also have $h^0(\\enm{\\cal{O}} _X)=1$ and $h^1(\\enm{\\cal{O}} _X)=0$. \n\n\\begin{lemma}\\label{c2}\nEach line bundle $\\enm{\\cal{O}} _X(t)$ with $t \\in \\enm{\\mathbb{Z}}$, is stable as an $\\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^3}$-sheaf with pure depth $2$.\n\\end{lemma}\n\n\\begin{proof}\nIt is enough to deal with the case $t=0$. Assume the contrary and take a subsheaf $\\enm{\\cal{A}} \\subsetneq \\enm{\\cal{O}} _X$ such that $\\enm{\\cal{B}}:=\\enm{\\cal{O}} _X\/\\enm{\\cal{A}}$ has depth $2$ and normalized Hilbert polynomial at least the one of $\\enm{\\cal{O}} _X$. Since $\\enm{\\cal{B}}$ is a quotient of $\\enm{\\cal{O}} _X$ with depth $2$ and $X$ has no embedded component, we get $\\enm{\\cal{B}} \\cong \\enm{\\cal{O}} _T$ for $T$ a union of some of the irreducible components of $X_{\\mathrm{red}}$ with at most the multiplicities appearing in $X$. This implies that $T\\in |\\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^3}(d)|$ for some integer $d$ with $1\\le d < m$. Now the Hilbert polynomial of $\\enm{\\cal{O}} _X$ is \n\\begin{align*}\n\\mathrm{P}_{\\enm{\\cal{O}}_X}(t)&=\\binom{t+3}{3} -\\binom{t-m+3}{3} \\\\\n&= \\left (\\frac{m}{2}\\right) t^2 + \\left ( 2m-\\frac{m^2}{2}\\right) t + \\left(\\frac{m^3}{6} -m^2 + \\frac{11m}{6} \\right). \n\\end{align*}\nSimilarly, we get the Hilbert polynomial $\\mathrm{P}_{\\enm{\\cal{O}}_T}(t)$ of $\\enm{\\cal{O}}_T$ by replacing $m$ in $\\mathrm{P}_{\\enm{\\cal{O}}_X}(t)$ by $d$. Then we see that $p_{\\enm{\\cal{O}}_X}(t)<p_{\\enm{\\cal{O}} _T}(t)$ for $t\\gg 0$, a contradiction.\n\\end{proof}\n\n\\begin{remark}\\label{c3}\nIf either $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}}_X(1)\\rangle$ or $X$ is integral, then every line bundle is stable. Note also that the proof of Lemma \\ref{c2} shows that the ideal sheaf $\\enm{\\cal{I}} _{Z,X}$ for any zero-dimensional subscheme $Z\\subset X$, is also stable. If $X$ is integral, then any sheaf of rank $1$ with positive depth is stable. Thus these sheaves are indecomposable.\n\\end{remark}\n\n\n\n\n\\begin{proposition}\\label{a1.1}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree $m\\ge 2$ with $X_{\\mathrm{reg}} \\ne \\emptyset$. Fix $p \\in X_{\\mathrm{reg}}$, and let $\\enm{\\cal{E}}_p$ be the unique non-trivial extension  \n\\begin{equation}\\label{eqb1}\n0 \\to \\enm{\\cal{O}} _X(m-3) \\to \\enm{\\cal{E}}_p \\to \\enm{\\cal{I}} _{p,X}\\to 0.\n\\end{equation}\nThen $\\enm{\\cal{E}}_p$ is an aCM vector bundle of rank two on $X$ and $\\enm{\\cal{E}} \\ncong \\enm{\\cal{O}} _X(a)\\oplus \\enm{\\cal{O}} _X(b)$ for any $a, b\\in \\enm{\\mathbb{Z}}$. If one of the following holds, then $\\enm{\\cal{E}}$ is indecomposable. \n\\begin{itemize}\n\\item [(i)] $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}} _X(1)\\rangle $,\n\\item [(ii)] $\\enm{\\cal{O}} _X(t)$ for $t\\in \\enm{\\mathbb{Z}}$ are the only aCM line bundles on $X$, or \n\\item [(iii)] $m\\ge 4$ and $X$ is integral.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nBy Proposition \\ref{e1} it remains to deal with indecomposability of $\\enm{\\cal{E}}_p$. First show that there are no integers $a, b$ such that $\\enm{\\cal{E}}_p \\cong \\enm{\\cal{O}} _X(a)\\oplus \\enm{\\cal{O}} _X(b)$. Assume that such $a,b$ exist, say $a\\ge b$. Since $h^0(\\enm{\\cal{E}}_p (3-m)) =1$ and $h^0(\\enm{\\cal{E}}_p (2-m)) =0$, we get $(a,b)=(m-3,0)$ and $m\\ge 3$. Then we get $h^0(\\enm{\\cal{E}}_p) = \\binom{m}{3} +1$, while (\\ref{eqb1}) gives $h^0(\\enm{\\cal{E}}_p )=\\binom{m}{3}$. \n\nNow assume that $\\enm{\\cal{E}}_p$ is decomposable. Since $\\enm{\\cal{E}}_p$ is locally free and it has rank $2$, we have $\\enm{\\cal{E}}_p \\cong \\enm{\\cal{A}}_1 \\oplus \\enm{\\cal{A}}_2$ with each $\\enm{\\cal{A}}_i\\in \\mathrm{Pic}(X)$. Since $\\enm{\\cal{E}}_p$ is aCM, each $\\enm{\\cal{A}}_i$ is aCM. In cases (i) and (ii) the assertion holds by above. Thus we assume the case (iii). By Lemma \\ref{c2} and Remark \\ref{c3}, (\\ref{eqb1}) is the HN filtration of $\\enm{\\cal{E}}_p$. Applying the functor $\\mathrm{Hom}_X(\\enm{\\cal{E}}_p, -)$ to (\\ref{eqb1}), we get \n$$0\\to \\mathrm{Hom}_X(\\enm{\\cal{E}}_p, \\enm{\\cal{O}}_X(m-3)) \\to \\mathrm{Hom}_X(\\enm{\\cal{E}}_p, \\enm{\\cal{E}}_p) \\to \\mathrm{Hom}_X(\\enm{\\cal{E}}_p, \\enm{\\cal{I}}_{p,X})\\to \\mathrm{Ext}_X^1(\\enm{\\cal{E}}_p, \\enm{\\cal{O}}_X(m-3)).$$\nNote that $\\hom_X(\\enm{\\cal{E}}_p, \\enm{\\cal{O}}_X(m-3))=h^2(\\enm{\\cal{E}}_p(-1))=h^0(\\enm{\\cal{E}}_p)=\\binom{m}{3}$ by Serre's duality. By applying the functor $\\mathrm{Hom}_X(-, \\enm{\\cal{I}}_{p,X})$ to (\\ref{eqb1}), we get \n$$\\hom_X(\\enm{\\cal{E}}_p, \\enm{\\cal{I}}_{p,X})=\\hom_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{I}}_{p,X})=1.$$ \nThus we have \n$$\\binom{m}{3}\\le \\hom_X (\\enm{\\cal{E}}_p ,\\enm{\\cal{E}}_p )\\le 1 + \\binom{m}{3}.$$\nSince $h^0(\\enm{\\cal{O}} _X) =1$, we have $\\hom_X (\\enm{\\cal{A}}_i ,\\enm{\\cal{A}}_i ) = 1$ for each $i$. So we get\n$$\\hom_X (\\enm{\\cal{E}}_p ,\\enm{\\cal{E}}_p ) =2+\\hom_X (\\enm{\\cal{A}}_1 ,\\enm{\\cal{A}}_2 )+\\hom_X (\\enm{\\cal{A}}_2,\\enm{\\cal{A}}_1 ).$$ \nSince $X$ is integral, each $\\enm{\\cal{A}}_i$ is stable and we get either $\\enm{\\cal{A}}_1\\cong \\enm{\\cal{A}}_2$ or $\\hom_X (\\enm{\\cal{A}}_i ,\\enm{\\cal{A}}_{3-i}) =0$ for each $i$. In the latter case we have $\\hom_X(\\enm{\\cal{E}}_p, \\enm{\\cal{E}}_p)=2<\\binom{m}{3}$, a contradiction. In the former case, we have $\\mathrm{hom}_X(\\enm{\\cal{E}}_p, \\enm{\\cal{E}}_p)=4$ and the only possibility is $m=4$. But this is also impossible, since we would get $\\enm{\\cal{A}}_1^{\\otimes 2} \\cong \\det (\\enm{\\cal{E}}_p) \\cong \\enm{\\cal{O}}_X(1)$.  \n\\end{proof}\n\n\n\n\\begin{proposition}\\label{z2}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree $m\\ge 2$ and let $Z\\subset X$ be a zero-dimensional subscheme of degree $3$, which is not collinear. Assume that $Z$ is a locally complete intersection inside $X$, i.e. for each $p\\in Z_{\\mathrm{red}}$ the ideal sheaf of $Z$ at $\\enm{\\cal{O}} _{X,p}$ is generated by two elements of $\\enm{\\cal{O}} _{X,p}$. Then there is a vector bundle $\\enm{\\cal{G}}$ of rank two fitting into an exact sequence\n\\begin{equation}\\label{eqz2}\n0 \\to \\enm{\\cal{O}} _X(m-4)\\to \\enm{\\cal{G}} \\to \\enm{\\cal{I}} _{Z,X}\\to 0\n\\end{equation}\nwith $h^1(\\enm{\\cal{G}} (t)) =0$ for all $t\\ne 0$ and $h^1(\\enm{\\cal{G}})=1$. There is also an exact sequence\n\\begin{equation}\\label{eqz3}\n0 \\to \\enm{\\cal{G}} \\to \\enm{\\cal{E}} \\stackrel{u}{\\to} \\enm{\\cal{O}} _X\\to 0,\n\\end{equation}\nwhere $\\enm{\\cal{E}}$ is an aCM vector bundle of rank three such that $\\enm{\\cal{E}} \\ncong \\enm{\\cal{O}} _X(a_1)\\oplus\n\\enm{\\cal{O}} _X(a_2)\\oplus \\enm{\\cal{O}} _X(a_3)$ for any $(a_1, a_2, a_3)\\in \\enm{\\mathbb{Z}}^{\\oplus 3}$. Moreover, if $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}}_X(1) \\rangle$, then $\\enm{\\cal{E}}$ is indecomposable. \n\\end{proposition}\n\n\\begin{proof}\nSince $\\omega _X\\cong \\enm{\\cal{O}} _X(m-4)$, we have $h^0(\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X)=1$ and $\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X$ is globally generated. Since $\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X$ is globally generated, we have $h^0(\\enm{\\cal{I}} _{p,X}\\otimes \\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X)=0$ for all $p\\in Z_{\\mathrm{red}}$. Since $Z$ is a locally complete intersection, the Cayley-Bacharach condition is satisfied and so there is a locally free $\\enm{\\cal{G}}$ fitting into (\\ref{eqz2}); see \\cite{cat}. From (\\ref{eqz2}) we immediately get $h^1(\\enm{\\cal{G}} (t)) =0$ for all $t>0$, because $Z$ is not collinear. Note that $\\det (\\enm{\\cal{G}} ) \\cong \\enm{\\cal{O}} _X(m-4)$ and $\\enm{\\cal{G}}$ is a vector bundle of rank two. This implies $\\enm{\\cal{G}} ^\\vee \\cong \\enm{\\cal{G}} (4-m)$. For $t<0$, we have $h^1(\\enm{\\cal{G}} (t)) = h^1(\\enm{\\cal{G}} ^\\vee (-t)\\otimes \\omega _X) = h^1(\\enm{\\cal{G}} (-t)) =0$ by Serre's duality. Now consider the coboudnary map $\\delta _1: H^1(\\enm{\\cal{I}} _{Z,X}) \\rightarrow H^2(\\enm{\\cal{O}} _X(m-4))\\cong \\mathbf{k}$ with $\\mathrm{ker}(\\delta _1) =H^1(\\enm{\\cal{G}})$. The dual of $\\delta_1$ is the map \n$$\\mathrm{Hom}_X(\\enm{\\cal{O}}_X(m-4), \\enm{\\cal{O}}_X(m-4)) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{Z,X}, \\enm{\\cal{O}}_X(m-4)) $$\nsending the identity map to the element corresponding to $\\enm{\\cal{G}}$. This implies that $\\delta_1$ is surjective and $h^1(\\enm{\\cal{G}})=1$. \n\nNow we apply Lemma \\ref{z1} to $\\enm{\\cal{G}}$ to obtain an aCM vector bundle $\\enm{\\cal{E}}$ of rank three fitting into (\\ref{eqz3}). Since $h^1(\\enm{\\cal{G}} )=1$ and $h^1(\\enm{\\cal{E}} )=0$, (\\ref{eqz2}) and (\\ref{eqz3}) give $h^0(\\enm{\\cal{E}} )=h^0(\\enm{\\cal{G}} )=\\binom{m-1}{3}$. Assume the existence of integers $a_1\\ge a_2\\ge a_3$ such that $\\enm{\\cal{E}} \\cong \\oplus_{i=1}^3 \\enm{\\cal{O}} _X(a_i)$. Since $\\det (\\enm{\\cal{E}} )\\cong \\enm{\\cal{O}} _X(m-4)$, we have $a_1+a_2+a_3 =m-4$. If $2 \\le m\\le 3$, then we have $a_1\\ge 0$ from $a_1+a_2+a_3 =m-4$. This implies that $h^0(\\enm{\\cal{O}} _X(a_1)) > 0 =\\binom{m-1}{3}=h^0(\\enm{\\cal{E}})$, a contradiction. If $m=4$, then we have $h^0(\\enm{\\cal{E}} )=1$. Since $a_1+a_2+a_3 =0$, we have $\\sum _{i=1}^{3} h^0(\\enm{\\cal{O}} _X(a_i)) >1$, a contradiction. Finally assume $m>4$. From (\\ref{eqz2}) and (\\ref{eqz3}) we see that $\\enm{\\cal{O}} _X(m-2)$ is the first non-trivial sheaf in the HN filtration of $\\enm{\\cal{E}}$. Thus $a_1 =m-4$ and $h^0(\\enm{\\cal{O}} _X(a_1)) =\\binom{m-1}{3}$. Since $a_2+a_3=0$, we have $h^0(\\enm{\\cal{O}} _X(a_2)) >0$ and so $h^0(\\enm{\\cal{E}} )> \\binom{m-1}{3}$, a contradiction. Hence we get $\\enm{\\cal{E}} \\ncong \\oplus_{i=1}^3\\enm{\\cal{O}}_X(a_i)$ for any triple of integers $(a_1, a_2, a_3)$. \n\nIt remains to show the last assertion. Assume $\\mathrm{Pic}(X) \\cong\\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}} _X(1)\\rangle$ and that $\\enm{\\cal{E}}$ is decomposable; by the previous assertion we have $\\enm{\\cal{E}} \\cong \\enm{\\cal{A}}_1 \\oplus \\enm{\\cal{A}}_2$ with $\\mathrm{rank}(\\enm{\\cal{A}}_i ) =i$ for each $i$ and $\\enm{\\cal{A}}_2$ indecomposable. Set $\\enm{\\cal{A}}_1 \\cong \\enm{\\cal{O}} _X(a)$ for $a\\in \\enm{\\mathbb{Z}}$. Since $h^0(\\enm{\\cal{E}} )=\\binom{m-1}{3}$, we have $a\\le m-4$. From (\\ref{eqz2}) and (\\ref{eqz3}) we get the existence of a subsheaf $\\enm{\\cal{F}} \\subset \\enm{\\cal{E}}$ such that $\\enm{\\cal{F}} \\cong \\enm{\\cal{O}} _X(m-4)$ and $\\enm{\\cal{E}} \/\\enm{\\cal{F}}$ is an extension $\\enm{\\cal{H}}$ of $\\enm{\\cal{O}} _X$ by $\\enm{\\cal{I}} _{Z,X}$. Note that $\\enm{\\cal{H}}$ is not locally free, because $\\enm{\\cal{I}} _{Z,X}$ has not depth $2$. In particular, $\\enm{\\cal{H}}$ is not isomorphic to $\\enm{\\cal{A}}_2$ and we get $\\enm{\\cal{A}}_1 \\ncong \\enm{\\cal{F}}$. So we have $a<m-4$. Now consider a restriction map\n$$u_{|\\{0\\}\\oplus \\enm{\\cal{A}}_2} : \\{0\\}\\oplus \\enm{\\cal{A}}_2 \\to \\enm{\\cal{O}}_X.$$\nIf this restriction map is surjective, then its kernel is a line bundle, say $\\enm{\\cal{O}}_X(b)$. Since $X$ is aCM, we get $\\enm{\\cal{A}}_2 \\cong \\enm{\\cal{O}}_X\\oplus \\enm{\\cal{O}}_X(b)$, a contradiction. Thus the restriction map is not surjecitve and so the other restriction map $u_{|\\enm{\\cal{A}}_1 \\oplus \\{0\\}} $ is not zero. In particular, we get $a\\le 0$. If $a=0$, then we have $\\enm{\\cal{A}}_1 \\cong \\enm{\\cal{O}}_X$ and the map $u_{|\\enm{\\cal{A}}_1 \\oplus \\{0\\}}$ is an isomorphism. Thus (\\ref{eqz3}) splits and we get $h^1(\\enm{\\cal{E}} )\\ge h^1(\\enm{\\cal{G}} )>0$, a contradiction. Hence we get $a<0$. Since there is no nonzero map $\\enm{\\cal{F}} \\rightarrow \\enm{\\cal{A}}_1$ from $a<m-4$, $\\enm{\\cal{F}}$ is isomorphic to a subsheaf $\\enm{\\cal{F}}_1$ of $\\enm{\\cal{A}}_2$ and we get $\\enm{\\cal{H}} \\cong \\enm{\\cal{O}}_X(a)\\oplus \\enm{\\cal{A}}_2\/\\enm{\\cal{F}}_1$. From $a<0$ we see that there is no nonzero map $\\enm{\\cal{I}}_{Z,X} \\rightarrow \\enm{\\cal{O}}_X(a)$. Since $\\enm{\\cal{H}}$ is an extension of $\\enm{\\cal{O}}_X$ by $\\enm{\\cal{I}}_{Z,X}$, we get that $\\enm{\\cal{I}}_{Z,X} \\cong \\enm{\\cal{A}}_2\/\\enm{\\cal{F}}_1$ and so $\\enm{\\cal{O}}_X(a)\\cong \\enm{\\cal{O}}_X$, a contradiction. \n\\end{proof}\n\n\\begin{remark}\nIn case $m=1$, i.e. $X=\\enm{\\mathbb{P}}^2$, we fail in obtaining an indecomposable aCM vector bundle of rank three, using the method in Proposition \\ref{z2}. Indeed, we get $\\enm{\\cal{G}} \\cong \\Omega_{\\enm{\\mathbb{P}}^2}^1$ and the corresponding  vector bundle of rank three is $\\enm{\\cal{E}} \\cong \\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^2}(-1)^{\\oplus 3}$.\n\\end{remark}\n\n\\begin{corollary}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be union of multiple planes in which at least one plane occurs with multiplicity $1$. Then there is an indecomposable aCM vector bundle of rank three on $X$. If $m>4$, we have a family of such aCM vector bundles of dimension $6$.\n\\end{corollary}\n\n\\begin{proof}\nAssume that $X$ has one component $H$ with multiplicity $1$. In this case we take as $Z$ a set of $3$ general points in $H$. Then the first assertion follows from Proposition \\ref{z2}. Note that the set of all such $Z$ has dimension $6$. Now assume that $X$ has a component $H$ with multiplicity $3$. Fix a general point $p\\in H$ and take a general line $L\\subset \\enm{\\mathbb{P}}^3$ with $p\\in L$. Then set $Z$ to be the connected component of the scheme $X\\cap L$ with $p$ as its reduction. Then we may get the assertion from Proposition \\ref{z2} and that $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}} _X(1)\\rangle$ by \\cite[Lemma 2.5]{BHMP}. \n\\end{proof}\n\n\\begin{proposition}\\label{z2+0}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree $m\\ge 4$ with an irreducible component $Y$ appearing with multiplicity $2$ in $X$. Fix $p\\in Y_{\\mathrm{reg}}$ so that $T$ is the only irreducible component of $X$ containing $p$. For a general line $L\\subset \\enm{\\mathbb{P}}^3$ containing $p$, let $Z\\subset X$ be the connected component of $L\\cap X$ with $p$ as its reduction. We have $\\deg (Z)=2$ and there is an aCM  vector bundle $\\enm{\\cal{E}}_Z$ of rank two fitting into an exact sequence\n\\begin{equation}\\label{eqz2+0}\n0 \\to \\enm{\\cal{O}} _X(m-4)\\to \\enm{\\cal{E}} _Z\\to \\enm{\\cal{I}} _{Z,X}\\to 0.\n\\end{equation}\nThe set of all isomorphism classes of $\\enm{\\cal{E}} _Z$ is uniquely parametrized by a $4$-dimensional irreducible quasi-projective variety $\\Delta$ satisfying the following. \n\\begin{itemize}\n\\item [(i)] For any $\\enm{\\cal{E}} _Z\\in \\Delta$, there are no integers $a, b$ with $\\enm{\\cal{E}} _Z \\cong \\enm{\\cal{O}} _X(a)\\oplus \\enm{\\cal{O}} _X(b)$.\n\\item [(ii)] A very general $\\enm{\\cal{E}} _Z\\in \\Delta$ is indecomposable.\n\\item [(iii)] If $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}} _X(1)\\rangle $, then each $\\enm{\\cal{E}} _Z\\in \\Delta$ is indecomposable.\n\\item [(iv)] If $\\enm{\\mathbb{Z}} \\langle \\enm{\\cal{O}} _X(1)\\rangle $ are the only aCM line bundles on $X$, then each $\\enm{\\cal{E}} _Z\\in \\Delta$ is indecomposable.\n\\end{itemize} \n\\end{proposition}\n\n\\begin{proof}\nSince no other component of $X$ than $Y$ contains $p$ and $p$ is a smooth point of $X$, we have $\\deg (Z)=2$; it is sufficient to take as $L$ any line through $p$ not contained in the tangent plane $T_pY$ of $Y$ at $p$. \n\nSince $\\omega _X\\cong \\enm{\\cal{O}} _X(m-4)$, we have $h^0(\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X)=1$ and $\\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X$ is globally generated. Thus we have $h^0(\\enm{\\cal{I}} _{p,X}\\otimes \\enm{\\cal{O}} _X(4-m)\\otimes \\omega _X)=0$. Since $Z$ is a locally complete intersection, the Cayley-Bacharach condition is satisfied for (\\ref{eqz2+0}) and so there is a locally free $\\enm{\\cal{E}}_Z$ fitting into (\\ref{eqz2+0}); see \\cite{cat}. \n\nSince $\\enm{\\cal{O}} _X(1)$ is very ample and $\\deg (Z) =2$, we get $h^1(\\enm{\\cal{E}}_Z (t)) =0$ for all $t>0$ by (\\ref{eqz2}). Note that $\\det (\\enm{\\cal{E}}_Z ) \\cong \\enm{\\cal{O}} _X(m-4)$ and $\\enm{\\cal{E}}_Z$ is a vector bundle of rank two. This implies $\\enm{\\cal{E}}_Z ^\\vee \\cong \\enm{\\cal{E}}_Z (4-m)$. For $t<0$, we have $h^1(\\enm{\\cal{E}}_Z (t)) = h^1(\\enm{\\cal{E}}_Z ^\\vee (m-t-4)) = h^1(\\enm{\\cal{E}}_Z (-t)) =0$ by Serre's duality. Now consider the coboudary map $\\delta _1: H^1(\\enm{\\cal{I}} _{Z,X}) \\rightarrow H^2(\\enm{\\cal{O}} _X(m-4))\\cong \\mathbf{k}$ with $\\mathrm{ker}(\\delta _1) =H^1(\\enm{\\cal{E}}_Z)$. The dual of $\\delta_1$ is the map \n$$\\mathrm{Hom}_X(\\enm{\\cal{O}}_X(m-4), \\enm{\\cal{O}}_X(m-4)) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{Z,X}, \\enm{\\cal{O}}_X(m-4)) $$\nsending the identity map to the element corresponding to $\\enm{\\cal{E}}_Z$. This implies that $\\delta_1$ is non-zero and hence and $h^1(\\enm{\\cal{E}}_Z)=0$. Thus $\\enm{\\cal{E}}_Z$ is aCM.\n\nThe set of all $p\\in Y_{\\mathrm{reg}}$ such that $Y$ is the only irreducible component of $X$ containing $p$ is an irreducible $2$-dimensional variety $\\Delta '$. For each $p\\in \\enm{\\mathbb{P}}^3$ the set of all lines through $p$ is a $\\enm{\\mathbb{P}}^2$. Define a variety $\\Delta$ as follows:\n$$\\Delta:=\\{(p,L)~|~p\\in \\Delta' \\text{ and }L \\text{ a line in }\\enm{\\mathbb{P}}^3 \\text{ with }p\\in L \\text{ and } L\\nsubseteq T_pY\\}.$$\nSince $m\\ge 4$, we have $h^0(\\enm{\\cal{I}} _{Z,X}(4-m)) =0$. Thus (\\ref{eqz2+0}) gives $h^0(\\enm{\\cal{E}} _Z(4-m)) =1$. Thus the isomorphism classes of $\\enm{\\cal{E}} _Z$ uniquely determines $Z$, i.e. if $\\enm{\\cal{E}} _Z\\ncong \\enm{\\cal{E}} _{Z'}$, then we get $Z\\ne Z'$. For two elements $(p_1, L_1), (p_2, L_2)\\in \\Delta$, let $Z_i$ be the subscheme of degree $2$ determined by $(p_i, L_i)$ for each $i=1,2$. Since each $p_i$ is the reduction of $Z_i$ and $L_i$ is the line spanned by $Z_i$, the variety $\\Delta$ uniquely parametrizes the isomorphism classes of the aCM vector bundles $\\enm{\\cal{E}}_Z$.\n\nAssume $\\enm{\\cal{E}}_Z \\cong \\enm{\\cal{O}} _X(a)\\oplus \\enm{\\cal{O}} _X(b)$ for some integers $a, b$ with $a\\ge b$. Since $\\det (\\enm{\\cal{E}}_Z ) \\cong \\enm{\\cal{O}} _X(m-4)$, we have $b= m-4-a$. But since $h^0(\\enm{\\cal{E}}_Z (4-m)) =1$, the only possibility is that $a=4-m$ and $b<0$, a contradiction. Thus we get (i). We may get (ii) as in the proof of Theorem \\ref{e2}. Now assume that $\\enm{\\cal{E}}_Z$ is decomposable, say $\\enm{\\cal{E}}_Z \\cong \\enm{\\cal{A}}_1 \\oplus \\enm{\\cal{A}}_2$ with each $\\enm{\\cal{A}}_i$ a line bundle. Since $\\enm{\\cal{E}}_Z$ is aCM, each $\\enm{\\cal{A}}_i$ is also aCM. Thus (iii) and (iv) follow from (i).\n\\end{proof}\n\n\\begin{remark}\nIn case $m=2$, i.e. $X=2H$ the double plane with a hyperplane $H\\subset \\enm{\\mathbb{P}}^3$, the vector bundle $\\enm{\\cal{E}} _Z$ described in Proposition \\ref{z2+0} is the vector bundle $\\enm{\\cal{O}} _X(-1)^{\\oplus 2}$.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{theorem}\\label{i1}\nLet $X\\subset \\enm{\\mathbb{P}}^3$ be a surface of degree $m\\ge 4$ with $X_{\\mathrm{reg}} \\ne \\emptyset$, i.e. $X$ has an irreducible component $Y$ appearing with multiplicity $1$.  We further assume that either $\\mathrm{Pic}(X) =\\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}} _X(1)\\rangle $ or $X$ is integral. For a fixed integer $s>0$ and a set $S\\subset X_{\\mathrm{reg}}\\cap Y$ with $\\sharp (S)=s$, a general sheaf $\\enm{\\cal{E}} _S$ fitting into an exact sequence\n\\begin{equation}\\label{eqi1}\n0 \\to \\enm{\\cal{O}} _X(m-3)^{\\oplus s} \\stackrel{v}{\\to} \\enm{\\cal{E}} _S \\to \\oplus _{p\\in S}\\enm{\\cal{I}}_{p,X}\\to 0, \n\\end{equation}\nis a locally free, indecomposable and aCM sheaf of rank $2s$. Moreover, if $S'\\subset X_{\\mathrm{reg}}\\cap Y$ is another set with $\\sharp (S')=s$ and $S'\\ne S$, then we have $\\enm{\\cal{E}} _{S'}\\ncong \\enm{\\cal{E}} _S$.\n\\end{theorem}\n\n\nWe have $\\mathrm{ext}_X^1(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X(m-3))=h^1(\\enm{\\cal{I}}_{p,X}(-1))=1$ for each $p\\in X_{\\mathrm{reg}}$ by Serre's duality. So the extension $\\enm{\\cal{E}}_S$ corresponds to an element in a finite dimensional vector space\n$$\\mathbb{E}(S):=\\mathrm{Ext}_X^1(\\oplus_{p\\in S}\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X(m-3)^{\\oplus s})\\cong \\mathbf{k}^{s^2}.$$\nIf $s=1$, say $S=\\{p\\}$, the dimension of $\\mathbb{E}$ is one. Thus there exists a unique non-trivial extension. Denote this non-trivial extension simply by $\\enm{\\cal{E}}_p$. \n\nIn Theorem \\ref{i1}, a ``general'' choice of $\\enm{\\cal{E}}_S$ means that there exists a non-empty Zariski open subset $\\mathbb{U}\\subset \\mathbb{E}(S)$ such that the middle term of any extension in $\\mathbb{U}$ is aCM, locally free and indecomposable.\n\n\\begin{proof}[Proof of Theorem \\ref{thth}:]\nThe family $\\Sigma$ of all $S\\subset X_{\\mathrm{reg}}$ with $\\sharp (S) =s$ clearly has dimension $2s$. By Theorem \\ref{i1}, if $S$ and $S'$ are two distinct sets in $\\Sigma$, then we get $\\enm{\\cal{E}}_{S} \\ncong \\enm{\\cal{E}}_{S'}$. Now there is a universal family on any $\\mathrm{Ext}^1$-group of families of sheaves with $\\Sigma \\times X$ as its base. Thus, we get a family of aCM locally free and indecomposable vector bundles with as a parameter space a rank $s^2$ vector bundle over $\\Sigma$; the fibre of this vector bundle over $S\\in \\Sigma$ is $\\mathbb{E}(S)$, corresponding to $S$. Taking a non-empty open subset $V$ of $\\Sigma$ on which this vector bundle is trivial we get a family of pairwise non-isomorphic sheaves, at least if we restrict $V$, so that all sheaves in the family are locally free, aCM and indecomposable.\n\\end{proof}\n\n\\begin{remark}\nFor a surface $X$ as in Theorem \\ref{i1} and Theorem \\ref{thth}, the algebraic group $\\mathrm{Aut}(X)$ has finite dimension; it is often zero-dimensional. Hence there exists an integer $t_0$ such that for every even integer $r$, $X$ has a family of dimension at least $r-t_0$, consisting of indecomposable aCM vector bundles of rank $r$ on $X$, such that for any two distinct elements $\\enm{\\cal{E}}$, $\\enm{\\cal{E}}'$ in the family there is no $f\\in \\mathrm{Aut}(X)$ with $f^\\ast (\\enm{\\cal{E}} )\\cong \\enm{\\cal{E}}'$.\n\\end{remark}\n\n\n\\section{Proof of Theorem \\ref{i1}}\\label{sec4}\n\nSet $\\mathbb{E}'(S)$ to be the set of all elements in $\\mathbb{E}(S)$ whose corresponding middle term is locally free and aCM.\n\n\\begin{lemma}\\label{x01}\n$\\mathbb{E}'(S)$ is a non-empty open subset of $\\mathbb{E}(S)$.\n\\end{lemma}\n\n\\begin{proof}\nSince being locally free and aCM are both open properties in a flat family, $\\mathbb{E}'(S)$ is an open subset of $\\mathbb{E}(S)$. Thus it is sufficient to prove that $\\mathbb{E}'(S)  \\ne \\emptyset$. Proposition \\ref{a1.1} gives the case $s=1$. For $s>1$, we may find a direct sum of aCM vector bundles of rank two fitting into (\\ref{eqi1}), i.e. take $\\oplus _{p\\in S} \\enm{\\cal{E}}_p$. This implies $\\mathbb{E}'(S) \\ne \\emptyset$.\n\\end{proof}\n\n\\begin{remark}\\label{x00} \nIn the set-up of (\\ref{eqi1}) set $\\enm{\\cal{A}} := v(\\enm{\\cal{O}} _X(m-3)^{\\oplus s})$. By Lemma \\ref{c2} and Remark \\ref{c3} together with the assumption $m\\ge 3$, we see that $\\enm{\\cal{A}}$ is the first term of the HN filtration of $\\enm{\\cal{E}}_S$. Thus we get $f(\\enm{\\cal{A}})\\subseteq \\enm{\\cal{A}}$ for any $f\\in \\op{End} (\\enm{\\cal{E}}_S )$.\n\\end{remark}\n\n\\begin{lemma}\\label{x0}\nIf $\\enm{\\cal{E}}$ is the middle term of an extension $\\epsilon \\in \\mathbb{E}'(S)$, then $\\enm{\\cal{E}}$ has no line bundle as a factor.\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\enm{\\cal{L}}$ is a line bundle that is a factor of $\\enm{\\cal{E}}$, i.e. $\\enm{\\cal{E}} = \\enm{\\cal{L}} \\oplus \\enm{\\cal{G}}$ for some aCM vector bundle $\\enm{\\cal{G}}$ of rank $2s-1$. Since $m\\ge 3$, we have \n$$h^0(\\enm{\\cal{L}} (3-m)) +h^0(\\enm{\\cal{G}} (3-m)) =h^0(\\enm{\\cal{E}}(3-m))=s.$$ \nFirst assume $h^0(\\enm{\\cal{L}} (3-m)) =0$ and $h^0(\\enm{\\cal{G}} (3-m)) =s$. Then we have $v(\\enm{\\cal{O}}_X(m-3)^{\\oplus s}) \\subset \\{0\\}\\oplus \\enm{\\cal{G}}$ in (\\ref{eqi1}) and so $\\enm{\\cal{L}} \\cong \\enm{\\cal{I}}_{p,X}$ for some $p\\in S$, a contradiction. Thus we have $h^0(\\enm{\\cal{L}} (3-m)) >0$ and so $h^0(\\enm{\\cal{G}} (3-m)) <s$. In particular, there is a nonzero map $u: \\enm{\\cal{O}} _X(m-3) \\rightarrow \\enm{\\cal{L}}$. Assume for the moment that $\\mathrm{Pic}(X) \\cong \\enm{\\mathbb{Z}}\\langle \\enm{\\cal{O}} _X(1)\\rangle$ and write $\\enm{\\cal{L}} \\cong \\enm{\\cal{O}} _X(a)$ for some $a\\in \\enm{\\mathbb{Z}}$. The map $u$ gives $a\\ge m-3$. Since $m\\ge 3$, (\\ref{eqi1}) is the HN-filtration of $\\enm{\\cal{E}}$ and we get $a=m-3$. Thus $\\enm{\\cal{G}}$ fits into an exact sequence\n$$0\\to \\enm{\\cal{O}}_X(m-3)^{\\oplus (s-1)} \\to \\enm{\\cal{G}} \\to \\oplus _{p\\in S}\\enm{\\cal{I}} _{p,X}\\to 0. $$ \nThen we get $h^1(\\enm{\\cal{G}}(-1))\\ge 1$ from $h^1(\\enm{\\cal{I}}_{p,X}(-1))=1$ and $h^2(\\enm{\\cal{O}}_X(m-4))=1$. Thus $\\enm{\\cal{G}}$ is not aCM, a contradiction. If $X$ is integral, then every line bundle is stable and so (\\ref{eqi1}) is the HN-filtration of $\\enm{\\cal{E}}$, we get either $\\enm{\\cal{L}} \\cong \\enm{\\cal{O}} _X(m-3)$; we get a contradiction as above, or $\\enm{\\cal{L}} $ is a factor of $\\oplus _{p\\in S} \\enm{\\cal{I}} _{p,X}$, which is not locally free, a contradiction.\n\\end{proof}\n\n\n\n\nLet $\\enm{\\mathbb{F}} (S)$ (resp. $\\enm{\\mathbb{F}} '(S)$) be the set of isomorphism classes of middle terms of extensions in $\\enm{\\mathbb{E}} (S)$ (resp. $\\enm{\\mathbb{E}} '(S)$). Let us denote by $\\enm{\\cal{E}}=\\enm{\\cal{E}}(\\epsilon)$ the middle term of the extension corresponding to $\\epsilon \\in \\enm{\\mathbb{E}}'(S)$.\n\n\\begin{lemma}\\label{x2}\nFor two non-empty finite sets $S_1, S_2\\subset X_{\\mathrm{reg}}$ with $\\sharp (S_i)=s_i$, take $\\enm{\\cal{E}}_i\\in \\enm{\\mathbb{F}}'(S_i)$ and call $\\enm{\\cal{A}}_i$ the subsheaf of $\\enm{\\cal{E}}_i$ isomorphic to $\\enm{\\cal{O}}_X(m-3)^{\\oplus s_i}$ for each $i=1,2$. If there exists a map $f: \\enm{\\cal{E}}_1 \\rightarrow \\enm{\\cal{E}}_2$ with $f(\\enm{\\cal{E}}_1) \\subset \\enm{\\cal{A}}_2$, then we have $S_1 \\cap S_2\\ne \\emptyset$. \n\\end{lemma}\n\n\\begin{proof}\nSince $\\op{Hom}_X (\\enm{\\cal{O}} _X(m-3),\\enm{\\cal{I}} _{p,X}) =0$ for all $p\\in X$, we have $f(\\enm{\\cal{A}} _1)\\subseteq \\enm{\\cal{A}} _2$. In particular, $f$ induces a nonzero map $\\tilde{f}: \\oplus _{p\\in S_1}\\enm{\\cal{I}} _{p,X} \\rightarrow \\oplus _{q\\in S_2} \\enm{\\cal{I}} _{q,X}$. This implies that $S_1\\cap S_2 \\ne \\emptyset$.\n\\end{proof}\n\n\\begin{lemma}\\label{x1}\nAssume that $\\enm{\\cal{E}}\\in \\enm{\\mathbb{F}}'(S)$ is decomposable; $\\enm{\\cal{E}} \\cong  \\enm{\\cal{E}} _1\\oplus \\cdots \\oplus \\enm{\\cal{E}} _h$ with each $\\enm{\\cal{E}} _i$ indecomposable. Then there is a partition $S = \\sqcup _{i=1}^{h} S_i$ with $\\enm{\\cal{E}} _i\\in \\enm{\\mathbb{F}} '(S_i)$ for each $i$. If there is another decomposition $\\enm{\\cal{E}} \\cong \\enm{\\cal{E}} _1'\\oplus \\cdots \\oplus \\enm{\\cal{E}} _k'$ with each $\\enm{\\cal{E}}_j'$ indecomposable, then we get $k=h$ and there is a permutation $\\sigma : \\{1,\\dots ,h\\}\\rightarrow\\{1,\\dots ,h\\}$ such that $\\enm{\\cal{E}} _{\\sigma (i)}' \\cong \\enm{\\cal{E}} _i$ for all $i$ and $\\enm{\\cal{E}} _{\\sigma (i)}'\\in \\enm{\\mathbb{F}} (S_{\\sigma\n(i)})$.\n\\end{lemma}\n\n\\begin{proof}\nWe use induction on $s$. The case $s=1$ is true, because each $\\enm{\\cal{E}} _p$ for $p\\in X_{\\mathrm{reg}}$ is indecomposable by Proposition \\ref{a1.1}. Since $\\enm{\\cal{E}}$ is aCM by the definition of $\\enm{\\mathbb{F}} (S)$, each $\\enm{\\cal{E}} _i$ is also aCM. We consider the subsheaf $\\enm{\\cal{A}} \\cong \\enm{\\cal{O}}_X(m-3)^{\\oplus s}\\subset \\enm{\\cal{E}}$ as in Remark \\ref{x00} and set $\\enm{\\cal{G}} _i:= \\enm{\\cal{A}} \\cap \\enm{\\cal{E}}_i$. Since the HN filtration of $\\enm{\\cal{E}}$ is obtained from the ones of each factors, we have \n$$\\enm{\\cal{A}} \\cong \\oplus _{i=1}^{h} \\enm{\\cal{G}} _i \\phantom{AA}\\text{and}\\phantom{AA}\\oplus _{p\\in S}\\enm{\\cal{I}}_{p,X}\\cong \\oplus_{i=1}^h \\enm{\\cal{E}} _i\/\\enm{\\cal{G}} _i.$$\nBy Lemma \\ref{x0} we have $\\enm{\\cal{G}} _i\\subsetneq \\enm{\\cal{E}} _i$ for all $i$. By Remark \\ref{c3} we may write $S = \\sqcup _{i=1}^{h} S_i $ with $\\enm{\\cal{E}}_i\/\\enm{\\cal{G}} _i\\cong \\oplus _{p\\in S_i}\\enm{\\cal{I}}_{p, X}$. Since $\\enm{\\cal{E}} _i\/\\enm{\\cal{G}} _i \\ne 0$, we have $S_i\\ne \\emptyset$ for all $i$. Thus the set $\\{S_1,\\dots ,S_h\\}$ gives a partition of $S$. To prove the first part of the lemma it is sufficient to prove that $\\sharp (S_i) =\\mathrm{rank}(\\enm{\\cal{G}} _i)\/2$ for all $i$. If this is not true, then there is $i\\in \\{1,\\dots ,h\\}$ with $\\sharp (S_i) > \\mathrm{rank}(\\enm{\\cal{G}} _i)\/2$, i.e. $\\mathrm{rank}(\\enm{\\cal{G}} _i\\cap \\enm{\\cal{A}})> \\sharp (S_i)$. The exact sequence\n$$0 \\to \\enm{\\cal{A}}\\cap \\enm{\\cal{G}} _i\\to \\enm{\\cal{G}} _i\\to \\oplus _{p\\in S_j}\\enm{\\cal{I}} _{p,X}\\to 0$$ \ngives $h^1(\\enm{\\cal{G}} _i)\\ge \\sharp (S_i)-\\mathrm{rank}(\\enm{\\cal{G}} _i\\cap\\enm{\\cal{A}}) >0$. In particular, $\\enm{\\cal{G}} _i$ is not aCM, a contradiction.\n\nNow we check the last assertion of the lemma. Take two partitions \n$$S = S_1\\sqcup \\cdots \\sqcup S_h= S_1'\\sqcup \\cdots \\sqcup S_k'$$ \nsuch that there is a decomposition \n$$\\enm{\\cal{E}} \\cong  \\enm{\\cal{E}} _1\\oplus \\cdots \\oplus \\enm{\\cal{E}} _h \\cong  \\enm{\\cal{E}} _1'\\oplus \\cdots \\oplus \\enm{\\cal{E}} _k'$$ \nwith $\\enm{\\cal{E}}_i\\in \\enm{\\mathbb{F}}'(S_i) $ and $\\enm{\\cal{E}} _j'\\in \\enm{\\mathbb{F}}'(S_j')$ indecomposable. By the Krull-Schmidt theorem in \\cite{Atiyah}, we get $h=k$ and there is a permutation $\\sigma : \\{1,\\dots ,h\\}\\rightarrow \\{1,\\dots ,h\\}$ such that $\\enm{\\cal{B}} _{\\sigma (i)} \\cong \\enm{\\cal{E}} _i$ for all $i$. By renaming $\\{\\enm{\\cal{E}} _1',\\dots ,\\enm{\\cal{E}}_h'\\}$, we may assume that $\\enm{\\cal{E}} _i' \\cong \\enm{\\cal{E}} _i$ for all $i$. This implies\n$$\\sharp (S_i)=\\mathrm{rank}(\\enm{\\cal{E}}_i)\/2=\\mathrm{rank}(\\enm{\\cal{E}}_i')\/2=\\sharp(S_i').$$\nNow fix an isomorphism $f_i: \\enm{\\cal{E}} _i\\rightarrow \\enm{\\cal{E}} _i'$ for each $i$. Since (\\ref{eqi1}) gives the HN filtrations of $\\enm{\\cal{E}} _i$ and $\\enm{\\cal{E}}_i'$, the map $f$ induces an isomorphism $\\tilde{f_i}: \\oplus _{p\\in S_i}\\enm{\\cal{I}} _{p,X}\\rightarrow  \\oplus _{p\\in S_i'} \\enm{\\cal{I}} _{p,X}$. Since $p$ is the unique point of $X$ at which $\\enm{\\cal{I}} _{p,X}$ is not locally free, we get $S_i=S_i'$. For each $i$, let $\\enm{\\cal{A}}_i$ be the unique subsheaf of $\\enm{\\cal{E}} _i$ isomorphic to $\\enm{\\cal{O}} _X(m-3)^{\\sharp (S_i)}$. Then for any embedding $u: \\enm{\\cal{E}} _i \\rightarrow  \\enm{\\cal{E}} _1\\oplus \\cdots \\oplus \\enm{\\cal{E}} _h$, the composition $v_j\\circ \\pi _j \\circ u$\n$$\\enm{\\cal{E}}_i \\stackrel{u}{\\to} \\enm{\\cal{E}}_1\\oplus \\cdots \\oplus \\enm{\\cal{E}}_h \\stackrel{\\pi_j}{\\to} \\enm{\\cal{E}}_j \\stackrel{v_j}{\\to} \\oplus_{p\\in S_j}\\enm{\\cal{I}}_{p,X}$$\nis zero for any $j\\ne i$ by Lemma \\ref{x2}, where $\\pi _j: \\enm{\\cal{E}} \\rightarrow \\enm{\\cal{E}} _j$ is the projection and $v_j: \\enm{\\cal{E}} _j\\rightarrow \\oplus _{p\\in S_j}\\enm{\\cal{I}} _{p,X}$ is the surjection in (\\ref{eqi1}) for $S_j$. Since $u$ is an embedding, we see that $v_i\\circ \\pi _i \\circ u$ is surjective. Thus $\\enm{\\cal{G}}:= \\pi _i(u(\\enm{\\cal{E}} _i))$ is a subsheaf with $v_i(\\enm{\\cal{G}} ) = \\oplus _{p\\in S_i}\\enm{\\cal{I}} _{p,X}$. \n\\end{proof}\n\n\\begin{lemma}\\label{ppo}\nWith the setting as in Theorem \\ref{i1}, we have $\\mathrm{ext}_X^1(\\enm{\\cal{E}}_p, \\enm{\\cal{E}}_q)\\ge 2$ for two points $p,q\\in X_{\\mathrm{reg}}$, possibly $p=q$. \n\\end{lemma}\n\n\\begin{proof}\nSet $\\enm{\\cal{F}} _o:= \\enm{\\cal{E}} _o(3-m)$ for $o\\in \\{p,q\\}$. Since $\\op{Ext} ^i_X(\\enm{\\cal{E}} _p,\\enm{\\cal{E}} _q) \\cong \\op{Ext}_X ^i(\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _q)$, we have $\\chi (\\enm{\\cal{E}} _p\\otimes \\enm{\\cal{E}} _q^\\vee ) = \\chi (\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _q^\\vee)$. Since Euler's characteristic is constant in a flat family of vector bundles and $p,q\\in X_{\\mathrm{reg}}$, it is sufficient to compute $\\chi (\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _q^\\vee)$ when $X$ is smooth. Since a smooth surface in $\\enm{\\mathbb{P}}^3$ is connected, the same observation applied to a family of vector bundles on $X$ shows $\\chi (\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _q^\\vee) =\\chi (\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee)$.\n\nWe have an exact sequence\n\\begin{equation}\\label{eqii1}\n0 \\to \\enm{\\cal{O}} _X \\stackrel{v}{\\to} \\enm{\\cal{F}} _p \\stackrel{w}{\\to} \\enm{\\cal{I}} _{p,X}(3-m)\\to 0\n\\end{equation}\nwith $\\det (\\enm{\\cal{F}} _p)\\cong \\enm{\\cal{O}} _X(3-m)$ and $c_2(\\enm{\\cal{F}} _p)=1$. Since $X\\subset \\enm{\\mathbb{P}}^3$ is a surface of degree $m$, we have $c_1(\\enm{\\cal{F}} _p)^2 = m(m-3)^2$. By Riemann-Roch for $\\mathcal{E}nd (\\enm{\\cal{F}} _p)$, we have \n\\begin{align*}\n\\chi (\\mathcal{E}nd (\\enm{\\cal{F}} _p ))& = c_1(\\enm{\\cal{F}} _p)^2 -4c_2(\\enm{\\cal{F}} _p) + 4\\chi (\\enm{\\cal{O}} _X) = m(m-3)^2 -4 +4\\binom{m-1}{3} +4\\\\\n&=\\frac{1}{6}\\left(10m^3-60m^2+98m-24\\right).\n\\end{align*}\nIn particular, we have $\\chi \\sim \\frac{5}{3}m^3$ for $m\\gg 0$. Note that by Serre's duality we have $h^2(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee )=h^0(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee (m-4))$.\n\n\\quad \\emph{Claim 1:} We have $\\mathrm{hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p) =1+\\binom{m}{3}$.\n\n\\quad \\emph{Proof of Claim 1:} We have $\\mathrm{hom}_X (\\enm{\\cal{I}} _{p,X}(3-m),\\enm{\\cal{O}} _X) =h^0(\\enm{\\cal{O}} _X(m-3)) = \\binom{m}{3}$ and any nonzero map $\\enm{\\cal{I}} _{p,X}(3-m) \\rightarrow \\enm{\\cal{O}} _X$ induces an element in $\\op{Hom}_X(\\enm{\\cal{F}}_p, \\enm{\\cal{F}}_p)$ with rank one as the following composition:\n$$\\enm{\\cal{F}}_p \\stackrel{w}{\\to} \\enm{\\cal{I}}_{p,X}(3-m) \\to \\enm{\\cal{O}}_X \\stackrel{v}{\\to} \\enm{\\cal{F}}_p.$$\nThe vector space $\\op{Hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p)$ also contains the nonzero multiples of the identity map $\\enm{\\cal{F}} _p\\rightarrow \\enm{\\cal{F}} _p$ and these maps have rank two. Thus we get $h^0(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee )\\ge 1+\\binom{m}{3}$. On the other hand, for any $f\\in \\op{Hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p)$ we get $w\\circ f \\circ (v(\\enm{\\cal{O}} _X)) \\subseteq v(\\enm{\\cal{O}} _X)$ from $h^0(\\enm{\\cal{I}} _{p,X}(3-m)) =0$. Thus $w\\circ f\\circ v$ induces a map $f_1: \\enm{\\cal{O}} _X\\rightarrow \\enm{\\cal{O}}_X$, which is induced by the multiplication by $c\\in \\mathbf{k}$. Hence $f-c{\\cdot}\\mathrm{Id} _{\\enm{\\cal{F}} _p}$ is induced by a unique $g\\in \\op{Hom}_X (\\enm{\\cal{I}} _{p,X}(3-m),\\enm{\\cal{F}} _p)$. Since $\\enm{\\cal{F}} _p$ is locally free and $X$ is smooth at $p$, we have $\\op{Hom}_X (\\enm{\\cal{I}} _{p,X}(3-m),\\enm{\\cal{F}} _p) = H^0(\\enm{\\cal{F}} _p(m-3))$. By\n(\\ref{eqii1}) we have $h^0(\\enm{\\cal{F}}_p (m-3)) =\\binom{m}{3}$ and so $\\mathrm{hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p) \\le 1+\\binom{m}{3}$. \\qed\n\n \\quad \\emph{Claim 2:} We have $\\mathrm{hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p(m-4)) \\ge \\binom{2m-4}{3} +2\\binom{m-1}{3} -\\binom{m-4}{3}-1$.\n\n\\quad \\emph{Proof of Claim 2:} For any $f \\in \\op{Hom}_X (\\enm{\\cal{F}} _p, \\enm{\\cal{F}} _p(4-m))$, set $f_1:= f_{|v(\\enm{\\cal{O}} _X)}$. Since $h^0(\\enm{\\cal{O}} _X(-1)) =0$, we have $w\\circ f_1 =0$ and so $f_1(v(\\enm{\\cal{O}} _X)) \\subset v(\\enm{\\cal{O}} _X(m-4)))$. Take $f$ with $f_1\\equiv 0$. Such a map $f$ is uniquely determined by an element in $\\op{Hom}_X (\\enm{\\cal{I}} _{p,X}(3-m), \\enm{\\cal{F}} _p(m-4))$ and the converse also holds. Since $\\enm{\\cal{F}} _p(m-4)$ is locally free and $X$ is smooth at $p$, we have $\\op{Hom}_X (\\enm{\\cal{I}} _{p,X}(3-m), \\enm{\\cal{F}} _p(m-4)) = \\op{Hom}_X (\\enm{\\cal{O}} _X(3-m),\\enm{\\cal{F}} _p(m-4)) = H^0(\\enm{\\cal{F}} _p(2m-7))$. Since $h^1(\\enm{\\cal{O}} _X(t)) =0$ for any $t\\in \\enm{\\mathbb{Z}}$, (\\ref{eqii1}) gives \n$$h^0(\\enm{\\cal{F}} _p(2m-7)) = h^0(\\enm{\\cal{O}} _X(2m-7)) +h^0(\\enm{\\cal{O}} _X(m-4))-1 = \\binom{2m-4}{3} -\\binom{m-4}{3}+\\binom{m-1}{3}-1.$$ \nNote that a map $f$ obtained by a composition\n$$\\enm{\\cal{F}}_p \\stackrel{w}{\\to} \\enm{\\cal{I}}_{p,X}(3-m) \\to \\enm{\\cal{O}}_X(m-4) \\stackrel{v}{\\to} \\enm{\\cal{F}}_p(m-4)$$\nhas $f_1 \\equiv 0$. Now for any linear subspace $W\\subset \\op{Hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p(m-4))$ such that $f_1\\not\\equiv 0$ for any $f\\in W\\setminus \\{0\\}$, we would get \n$$\\mathrm{hom}_X (\\enm{\\cal{F}} _p,\\enm{\\cal{F}} _p(m-4)) \\ge \\binom{2m-4}{3} -\\binom{m-4}{3}+\\binom{m-1}{3}-1+\\dim W.$$\nWe may choose $W$ to consist of the compositions of the identity map $\\enm{\\cal{F}} _p\\rightarrow \\enm{\\cal{F}} _p$ with the multiplication by an element of $H^0(\\enm{\\cal{O}} _X(m-4))$. Then we have $\\dim W=\\binom{m-1}{3}$.   \\qed\n\nCombining Claims 1 and 2, we get\n\\begin{align*}\nh^0(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee) +h^2(\\enm{\\cal{F}} _p\\otimes \\enm{\\cal{F}} _p^\\vee )&\\ge \\binom{2m-4}{3} +\\binom{m}{3}+2\\binom{m-1}{3}-\\binom{m-4}{3}\\\\\n&=\\frac{1}{6}\\left( 10m^3-60m^2+98m-12\\right).\n\\end{align*}\nThus we have \n$$\nh^1(\\enm{\\cal{F}}_p\\otimes \\enm{\\cal{F}}_p^\\vee)=h^0(\\enm{\\cal{F}}_p\\otimes \\enm{\\cal{F}}_p^\\vee)+h^2(\\enm{\\cal{F}}_p\\otimes \\enm{\\cal{F}}_p^\\vee)-\\chi(\\mathcal{E}nd(\\enm{\\cal{F}}_p))\\ge 2\n$$\nand so we get the assertion. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{i1}:]\nBy Remark \\ref{x00} (\\ref{eqi1}) is the HN filtration of $\\enm{\\cal{E}}_S$. Proposition \\ref{a1.1} gives the case $s=1$. For $s>1$, we may find a direct sum of $s$ vector bundles of rank $2$ from the case $s=1$, fitting into (\\ref{eqi1}): just take $\\oplus _{p\\in S} \\enm{\\cal{E}}_p$. So a general extension in $\\mathbb{E}(S)$ has a locally free and aCM middle term, because being local free and aCM are both open conditions. \n\nNote that $h^0(\\enm{\\cal{E}}_S(3-m))=s$ from (\\ref{eqi1}). In particular there is a unique subsheaf $\\enm{\\cal{A}}\\subset \\enm{\\cal{E}}_S$ isomorphic to $\\enm{\\cal{O}} _X(m-3)^{\\oplus s}$ and for each $f\\in \\mathrm{Hom}(\\enm{\\cal{O}} _X(m-3),\\enm{\\cal{E}}_S)$ we have $f(\\enm{\\cal{O}} _X(m-3)) \\subseteq \\enm{\\cal{A}}$. Now by Lemma \\ref{c2} and Remark \\ref{c3}, the extension (\\ref{eqi1}) is the HN filtration of $\\enm{\\cal{E}}_S$. By uniqueness of the HN filtration, we get $\\enm{\\cal{E}}_S \\ncong \\enm{\\cal{E}}_{S'}$ for $S\\ne S'$. \n\nNow it remains to show the indecomposability of $\\enm{\\cal{E}}_S$. By Lemma \\ref{x0}, there is no rank one factor of $\\enm{\\cal{E}}_S$.\n\n\\quad \\emph{Claim 1:} For two distinct points $p, q$ in $X_{\\mathrm{reg}}$, we have \n$$\\mathrm{Hom}_X(\\enm{\\cal{I}} _{p,X},\\enm{\\cal{I}} _{q,X}) =0, \\mathrm{Hom}_X(\\enm{\\cal{E}}_p,\\enm{\\cal{I}} _{q,X}) =0 \\text{ and } \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{p,X} , \\enm{\\cal{I}}_{q,X})=0.$$  \n\n\\quad \\emph{Proof of Claim 1:} By an extension theorem for locally free sheaves in \\cite[Exercise I.3.20]{Hartshorne}, we have $\\mathrm{Hom}_X(\\enm{\\cal{I}} _{p,X},\\enm{\\cal{I}} _{q,X}) = \\mathrm{Hom}_X(\\enm{\\cal{O}} _X,\\enm{\\cal{I}} _{q,X}) =0$. The second vanishing is obtained from the first vanishing and $\\mathrm{Hom}_X(\\enm{\\cal{O}} _X(m-3),\\enm{\\cal{I}} _{q,X}) =0$. For the last vanishing, we apply the functor $\\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, -)$ to the standard exact sequence for $\\enm{\\cal{I}}_{q,X}\\subset \\enm{\\cal{O}}_X$ and obtain an exact sequence\n$$0 \\to \\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X) \\to \\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_q) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{I}}_{q,X}) \\to \\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X)$$\nby the first vanishing in the Claim. Here we have\n$$\\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X) \\cong \\mathrm{Hom}_X(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_q) \\cong \\mathbf{k}$$\nand $\\mathrm{Ext}_X^1(\\enm{\\cal{I}}_{p,X}, \\enm{\\cal{O}}_X) \\cong H^1(\\enm{\\cal{I}}_{p,X}(m-4))^\\vee$ by Serre's duality. Then we get the assertion from the assumption that $m\\ge 4$.  \\qed\n\n\\quad {(a)} First assume $s=2$ and take  two distinct points $p, q$ in $X_{\\mathrm{reg}}$.\n\n\\quad \\emph{Claim 2:} If there exists a sheaf $\\enm{\\cal{G}}\\ncong \\enm{\\cal{E}} _p\\oplus \\enm{\\cal{E}} _q$ fitting into the exact sequence\n\\begin{equation}\\label{eqa3.1}\n0 \\to \\enm{\\cal{E}} _p\\stackrel{u}{\\to} \\enm{\\cal{G}} \\stackrel{v}{\\to} \\enm{\\cal{E}} _q\\to 0,\n\\end{equation}\nthen the case $s=2$ is true.\n\n\\quad \\emph{Proof of Claim 2:} Such a sheaf $\\enm{\\cal{G}}$ would be locally free and aCM with rank $4$. Since $h^1(\\enm{\\cal{O}} _X)=0$ and (\\ref{eqi1}) gives the HN filtrations of $\\enm{\\cal{E}} _p$ and $\\enm{\\cal{E}} _q$ by Lemmas \\ref{c2}\nand Remark \\ref{c3}, $\\enm{\\cal{G}}$ has a subsheaf $\\enm{\\cal{F}} \\cong \\enm{\\cal{O}} _{X}(m-3)^{\\oplus 2}$ such that $\\enm{\\cal{G}} \/\\enm{\\cal{F}}$ is an extension of $\\enm{\\cal{I}} _{q,X}(1)$ by $\\enm{\\cal{I}} _{p,X}(1)$. Claim 1 gives $\\enm{\\cal{G}}\/\\enm{\\cal{F}} \\cong \\enm{\\cal{I}}_{p,X}\\oplus \\enm{\\cal{I}}_{q,X}$ and so we get $\\enm{\\cal{G}} \\cong \\enm{\\cal{E}} _S$ with $S =\\{p,q\\}$.\\qed\n\n\\quad \\emph{Claim 3:} If $\\enm{\\cal{G}} \\cong \\enm{\\cal{E}} _p\\oplus \\enm{\\cal{E}} _q$ for all $\\enm{\\cal{G}}$ in (\\ref{eqa3.1}), then we have $\\op{Ext}_X ^1(\\enm{\\cal{E}} _q,\\enm{\\cal{E}} _p)=0$.\n\n\\quad \\emph{Proof of Claim 3:} Let $\\enm{\\cal{G}} \\cong \\enm{\\cal{E}} _p\\oplus \\enm{\\cal{E}} _q$ fitting into (\\ref{eqa3.1}) correspond to $\\epsilon \\in \\op{Ext}_X ^1(\\enm{\\cal{E}} _q,\\enm{\\cal{E}} _p)$. Then it is sufficient to prove that $\\epsilon =0$, or $\\mathrm{ker}(v) \\cong \\enm{\\cal{E}}_p\\oplus \\{0\\}$. But since $\\mathrm{ker}(v) \\cong \\enm{\\cal{E}} _p$, it is sufficient to prove that either $\\enm{\\cal{E}} _p\\oplus \\{0\\} \\supseteq \\mathrm{ker}(v)$ or $\\enm{\\cal{E}} _p\\oplus \\{0\\} \\subseteq \\mathrm{ker}(v)$. Assume $v(\\enm{\\cal{E}} _p\\oplus \\{0\\}) \\ne 0$. Since $\\mathrm{Hom}_X(\\enm{\\cal{E}}_p,\\enm{\\cal{I}} _{q,X}) =0$ by Claim 1, we have $v(\\enm{\\cal{E}} _p\\oplus \\{0\\})\\subseteq \\enm{\\cal{O}} _X(m-3)$. This implies that the restriction of the surjection $\\enm{\\cal{E}} _q\\rightarrow \\enm{\\cal{I}} _{q,X}$ to $v(\\{0\\}\\oplus \\enm{\\cal{E}} _q)$ is surjective. Since $h^0(\\enm{\\cal{O}} _X)=1$ and $\\op{Hom}_X (\\enm{\\cal{O}} _X(m-3),\\enm{\\cal{I}} _{q,X}) =0$, we get either $v(\\{0\\}\\oplus \\enm{\\cal{O}} _X(m-3)) =0$ or $v$ induces an isomorphism $\\{0\\}\\oplus \\enm{\\cal{O}} _X(m-3)\\rightarrow \\enm{\\cal{O}} _X(m-3)$. Assume for the moment $v(\\{0\\}\\oplus \\enm{\\cal{O}} _X(m-3)) =0$. Since $v(\\enm{\\cal{E}} _p\\oplus \\{0\\})$ maps to $0$ in $\\enm{\\cal{I}} _{q,X}$, we get that $v(\\{0\\}\\oplus \\enm{\\cal{E}} _q)$ is a subsheaf of $\\enm{\\cal{E}} _q$ which maps isomorphically onto $\\enm{\\cal{I}} _{q,X}$. So we get $\\enm{\\cal{E}} _q\\cong \\enm{\\cal{O}} _X(m-3)\\oplus \\enm{\\cal{I}} _{q,X}$, a contradiction. Now assume $v(\\{0\\}\\oplus \\enm{\\cal{O}} _X(m-3)) =\\enm{\\cal{O}} _X(m-3)$. Since $v(\\{0\\}\\oplus \\enm{\\cal{E}} _q)$ maps surjectively onto $\\enm{\\cal{I}} _{q,X}$, the surjection $v$ induces an isomorphism $\\{0\\}\\oplus \\enm{\\cal{E}} _q\\rightarrow \\enm{\\cal{E}} _q$. Hence we get $\\enm{\\cal{E}} _p\\oplus \\{0\\} \\subseteq \\mathrm{ker}(v)$. \\qed\n\n\\noindent Since $\\op{Ext}_X ^1(\\enm{\\cal{E}}_q,\\enm{\\cal{E}} _p) \\ne 0$ by Lemma \\ref{ppo}, Claim 3 concludes the proof of the case $s=2$.\n\n\\quad {(b)} Assume $s>2$ and that Theorem \\ref{i1} holds for smaller numbers. On $\\mathbb{E}(S)$ there is a universal family of extensions, i.e. a coherent sheaf $\\enm{\\cal{V}}$ over $\\mathbb{E}(S)\\times X$ such that\nfor each $\\epsilon \\in \\mathbb{E}(S)$ the sheaf $\\enm{\\cal{V}} _{|\\{\\epsilon \\}\\times X}$ is the middle term $\\enm{\\cal{E}} (\\epsilon )$ of the extension corresponding to $\\epsilon$; in general, if we take $\\enm{\\mathbb{P}}(\\mathbb{E}(S))$ as a parameter space, then no such a universal sheaf exists. We call $\\enm{\\cal{V}} '$ the restriction of of $\\enm{\\cal{V}}$ to $\\enm{\\mathbb{E}} '(S)\\times X$; we thus consider the family of aCM vector bundles induced from the extensions in $\\enm{\\mathbb{E}}'(S)$. \n\nDefine a set $\\Gamma(S)$ as follows:\n$$\\Gamma(S) :=\\left\\{(\\epsilon ,\\phi)~|~\\epsilon \\in \\enm{\\mathbb{E}}' (S) \\text{ and } \\phi \\in \\op{End} (\\enm{\\cal{E}} (\\epsilon )) \\text{ with }\\phi ^2 = \\phi \\right \\}.$$ \nNote that $\\phi$ is a projection of $\\enm{\\cal{E}} (\\epsilon)$ onto a factor of $\\enm{\\cal{E}} (\\epsilon)$, with the exception when $\\phi = \\mathrm{Id}_{\\enm{\\cal{E}} (\\epsilon)}$ or $\\phi \\equiv 0$; if $\\enm{\\cal{E}} (\\epsilon )$ is indecomposable, only $(\\epsilon ,\\mathrm{Id}_{\\enm{\\cal{E}} (\\epsilon)})$ and $(\\epsilon ,0)$ are contained in $\\Gamma(S)$. Indeed, for any vector bundle $\\enm{\\cal{G}}$, there exists a one-to-one correspondence:\n$$\\{ \\phi \\in \\op{End}(\\enm{\\cal{G}}) ~|~ \\phi^2=\\phi\\} \\leftrightarrow \\{ \\text{factors of }\\enm{\\cal{G}}\\}$$ \nvia $\\phi \\mapsto \\mathrm{Im}(\\phi) = \\mathrm{ker}(\\mathrm{Id}_{\\enm{\\cal{G}}} -\\phi)$, with $\\enm{\\cal{G}}$ being associated to $\\mathrm{Id}_{\\enm{\\cal{G}}}$ and $0$ associated to the zero map. Thus $\\enm{\\cal{G}}$ is decomposable if and only if $\\op{End} (\\enm{\\cal{G}} )$ has a non-trivial idempotent. Note that $\\Gamma(S)$ is a closed in the total space of the vector bundle $\\mathcal{H}om(\\enm{\\cal{V}}' , \\enm{\\cal{V}}')$ over $\\enm{\\mathbb{E}} '(S)\\times X$. By Lemma \\ref{x1}, for each $\\enm{\\cal{E}} (\\epsilon )$ there is a unique partition of $S$ associated to any decomposition of $\\enm{\\cal{E}} (\\epsilon )$ with only finitely many indecomposable factors by the Krull-Schmidt theorem in \\cite{Atiyah}. By Lemma \\ref{x1} for each $\\enm{\\cal{E}} \\in \\enm{\\mathbb{F}}' (S)$ each isomorphism class of factors of $\\enm{\\cal{E}}$ corresponds to a unique subset of $S$; $\\enm{\\cal{E}}$ and $0$ correspond to $S$ and $\\emptyset$, respectively. For each $(\\epsilon,\\phi)\\in \\Gamma(S)$, let $S(\\phi)$ be the subset of $S$ associated to $\\mathrm{Im}(\\phi)$ by Lemma \\ref{x1}. Set \n$$\\Gamma _0(S) :=\\left\\{(\\epsilon, \\phi)\\in \\Gamma(S) ~\\big|~ \\phi\\ne 0 \\text{ and } \\phi\\ne \\mathrm{Id}_{|\\enm{\\cal{E}}(\\epsilon)}\\right\\}. $$\nThe goal is to show that $\\Gamma_0(S)$ is not dominant over $\\enm{\\mathbb{F}}(S)$ for a general $S$. \n\nNote that up to now we did not use that $S$ is contained in the same connected component $Y\\cap X_{\\mathrm{reg}}$ of $X_{\\mathrm{reg}}$. In particular the case $s=2$ holds even if $X$ has more than one irreducible components with multiplicity one and the two points of $S$ belong to different connected components of $X_{\\mathrm{reg}}$. \n\nNow we use a monodromy argument, which requires that $S$ is contained in a connected component of $T:=X_{\\mathrm{reg}}\\cap Y$ and that $S$ is general in $Y$. Set $S=\\{p_1, \\ldots, p_s\\}$ and fix an ordering of the points in $S$, along which we get an ordering of the indecomposable factors of the sheaf $\\oplus _{p\\in S} \\enm{\\cal{I}}_{p,X}$. Together with the usual ordering on the factors of $\\mathcal {O}_X(m-3)^{\\oplus s}$, we may see any $\\epsilon \\in \\mathbb{E}(S)$ as an $(s\\times s)$-square matrix, say $\\epsilon =(\\epsilon_{ij})$ with $1\\le i, j\\le s$, where $\\epsilon _{ij}$ is an element of the $1$-dimensional vector space $\\op{Ext}_X ^1(\\enm{\\cal{I}} _{p_j,X},\\enm{\\cal{O}}_X(m-3))$. Note that for a fixed integer $j$, each $\\epsilon_{ij}$ with $i=1,\\dots ,s$, is an element of the same $1$-dimensional vector space. We write $\\enm{\\cal{O}} _X(m-3)^{\\oplus s} = \\enm{\\mathbb{C}} ^s\\otimes \\enm{\\cal{O}} _X(m-3)$. \n\n\\quad \\emph{Claim 4:} $\\enm{\\cal{E}}=\\enm{\\cal{E}}(\\epsilon)$ has two indecomposable factors, one of them being $\\mathrm{Im}(\\phi)$ and the other one being $\\mathrm{ker}(\\phi)$.\n\n\\quad \\emph{Proof of Claim 4:} Since $\\phi^2=\\phi$, we have $\\enm{\\cal{E}} \\cong \\enm{\\cal{F}}_1 \\oplus \\enm{\\cal{F}}_2$ with $\\enm{\\cal{F}} _1:= \\mathrm{Im}(\\phi)$ and $\\enm{\\cal{F}}_2= \\mathrm{ker}(\\phi)$. By the definition of $A$, we get an exact sequence\n\\begin{equation}\\label{eert1}\n0\\to \\enm{\\cal{O}}_X(m-3)^{\\oplus k} \\to \\enm{\\cal{F}}_1 \\to \\oplus_{p\\in A}\\enm{\\cal{I}}_{p,X} \\to 0,\n\\end{equation}\nwith $k:= \\sharp (A)$. Since neither $\\phi \\equiv 0$ nor $\\phi =\\mathrm{Id} _{\\enm{\\cal{E}}}$, we have $0<k<s$. Then by Lemma \\ref{x1} we get an exact sequence\n\\begin{equation}\\label{eert2}\n0\\to \\enm{\\cal{O}}_X(m-3)^{\\oplus (s-k)} \\to \\enm{\\cal{F}}_2 \\to \\oplus_{p\\in S\\setminus A}\\enm{\\cal{I}}_{p,X} \\to 0.\n\\end{equation}\nNow we need to prove that each $\\enm{\\cal{F}}_i$ is indecomposable. By the inductive assumption it is sufficient to prove that $\\enm{\\cal{F}}_1$ and $\\enm{\\cal{F}}_2$ are the middle terms of general extensions (\\ref{eert1}) and (\\ref{eert2}), respectively. Since (\\ref{eqi1}) gives the HN filtration of each $\\enm{\\cal{F}}_i$, there are linear subspaces $V_1,V_2\\subset \\enm{\\mathbb{C}}^s$ such that $\\dim V_1 =k$, $\\dim V_2 =s-k$ and \n$$v(\\enm{\\mathbb{C}} ^s\\otimes \\enm{\\cal{O}} _X(m-3))\\cap \\enm{\\cal{F}}_i = V_i\\otimes \\enm{\\cal{O}} _X(m-3)$$\nfor each $i$. From $\\enm{\\cal{E}} \\cong \\enm{\\cal{F}}_1\\oplus \\enm{\\cal{F}}_2$ we see that $\\enm{\\mathbb{C}}^{s} =V_1\\oplus V_2$. Now we reorder  the points in $S$ so that all points of $A$ are smaller than any points of $S\\setminus A$. Then $\\epsilon$ can be understood as an $(s\\times s)$-square matrix in a block form:\n\\[\n\\epsilon=\n\\left[\n\\begin{array}{c|c}\nB_{11} & B_{12}\\\\\n\\hline\nB_{21} & B_{22}\n\\end{array}\n\\right]\n\\]\nHere the $(k\\times k)$-matrix $B_{11}$ in the upper left corner, is associated to the extension (\\ref{eert1}) and similarly the $((s-k)\\times (s-k))$-matrix $B_{22}$ in the lower right corner, is associated to the extension (\\ref{eert2}). The matrix of $\\epsilon$ also has a $(k\\times (s-k))$-submatrix $B_{12}$ and an $((s-k)\\times k)$-submatrix $B_{21}$. Since $\\epsilon$ is general, all the entries in each $B_{ij}$ are also general. In particular, $B_{11}$ and $B_{22}$ are general and this implies that each  $\\enm{\\cal{F}}_i$ is general. The inductive assumption gives that each $\\enm{\\cal{F}}_i$ is indecomposable.\\qed\n\nAssume that a general $\\enm{\\cal{E}}=\\enm{\\cal{E}}(\\epsilon)$ has two indecomposable factor, i.e. the set $\\Gamma_0(S)$ is dominant over $\\enm{\\mathbb{F}}(S)$. Let $\\Gamma'(S)$ be an irreducible component of $\\Gamma_0(S)$ dominant over $\\enm{\\mathbb{F}} (S)$ and set $A:= S(\\phi)$, where $(\\epsilon ,\\phi )$ is any element of $\\Gamma'(S)$. Now assume that $(\\epsilon ,\\phi)$ is general in $\\Gamma'(S)$ and set $\\enm{\\cal{E}} := \\enm{\\cal{E}} (\\epsilon)$. Note that the subset $A\\subset S$ is invariant as $(\\epsilon, \\phi)$ varies in $\\Gamma_0(S)$, due to the irreducibility of $\\Gamma_0(S)$. Below we find a contradiction under the assumptions that $\\enm{\\cal{E}}$ is decomposable and that $S$ is general in $\\mathrm{Sym}^s(T)$. \n\nLet $\\widetilde{\\Gamma}$ be the set of all triples $(S,\\enm{\\cal{E}} ,\\phi)$ with $S\\in \\mathrm{Sym} ^s(T)$ and $(\\enm{\\cal{E}} ,\\phi )\\in \\Gamma_0 (S)$. Then $\\widetilde{\\Gamma}$ is an algebraic subset whose fibre over $S\\in \\mathrm{Sym}^s(T)$ is $\\Gamma_0(S)$, with a projection map $u: \\widetilde{\\Gamma} \\rightarrow  \\mathrm{Sym} ^s(T)$. If $u$ is not dominant, then it would imply that there exists a $2s$-dimensional family of pairwise not isomorphic indecomposable aCM vector bundles of rank $2s$ on $X$. Thus we may assume that $u$ is dominant. We fix a general $S\\in \\mathrm{Sym} ^s(T)$ and fix an irreducible component $\\Gamma '(S)$ of $\\Gamma (S)$ to which we apply the previous construction with the partition $A\\sqcup (S\\setminus A)$ of $S$ attached to $\\Gamma'(S)$. Let $\\widetilde{\\Gamma}'$ be any irreducible component of $\\widetilde{\\Gamma}$ containing $\\Gamma '(S)$ such that $u_{|\\widetilde{\\Gamma} '}$ is dominant.\n\nLet $\\enm{\\cal{V}}$ denote a non-empty Zariski open subset of $\\mathrm{Sym}^s(T)$ containing $S$ such that for every $T\\in \\enm{\\cal{V}}$ a general $\\enm{\\cal{E}}_T \\in \\enm{\\mathbb{E}} (T)$ has exactly two indecomposable factors, one associated to a subset $F$ of $T$ with $|F| = |A|=k$ and the other one associated to $T\\setminus E$. Now we fix $p\\in A$ and $q\\in S\\setminus A$. Since $Y_{\\mathrm{reg}}$ is a connected manifold and $p, q\\in Y_{\\mathrm{reg}}$, there exists a connected smooth affine curve $U\\subset \\AA^1(\\mathbf{k})$ with a map $\\phi :  U\\rightarrow  Y_{\\mathrm{reg}}$ such that $\\phi (t_0) = p$ and $\\phi (t_1) =q$ for some $t_0, t_1 \\in U$, and $\\phi(U)$ passes no other points of $S$. Similarly we may consider a map $\\phi' : U \\rightarrow Y_{\\mathrm{reg}}$ with $\\phi'(t_1)=p$ and $\\phi'(t_0)=q$ such that $\\phi(t) \\ne \\phi'(t)$ for any $t\\in U$. For each $t\\in U$, set \n$$A_t:= (A\\setminus \\{p\\})\\cup \\{\\phi(t)\\} \\phantom{A}, \\phantom{A}S_t:= (S\\setminus \\{p,q\\}) \\cup \\{\\phi(t) , \\phi'(t)\\},$$\ne.g. $(A_{t_0}, S_{t_0})=(A_{t_1}, S_{t_1})=(A,S)$. Restricting $U$ to an open neighborhood of $\\{t_0, t_1\\}$, we may assume that $S_t\\in \\enm{\\cal{V}}$ for all $t\\in U$. Then for each $t\\in U$ we have a partition $S_t = A_t\\sqcup (S_t\\setminus A_t)$ such that a general $\\enm{\\cal{E}} _{S_t}\\in \\Gamma' (S_t)$ has exactly two indecomposable factors, one associated to $A_t$ and the other associated to $S_t\\setminus A_t$, due to the choice of $\\widetilde{\\Gamma}'$. \n\nWe start from $t=t_0$ and vary $t$ in $U$ to arrive at $t=t_1$, where we have $S_{t_1} =S=A_q\\sqcup (S\\setminus A_q)$ with $A_q=(A\\setminus \\{p\\})\\cup \\{q\\}$. Since $s>2$, we have $\\{A,S\\setminus A\\} \\ne \\{A_q,S\\setminus A_q\\}$, contradicting the assumption that $\\enm{\\cal{E}} _S$ has exactly two indecomposable factors.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Non-locally free aCM sheaf}\\label{sec5}\nIn this section, we let $X\\subset \\enm{\\mathbb{P}}^N$ be a closed subscheme with pure dimension $n$ at least two.  Assume that each local ring $\\enm{\\cal{O}} _{X,x}$ with $x\\in X$, has depth $n$ and that $X$ is aCM with respect\nto $\\enm{\\cal{O}} _X(1)$, i.e. $h^i(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$ and all $1\\le i\\le n-1$. The exact sequence\n$$0\\to \\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(t) \\to \\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^N}(t) \\to \\enm{\\cal{O}} _X(t)\\to 0$$\nshows that $h^i(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(t)) = h^{i-1}(\\enm{\\cal{O}} _X(t))$ for all $i\\ge 2$. Hence we may restate our assumption as $h^1(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(t)) =0$ and $h^i(\\enm{\\cal{O}} _X(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$ and $i=1,\\dots ,n-2$. By a theorem of Serre, the condition that $h^i(\\enm{\\cal{O}} _X(-x)) =0$ for $x\\gg 0$ and $i=1,\\dots ,n-2$, plus having positive depth at each $x\\in X$, is equivalent to all $\\enm{\\cal{O}} _{X,x}$ having depth $n$. Since $h^1(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N})=0$, we have $h^0(\\enm{\\cal{O}} _X)=1$ and in particular $X$ is connected. Since $h^1(\\enm{\\cal{I}} _{X,\\enm{\\mathbb{P}}^N}(1)) =0$, $X$ is linearly normal in the linear subspace of $\\enm{\\mathbb{P}}^N$ spanned by $X$. Since $n\\ge 2$ we have $h^1(\\enm{\\cal{O}} _X)=0$ an so $\\mathrm{Pic}(X)$ is a finitely generated abelian group. \n\nFix an irreducible component $Y$ of $X_{\\mathrm{red}}$. If $X$ is a hypersurface in $\\enm{\\mathbb{P}}^N$, then the multiplicity $\\mu \\ge 1$ is well-defined. In the general case we do not need the notion of the multiplicity $\\mu$ of $Y$ in $X$ at a general point of $Y$. In this section we need knowledge only on whether $\\mu =1$ or $\\mu >1$. We say that $Y$ has multiplicity $\\mu=1$ if $X$ is reduced at a general $x\\in Y$, i.e. there is a non-empty open subset $U\\subseteq Y$ such that $\\enm{\\cal{O}}_{X,x}=\\enm{\\cal{O}}_{Y,x}$ for all $x\\in U$. Otherwise we say that $Y$ has multiplicity $\\mu>1$. We are interested only in the case $X$ not integral; if $Y$ has multiplicity $1$, then we have other irreducible components of $X_{\\mathrm{red}}$.\n\n\n\\begin{lemma}\\label{zz1}\nLet $C\\subset X$ be a reduced aCM subvariety of pure dimension $n-1$. Then its ideal sheaf $\\enm{\\cal{I}} _{C,X}$ is an aCM $\\enm{\\cal{O}} _X$-sheaf such that\n\\begin{itemize}\n\\item it is locally free outside $C$ and \n\\item for any closed subscheme $Y\\subsetneq X$, it is not an $\\enm{\\cal{O}} _Y$-sheaf. \n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nSince $C$ is aCM as a closed subscheme of $\\enm{\\mathbb{P}}^N$ and $C$ has pure dimension $n-1$, we have $h^1(\\enm{\\cal{I}} _{C,\\enm{\\mathbb{P}}^N}(t)) =0$ for all $t\\in \\enm{\\mathbb{Z}}$. Thus the restriction map $\\rho _t: H^0(\\enm{\\cal{O}} _{\\enm{\\mathbb{P}}^N}(t)) \\rightarrow H^0(\\enm{\\cal{O}} _C(t))$ is surjective for any $t\\in \\enm{\\mathbb{Z}}$. Since $\\rho _t$ factors through the restriction map $\\eta _t: H^0(\\enm{\\cal{O}} _X(t)) \\rightarrow H^0(\\enm{\\cal{O}} _C(t))$, $\\eta _t$ is surjective. Since $\\eta _t$ is surjective and $h^1(\\enm{\\cal{O}} _X(t))=0$, we have $h^1(\\enm{\\cal{I}} _{C,X}(t)) =0$. This implies that $\\enm{\\cal{I}} _{C,X}$ is aCM. From $\\enm{\\cal{I}} _{C,X\\setminus C} \\cong \\enm{\\cal{O}} _{X\\setminus C}$, we see that $\\enm{\\cal{I}} _{C,X}$ is locally free and of rank $1$ outside $C$. Since $C$ is not an irreducible component of $X_{\\mathrm{red}}$ and $\\enm{\\cal{I}} _{C,X}$ is locally free of positive rank outside $C$, there is no closed subscheme $Y\\subsetneq X$ with $\\enm{\\cal{I}} _{C,X}$ an $\\enm{\\cal{O}} _Y$-sheaf.\n\\end{proof}\n\n\\begin{proposition}\\label{zz2}\nFix an irreducible component $Y$ of $X_{\\mathrm{red}}$. For a fixed integer $e>0$ and any integral divisor $C\\in |\\enm{\\cal{O}}_Y(e)|$, define\n$$\\Sigma _C:=\\left\\{ p\\in Y~|~ \\enm{\\cal{I}}_{C,X} \\text{ is not locally free at }p\\right\\}.$$\n\\begin{itemize}\n\\item [(i)] If $Y$ has multiplicity $\\mu>1$ in $X$, then we have $\\Sigma _C = C$, i.e. for all $p\\in C$ the sheaf $\\enm{\\cal{I}} _{C,X}$ is not locally free at $p$. For any two integral curves $C_1,C_2\\in |\\enm{\\cal{O}}_Y(e)|$, we have $\\enm{\\cal{I}} _{C_1,X}\\cong \\enm{\\cal{I}} _{C_2,X}$ if and only if $C_1=C_2$.\n\\item [(ii)] Assume that $Y$ has multiplicity $\\mu=1$ and that $X$ is not integral. Let $F\\in |\\enm{\\cal{O}} _Y(m-1)|$ be the complete intersection of $Y$ with the other components of $X$, counting multiplicities. If $F\\ne \\emptyset$, then $F$ has pure dimension $n-1$ and $F\\cap C \\ne \\emptyset$ with $\\Sigma _C =(F\\cap C)_{\\mathrm{red}}$.\n\\item [(iii)] For any two integral divisors $C_1,C_2\\in |\\enm{\\cal{O}}_Y(e)|$ such that $\\enm{\\cal{I}} _{C_1,X}\\cong \\enm{\\cal{I}} _{C_2,X}$, we have $\\Sigma_{C_1}=\\Sigma_{C_2}$; in case (i) we have the converse.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nBy Lemma \\ref{zz1} the sheaf $\\enm{\\cal{I}} _{C,X}$ is aCM and locally free with rank $1$ at all $p\\in X\\setminus C$.  Fix $p\\in C$ and assume that $\\enm{\\cal{I}} _{C,X}$ is locally free at $p$. Then there is $w\\in (\\enm{\\cal{I}} _{C,X})_p$ such that $w$ is not a zero-divisor of $\\enm{\\cal{O}} _{X,p}$ and $(\\enm{\\cal{I}} _{C,X})_p \\cong w\\enm{\\cal{O}} _{X,p}$ as a module over the local ring $\\enm{\\cal{O}} _{X,p}$. We get that in a neighborhood of $p$ the divisor $C$ is a Cartier divisor of $X$. Let $I\\subset \\enm{\\cal{O}} _{X,p}$ be the ideal of $Y$ and $J\\subset \\enm{\\cal{O}} _{X,p}$ the ideal of $C$. We have $I \\subset J$. First assume that $X$ is not reduced at a general point of $X$. Since the support of the nilradical $\\eta \\subset \\enm{\\cal{O}} _X$  of the structural sheaf $\\enm{\\cal{O}}_Y$ is a closed subset of $X_{\\mathrm{red}}$, $X$ is not reduced at any point of $Y$ and in particular it is not reduced at $p$. Thus there is a nonzero $h\\in I$ such that $h^m=0$ for some $m>0$. Since $I\\subset J$, we have $h\\in J$ and so $h$ is divided by $w$. Thus we get $w^m=0$ and so $w$ is a zero-divisor, a contradiction.\n\nNow assume that $X$ is reduced at a general point of $Y$. Since $X$ is not integral and it has pure depth $n$, $X_{\\mathrm{red}}$ has at least one another irreducible component. Since $h^0(\\enm{\\cal{O}} _X)=1$, $X$ is connected and so $F\\ne \\emptyset$. Fix any $x\\in F$. Since $\\enm{\\cal{O}} _{X,x}$ has depth $n\\ge 2$, it is connected in dimension $\\le n-1$, i.e. for any open neighborhood $W$ of $x$ in $X$ and any closed subscheme $V$ of $W$, there is a neighborhhod $U$ of $x$ in $W$ such that $U\\setminus (U\\cap V)$ is connected. Thus $F$ has pure dimension $n-1$. Since $C\\in| \\enm{\\cal{O}} _Y(e)|$, $C$ is a Cartier divisor of $Y$. Thus $C$ is a Cartier divisor of $X$ at all points of $C\\setminus (C\\cap F)$. Since $e>0$, $C$ is an ample divisor of $Y$. In particular, we get $F\\cap C \\ne \\emptyset$. Fix $p\\in F\\cap C$. Any local equation $w$ of $C$ at $p$ vanishes on each irreducible component of $X_{\\mathrm{red}}$ containing $p$, because $w$ is assumed to be a non-zero divisor of $\\enm{\\cal{O}} _{X,p}$. There is at least one another irreducible component of $X_{\\mathrm{red}}$ containing $p$, because $p\\in F$.\n\nPart (iii) is obvious.\n\\end{proof}\n\nAs a corollary of Proposition \\ref{zz2} we get the following result, which shows that $X$ is of wild representation type in a very strong form.\n\n\\begin{proposition}\\label{zz3}\nTake $X$ as above. For a fixed integer $w>0$, there is an integral quasi-projective variety $\\Delta$ and a flat family $\\{\\enm{\\cal{F}} _a\\}_{a\\in \\Delta}$ of aCM sheaf on $X$ with each $\\enm{\\cal{F}} _a$ locally free outside a one-codimensional subscheme $C_a$ and for each $a\\in \\Delta$ the set of all $b\\in \\Delta$ such that\n$\\enm{\\cal{F}} _b \\cong \\enm{\\cal{F}} _a$ is contained in an algebraic subscheme $\\Delta _a\\subset \\Delta$ with $\\dim \\Delta -\\dim \\Delta _a\\ge w$.\n\\end{proposition}\n\n\\begin{proof}\nFirst assume that $X$ has at least one irreducible component $Y$ with multiplicity at least $2$. Fix a positive integer $e$ such that $\\dim |\\enm{\\cal{O}} _Y(e)| \\ge w$ and take as $\\Delta$ the family of all integral $C\\in |\\enm{\\cal{O}} _Y(e)|$. Then we may apply (i) of Proposition \\ref{zz2}. In this case we may find $\\Delta$ with the additional condition that for all $a,b\\in \\Delta$ we have $\\enm{\\cal{F}} _a\\cong \\enm{\\cal{F}} _b$ if and only if $a=b$.\n\nNow assume that each irreducible component of $X$ has multiplicity $1$ and fix one of them, say $Y$. Write $F\\subset Y$ as in (ii) of Proposition \\ref{zz2}. Fix an integer $e >0$ such that $h^0(\\enm{\\cal{O}} _X(e)) -h^0(\\enm{\\cal{O}} _X(e)(-F) )> w$ and let $\\Delta$ be the set of all integral divisors $C\\in |\\enm{\\cal{O}} _X(e)|$ not contained in $F$\nand such that the scheme $F\\cap C$ is reduced. Since $F$ has pure dimension $n-1$\nand $C$ is an ample divisor,\nthe set $(F\\cap C)_{\\mathrm{red}}$ has pure dimension $2$. Note that if $C, D\\in \\Delta$ and $(C\\cap F)_{\\mathrm{red}} = (D\\cap F)_{\\mathrm{red}} $, then any equation of $C$ in $H^0(\\enm{\\cal{O}} _X(e))$ differs from an equation of $D$ by an element of $H^0(\\enm{\\cal{O}} _X(e)(-F))$. Then we may apply (ii) of Proposition \\ref{zz2}.\n\\end{proof}\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe fourfold spin-valley degenerate degrees of freedom in bulk graphene can support rich physics and novel applications\nassociated with multicomponent quantum Hall effects \\cite{young2012spin,yang2010hierarchy,islam2016scheme}\nand linear conductance filtering \\cite{grujic2014spin,wu2016full,soodchomshom2011strain,wang2016valley}.\nFor the spin-valley quantum Hall effects,\nit is experimentally demonstrated that, the approximate SU(4) isospin\nsymmetry of Landau levels (LL) in graphene can be broken\nby interactions such as strong Coulomb interaction, Zeeman effect, and lattice scale interactions,\nmanifesting as quantum Hall isospin ferromagnetic states \\cite{young2012spin}.\nSeveral other schemes have also been proposed theoretically, such as\nbreaking the fourfold degeneracy of the central LL by valley-scattering random potential, Zeeman interaction,\nand electron-phonon coupling \\cite{yang2010hierarchy};\nor generating quantum spin-valley Hall effect\nvia quantum pumping by adiabatically modulating a magnetic impurity and an electrostatic potential\nin strained graphene \\cite{islam2016scheme}.\n\n\nFor the spin-valley filtering effect,\nit is demonstrated that, strain in a graphene barrier with carrier mass and spin-orbit coupling\ncan enforce opposite cyclotron trajectories\nfor the filtered valleys in a spin-valley dependent gap, demonstrating simultaneous filtering\nof both valley and spin \\cite{grujic2014spin}.\nIn addition, in strained graphene with Rashba spin-orbit coupling and magnetic barrier,\nfull valley- and spin-polarization currents can be accessed simultaneously,\ndue to the coexistence of valley and single spin band gaps \\cite{wu2016full}.\nAlso, the interplay of modulation fields in a graphene ferromagnetic\/strained (spin splitting barrier)\/ferromagnetic\nstructure \\cite{soodchomshom2011strain,wang2016valley}\nare shown as possible schemes. \n\n\nWhile almost all these works concern about\nconductances (quantum Hall or linear response),\nalmost nothing is known about how to break the spin-valley degeneracy of electron \\emph{beams} in graphene.\n\n\n\n\n\n\n\n\n\nIn this work, we\npropose a spin-valley beam splitter\nbased on Goos-H\\\"{a}nchen (GH) shifts \\cite{note1} and their difference between spin-valley flavours.\nThe device we considered is based on a proximity-induced ferromagnetic interaction\nfrom a EuO film covered by a top gate \\cite{haugen2008spin,yang2013proximity,song2015spin},\nwhich lifts the spin degeneracy,\ncombined with a uniaxial strain, which lifts the valley degeneracy (see Fig. \\ref{fig2}(a)).\nWe first consider the full quasi-ballistic transport regime.\nIt is shown that, the spin band-gaps and valley wave-vectors in the ferromagnetic and strained  regions\nrespectively mimic spin- and valley-dependent refractive indices,\nacting as a spin or valley beam splitter near total reflections.\nWe subsequently consider the full quantum tunneling regime.\nIt is found that, resonant tunnelings can lead $\\sim-\\pi$ sudden phase jumps of the transmission components,\nresulting four giant GH shifts as large as the transverse beam width \\cite{note2}.\nMore importantly, the interplay of the spin- and valley-dependent imaginary wave vectors\nin the ferromagnetic and strained regions lead to four different resonant angles (energies) for\neach spin-valley flavours, thus demonstrating a spin-valley beam splitter\nthat lies in the intersection of spintronics and valleytronics.\nThe beam splitting behavior is found to be controllable by the gating as well as the strain.\nSpin \\emph{or} valley beam splitter in graphene devices was studied before \\cite{zhai2011valley,zhang2015valley,wang2014spin,zhang2014spin};\nhowever, the mechanism proposed in this work is novel, and previously unexplored.\n\n\n\n\\begin{figure}[b]\n\\centering  \\includegraphics[width=\\linewidth]{model-snell-new}\n\\caption{Schematic diagrams of (a) the proposed spin-valley beam splitter,\n(b) modulation profiles felt by electrons from different spin-valley flavours,\nand (c) the motion paths in the device for electrons from different spin-valley flavours.\nIn (b) the modulations along the $z$ and $y$ axis stand for energy and wave vector, respectively.\n}\\label{fig2}\n\\end{figure}\n\n\n\n\n\n\nFigure \\ref{fig2}(a) shows the proposed device.\nA graphene of length $L$ and width $W$ is grown on a substrate placed\nin the $x$-$y$ plane.\nAn EuO film is deposited upon the graphene \\cite{swartz2012integration} in the region of $(0,l_f)$,\nwith a top gate further grown on top of it.\nA uniaxial strain is applied on the substrate in the region of $(l-l_{st},l)$.\n$W$ is several times of $l$ to ensure that the edge effect is negligible \\cite{tworzydlo2006sub}.\nThe spin- and valley-resolved trajectories of the reflection\nand transmission beam components are shown in Fig. \\ref{fig2}(a), with the line width approximately\nstanding for the beam strength.\nFor clearness, the sublattices position differences of $1\/(k_F\\cos\\alpha)$ \\cite{beenakker2009quantum}\nare not shown ($k_F$ stands for the Fermi wave vector).\nA detector placed at a proper position in the transmission region can be used to receive the splitting beam\ncomponent.\n\n\n\\begin{figure}[t]\n\\centering  \\includegraphics[width=\\linewidth]{Snell-shift}\n\\caption{(a) GH and (b) Snell shifts (in units of Fermi wave length) of the four beam components\nas a function of incident angle at beam energy of $E$=20.\nThe parameters of the spin-valley beam splitter are $(l_f,l_w,l_{st})=(1,2,1), V_g=135.6$, and $A_{Sy}=4$.\n}\\label{fig7}\n\\end{figure}\n\nAn electron beam can be represented\nas a wave packet of a weighted superposition of plane-wave\nspinors \\cite{li2003negative,beenakker2009quantum,song2013ballistic},\n$\\boldsymbol{\\Psi}_{i}(x,y)=\\frac{1}{\\sqrt{2}}\\int dq f(q-\\bar{q})\\Phi_p^+(x,y),$\n$\\boldsymbol{\\Psi}_{r}=\\frac{1}{\\sqrt{2}}\\int dq r_{\\xi s}(q)f(q-\\bar{q})\\Phi_p^-(x,y),$\n$\\boldsymbol{\\Psi}_{t}=\\frac{1}{\\sqrt{2}}\\int dq t_{\\xi s}(q)f(q-\\bar{q})\\Phi_p^+(x,y).$\nHere the subscript $i,r,t$ stands for incident, reflection, and transmission, respectively,\n$\\bar{q}$ is the central transverse wave vector,\nand $f()$ is the spectral distribution of the transverse wave vector,\nwhich can be assumed to be of Gaussian profile.\nFor reflection and transmission, the weights\nare further modulated by reflection\/transmission coefficients that\nare spin (with index $s=\\pm1$ for spin up\/down ($\\alpha$\/$\\beta$))\nand valley (with index $\\xi =\\pm1 $ for valley $K$ and $K^\\prime$) dependent.\nIn the steady state approximation \\cite{li2003negative,beenakker2009quantum,song2013ballistic},\nthe locus of the beam components can be given by\n\\begin{equation}\n\\sigma_{s\\xi}^{r,t}=-\\left(\\frac{\\partial \\phi_{s\\xi}^{r,t}}{\\partial q_p}\\right)_{E_p},\\label{eq4}\n\\end{equation}\nwhere $\\phi_{s\\xi}^{r}=\\bar\\phi_{s\\xi}^{r}$ and $\\phi_{s\\xi}^{t}=\\bar\\phi_{s\\xi}^{t}+k_p l$.\nThe phases relate with the coefficients through\n$r_{\\xi s}= |r_{\\xi s}|e^{i\\bar\\phi^r_{\\xi s}}$ and\n$t_{\\xi s}=|t_{\\xi s}|e^{i\\bar\\phi^t_{\\xi s}}$.\n\n\nTo obtain the phases, we calculate the transmission\ncoefficients by solving the right- and left-going\nspinor eigenstates in the pristine ($\\Phi_p^\\pm$ shown in the wave packet),\nferromagnetic, and strained regions,\nand subsequently using the well-know transfer matrix method \\cite{born2000principles}\nhandling the continuity of them.\nThe eigenstates in each uniform region can be exactly resolved by decoupling\nthe two-order differential equation $H_j\\Phi_j=E_j\\Phi_j$.\nThe Hamiltonian in the pristine and strained ($j=p,s$) regions are well known \\cite{katsnelson2006chiral,pereira2009strain}.\nFor the ferromagnetic ($j=f$) region, we adopt a half-metal model \\cite{yang2013proximity,song2015spin,ang2016nonlocal}.\nRather than a simple Zeeman effect, this Hamiltonian roundly describes \\cite{yang2013proximity,song2015spin}\nthe induced charge mass or opening energy gaps ($\\Delta_s=(58+9s)$ meV),\nre-normalized Fermi velocities ($v_s=(1.4825-0.1455s)v_F$),\nand shifted Dirac points ($D_s=(-1.356+0.031 s)$ eV)\ndue to the proximity interaction,\nthat are all spin resolved.\nThe eigenstates in the pristine, ferromagnetic, and strained regions can be written in an uniform form\n\\begin{equation}\n\\Phi_j^\\pm=e^{\\pm ik_jx+iq_jy}\n\\left(\n\\begin{array}{cc}\n\\sqrt{E_j\/2(\\pm k_j+iq_j)}\\\\\n\\sqrt{(\\pm k_j+iq_j)\/2E_j}\n\\end{array}\n\\right),\\label{eq2}\n\\end{equation}\nwhere $+ (-)$ stands for the right- (left-) going propagation,\n$E_p=E_{st}=E$ and $E_f=(E-\\widetilde{D}_s+\\Delta_s)\/v_s$ with\n$\\widetilde{D}_s=D_s+V$ and $V$ the gate voltage.\nIn Eq. (2), $q_p=q_f=E\\sin\\alpha$ is the conserved transverse wave vector,\n$q_{st}=q+\\xi A_{Sy}$ with $A_{Sy}$ the $y$-component of the pseudo magnetic vector potentials \\cite{pereira2009strain};\n$k_p=\\textmd{sign}(E_p)\\sqrt{E_p^{2}-q_p^{2}}$,\n$k_f=\\textmd{sign}(E_f)\\sqrt{E_f E_f^\\prime-q_f^{2}}$ with $E_f^\\prime=(E-\\widetilde{D}_s-\\Delta_s)\/v_s$,\nand $k_{st}=\\textmd{sign}(E_{st})\\sqrt{E_{st}^{2}-q_{st}^{2}}$.\nFor brevity, we express all quantities in dimensionless form by means of\na characteristic length $l_0 = 56.5$nm and energy unit $E_0 = 10$meV.\n\n\n\n\n\n\n\n\n\nIn Fig. \\ref{fig7}, we look at the behavior of GH shifts in detail for the quasi-ballistic transport regime\n(the beam energy is so high that all components are transported quasi-ballistically).\nAlso shown are the `electron paths' (see Fig. \\ref{fig2}(c)) calculated in the geometric optics using Snell's Law.\nThe sine of refractive angles in the three regions read $\\sin\\alpha_p=q_p\/E_p$,\n$\\sin\\alpha_f=q_f\/\\sqrt{E_fE^\\prime_f}$,\nand $\\sin\\alpha_{st}=q_{st}\/E_{st}$.\nSpin- or valley-dependent refractive indices can be defined as \\cite{cheianov2007focusing,moghaddam2010graphene}\n$n^f_s=\\sqrt{(E_p-D_s-V)^2-\\Delta_s^2}\/(v_s|E_p|)$ and $n^{st}_\\xi=q_p\/(q_p+\\xi A_{Sy})$.\nThen the four Snell shifts are\n$\\sigma^{Snell}_{\\xi s}=l_f\\tan(\\sin^{-1}(\\sin\\alpha\/n_s))+l_w\\tan\\alpha+l_s\\tan(\\sin^{-1}(\\sin\\alpha\/n_\\xi))$.\nIt is clear that, the GH shifts simply oscillate around corresponding Snell shifts.\nIt is seen in Fig. \\ref{fig7}(b) that, there are four critical angles corresponding to the four refractive indices,\ni.e., $\\sin\\alpha_{c1}=n^f_\\alpha$, $\\sin\\alpha_{c2}=n^f_\\beta$, $\\sin\\alpha_{c3}=n^{st}_K$, and $\\sin\\alpha_{c4}=n^{st}_{K^\\prime}$\n($\\alpha_{c4}$ is negative and not shown).\nNear each critical angle, GH shifts\nof two beam components with a same spin or valley increases dramatically.\nThis is an enhancement effect due to total reflection.\nAs a result, spin beam splitter of $\\alpha$\/$\\beta$ can be achieved near $\\alpha_{c1}$\/$\\alpha_{c2}$;\nwhile valley beam splitter of $K$\/$K'$ near $\\alpha_{c3}$\/$\\alpha_{c4}$.\nIn Fig. \\ref{fig7}(a) it is also observed that,\nthe four Snell shifts are spin- and valley-dependent,\nespecially for incident angle smaller than $\\alpha_{c2}$.\nHowever, it is hard to achieve enough GH shift difference\nbetween any component and the other three ones.\nThus, spin-valley beam splitting is rather challenging in this transport regime.\n\n\n\n\n\n\\begin{figure}[t]\n\\centering  \\includegraphics[width=\\linewidth]{shift-angle}\n\\caption{The GH shifts (in units of Fermi wave length) of the four\nbeam components as a function of the incident angle at $E=2$.\nThe device parameters are the same as in Fig. \\ref{fig7}.}\\label{fig5}\n\\end{figure}\n\n\n\n\n\nIn Fig. \\ref{fig5} we consider the behavior of GH shifts in the full resonant tunneling regime.\nOne can see that, as the incident angle increases,\nthe two spin $\\alpha$ (spin $\\beta$) components undergo increasing\n(decreasing) GH shifts approximatively proportional to\n$\\tan\\alpha$ ($-\\tan\\alpha$).\nThis is still a quasi-ballistic transport behavior.\nIt is distinct that, around some specific angles, the four GH shifts display sharp peaks.\nThe corresponding phases in the insert show sudden jumps of about $-\\pi$.\nThis is a clear signal for a formation of a standing wave in resonant tunneling.\nRecalling the condition for ideal standing waves in an infinite potential well, i.e.,\n$\\sin k_p l_w=0$,\nthe resonant angles can be solved as $\\alpha=38.2^\\circ$ with an only allowed $n=1$.\nThe resonant angles for the four spin-valley flavours\n($K\\alpha$, $K\\beta$, $K'\\alpha$, and $K'\\beta$) are 49.1$^\\circ$, 53.5$^\\circ$, 32.7$^\\circ$, and 43.4$^\\circ$, respectively.\nThey all deviate from the ideal resonant angle, because the standing waves are formed in\nnon-ideal potential wells formed by walls of spin band-gaps and valley wave-vectors, see Fig. \\ref{fig2}(b).\nIn this case, electron waves can leak out through evanescent waves\nin the walls.\nThis generally lowers the quantization energies and wave vectors, resulting larger resonant angles\n(this is true except for the $K'\\alpha$ case).\nMore importantly,\ninterplay of the spin- and valley-dependent imaginary wave vectors\nin the modulation regions (see Fig. \\ref{fig2}(b)) lead different standing waves,\nimplying different wave vectors and hence different resonant angles\nfor different spin-valley flavours.\nAt each resonant angle,\nthe resonant GH shift exceeds the transverse beam width\nand the other three ones are small.\nWhen we tune the incident angle, the four resonant spin-valley beam components can be splitting in sequence,\nmanifesting a novel spin-valley beam splitting effect.\n\n\n\\begin{figure}[t]\n  \\centering\n \n  \\includegraphics[width=\\linewidth]{k2}\\\\\n  \\caption{Square of the four wave vectors in the two modulated regions\n  as a function of the beam energy.\n  $\\alpha$=30$^\\circ$, $V_g=135.6$, and $A_{Sy}=4$.\n}\\label{fig4}\n\\end{figure}\n\nIt is surprising that, the spin band-gaps and valley wave-vectors play such different roles\nfor beam splitter in different transport regimes.\nIn Fig. \\ref{fig4}, we plot the square of the four wave vectors as a function of beam energy for a fixed incident angle\n(note that a beam component is related to one spin wave vector plus one valley wave vector).\nThree energy ranges can be defined.\ni) The energy rang that exceeds 15 or -21.\nAll wave vectors are real,\nmeaning that all the four beam components transport through the sandwich structure quasi-ballistically.\nThis is the case we considered in Fig. \\ref{fig7}, giving behaviors similar to geometric optics\nand can be applied as spin or valley beam splitter near total reflection.\nii) The range between -2.7 and 2.1.\nIn this range, the four wave vectors all become imaginary,\nimplying that the four spin-valley beams all undergo evanescent transport or resonant tunneling.\nThis is the case we considered in Fig. \\ref{fig5}, showing spin-valley beam splitter\ndue to flavours dependent standing waves.\nThis is the central result of the present work.\niii) The energy range between.\nAt least one wave vector (say $k_\\beta$) is real.\nThe two related components ($K\\beta$ and $K'\\beta$) cannot undergo resonant tunneling.\nAs a result, they cannot be splitting from each other (although can be\nsplitting from the $K\\alpha$ and $K'\\alpha$ components).\nIn this case, only components $K\\alpha$ and $K'\\alpha$ can be splitting,\ndemonstrating a partial spin-valley beam splitting effect.\n\n\n\n\\begin{figure}[t]\n\\centering  \\includegraphics[width=\\linewidth]{shift-energy-30}\n\\caption{The GH shifts (in units of Fermi wave length) of the four beam components as a function of the incident energy at $\\alpha=30^\\circ$.\nThe device parameters are the same as in Figs. \\ref{fig7} and \\ref{fig5}.\n}\\label{fig3}\n\\end{figure}\n\n\n\n\n\nThe full spin-valley beam splitting is further shown in the energy space in Figure \\ref{fig3} .\nOne can see clear huge resonant peaks on background of GH shifts of the order of the Fermi wavelength.\nThe resonant energies are around the one of an ideal standing wave with $n=\\pm1$,\ni.e., $E_{resonant}=\\pm1.81$.\nThe sequence of components is found to be in opposite orders\ncompared with that in the incident angle space (Fig. \\ref{fig5}).\nThese behaviors mean that,\nin addition to by tuning incident angle at fixed beam energy,\nspin-valley beam splitting can also be achieved by tuning the beam energy at fixed incident angle.\nThis can have potential applications in splitting beam of wide energy distribution.\n\n\n\n\nWe at last consider the tunability of the full spin-valley beam splitter.\nFigure \\ref{fig6} shows the GH shifts of $K\\beta$ and $K^\\prime\\alpha$ flavours\nfor different gate voltages or strains.\nAs can be seen, the resonant GH shift moves to a smaller incident angle\nwhen a higher gate voltage or a bigger strain is applied.\nThis is because, the wave leaks become weaken and the energy of the quasi-standing\nwave increases.\nOn the other hand, the resonant angle becomes bigger when a lower gate voltage or a smaller\nstress is applied.\nAll these trends are the same for the other three components\n(components $K\\alpha$ and $K^\\prime\\beta$ are not shown for clearness).\nThe GH shifts of $\\beta$-related flavours decrease at a lower gating\nbecause the wave leaks increase.\nThe above results imply that,\nthe resonant angles of the splitting components can be chosen by tuning the gating or strain,\nprovided the transport remains in the full resonant tunneling regime.\nOn the other hand, the bigger the strain, the wider the intersection set\n(also limited by the two $K^f_s$ sets),\nbenefiting to the full resonant tunneling.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{parameters}\\\\\n\\caption{GH shifts of the $K\\beta$ and $K^\\prime\\alpha$ components\nas a function of the incident angle at $E=2$ for\ndifferent gate voltages or strains. $l_f=1$, $l_w=2$, and $l_s=1$.\n}\\label{fig6}\n\\end{figure}\n\n\nIn summary, we have demonstrated that the spin-valley fourfold degeneracy of electron beam in\ngraphene can be totally eliminated by flavour-dependent giant GH shifts\nin a full resonant tunneling regime.\nThis manifests a novel application of spin-valley beam splitter, lying in\nthe intersection of spintronics and valleytronics.\nThe formation of standing waves in the potential wells, \nwhich leads sudden jump of phase of about $-\\pi$,\nand the interplay of the spin- and valley-dependent wave leaks, \nwhich leads difference in resonant angles,\nare at the heart of the mechanism of the beam splitting effect.\nThe spin-valley beam splitting can be achieved by tuning incident angle\n at fixed beam energy or vice versa,\nand can be controlled by gating or strain.\n\nWe have also shown that, the spin band-gaps and valley wave-vectors play\ntotally different roles in the full ballistic transport regime.\nThey act as medias with spin or valley dependent refractive index\nand can be applied as spin or valley beam splitter near total reflections.\n\nWe encourage experimental works on the proposed device and mechanism.\n\n\n\n\n\n\n\nThis work was supported by the National Natural Science Foundation of\nChina (NSFC) under Grant No. 11404300\nand the Science Challenge Project (SCP) under Grant No. JCKY2016212A503.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLinear optical architectures  offer\nthe potential for reliable realizations\nof small-scale quantum computing~\\cite{KLM01}\nIn the recent past, numerous \nproof-of-principle demonstrations have relied on the precise state\nmanipulation that is available using linear optical elements. The \ndevelopment of better and brighter sources with good mode \nquality as well as new types of detectors have opened up new \nperspectives~\\cite{Further} in state preparation\nand manipulation, for six or more photons. \nSpecifically, integrated optical circuits allow for\nstate manipulation with little mode matching problems in interferometers~\\cite{Integrated}. \n\nNaturally, significant efforts has been devoted towards realizing instances\nof quantum gates. As primitives for such small-scale computing, two-partite quantum gates delivering a \ncontrolled phase-shift of $\\varphi$ have already been experimentally\ndemonstrated (see Refs.~\\cite{HT02,RLBW02,PPBS} for $\\varphi=\\pi$ and Ref.~\\cite{LBA+09} \nfor general phases). In this note, we focus on linear\noptical implementations of phase gates with arbitrary phases. In particular,\nwe will ask when arbitrary phases can \nbe realised in the first place, and---one of the main figures of merit in linear optical\napplications---what the optimum probabilities of success are, as any non-linear map is necessarily\nprobabilistic.\nA realisation of such gates seems interesting from the perspective of \n\\begin{itemize}\n\\item[(i)] gaining an understanding of the probabilistic character of \nquantum gates as well as \n\\item[(ii)] serving as a proof of principle\nrealisation of a kind of quantum gate that has several \napplications in linear optical quantum information processing.\n\\end{itemize}\n\nAs far as the first aspect is concerned, \none may well expect that there\nis a trade-off between the notorious problem of having a small\nprobability of success and the phase that is being realised in\nthe gate. In fact, the study in Ref.~\\cite{Eisert04}\nsuggests exactly such a behavior: \nthe presented upper bounds to the probability of success\nincrease from the \nminimum at $\\ensuremath{p_{\\mathrm{s}}}(\\varphi=\\pi)=1\/4$ to $\\ensuremath{p_{\\mathrm{s}}}(0)=1$. To\ninvestigate such a trade-off is interesting in its own right\nand helps in building intuition concerning the probabilistic behavior\nof linear optics. One may well develop the intuition that ``large phases are costly'' as far\nas the probability of success and hence the overhead or repetition are concerned.\n\n\nFurther, concerning the second aspect, there are several applications \nfor which such a trade-off is relevant. In linear optical architectures,\nit may be a good idea to have a smaller phase, if one only\nhas higher success probabilities. The new measurement-based\nquantum computational models~\\cite{GE07a} for example offer this \nperspective: One does not have to have controlled $\\pi$ phase gates\nto prepare cluster states, but one would in principle also get\naway with smaller phases. This may well (but does not have to be)\na significant advantage when preparing resources for\nmeasurement-based quantum computing different from cluster states~\\cite{RB01,GE07a}.\n\nOf course, in standard gate-based quantum computing, \none will typically encounter all kinds of controlled phase gates. For example,\nin the quantum Fourier transform~\\cite{nielsen}, \none has to implement several controlled phase gates.\nThey can again be decomposed into other sets of universal \ngates (like CNOT or CZ and local unitaries). But,\nin terms of resource requirements, it is obviously an advantage to directly\nimplement the relevant quantum gates with phases in the range \n$0<\\varphi<\\pi$.\nThere are also interesting trade-offs\nbetween resource requirements and success probabilities in a number of\nrelated contexts, like non-local gates in distributed quantum computation~\\cite{EJPP00,CDKL01Berry07}. Refs.~\\cite{CDKL01Berry07}, for example,\nstudy distributed controlled phase-gates which would need less entanglement\nand succeed with a higher probability.\nIn the field of linear optics gates, numerical results on direct implementation of arbitrary two-qubit gates\nare known (see Ref.~\\cite{dmitry} and references therein).\n\nInstead of resorting to decompositions in the circuit model, one could gain\nfrom implementing unitaries in a fashion ``natural'' to the respective architecture at hand.\nIn the case of linear optics it means to leave the computational sub-space given by the\nencoding of the qubits for the sake of taking a ``shortcut'' through higher dimensions~\\cite{LBA+09}.\nGiven that the fundamental information carriers are implemented using bosonic modes, this will\noccur when mixing those modes in beam splitter networks and is an inherent feature of\ngenuine linear optics implementations, in contrast to decompositions into standard gate sets.\n\nHere we will study post-selected gates, so not genuine ``event-ready'' quantum\ngates, but---as is common in linear optical architectures at least to date---those \nwhere one measures the output modes and whether the gate\nactually succeeded is determined only {\\it a posteriori} by accepting only those\noutcomes which lie in the computational dual-rail subspace of the Hilbert space of $n$ photons on $2n$ modes.\nIncorporating less constraints, these gates concern only a smaller number of modes\nand are still within reach of current experiments.\nIn principle non-demolition measurements of the output would be required\nfor an event-ready gate.\n\n\\section{Controlled phase gates}\n\\subsection{Single beam splitter}\nIn a post-selected phase gate on four modes in \nthe standard dual-rail encoding, two of the modes\nare merely involved as ``by-standers'', in that their amplitude is \ncompensated in exactly the same fashion as in Refs.~\\cite{HT02,RLBW02,PPBS}.\nIn this section, we will hence \nconcentrate on two modes forming the ``core'' of the\nscheme, giving rise to a two-qubit dual-rail phase gate on four physical modes. The core\nitself may be regarded as a single-rail phase gate in its own right. Later we will see that\nnot breaking the network into a core and by-stander modes will not give any advantage.\n\nSimilar to Ref.~\\cite{HT02} we will briefly investigate the consequences of simply having a single beam\nsplitter forming the core of the quantum gate.\nThe action on the photonic creation operators of the two involved\nmodes it is mixing is described by the matrix\n\\begin{equation}\n  U = \\mathrm{diag} (\\mathrm{e}^{\\imath\\phi_1},\\mathrm{e}^{-\\imath\\phi_1}) \\cdot B \\cdot \\mathrm{diag} (\\mathrm{e}^{\\imath\\phi_2},\\mathrm{e}^{-\\imath\\phi_2} )\n\\end{equation}\nwith\n\\begin{equation} \\label{eqn:orthogonal}\n  B = \\left[\\begin{array}{cc} \\sin(\\vartheta) & \\cos(\\vartheta) \\\\ -\\cos(\\vartheta) & \\sin(\\vartheta) \\end{array}\\right] \n\\end{equation}\nand appropriate phases $\\phi_1,\\phi_2\\in[0,2\\pi)$\nand mixing angle $\\vartheta\\in [0,2\\pi)$.\nThe phases can also be realised deterministically by local operations on the dual-rail qubits, which leaves\nthe relevant part of the gate $U'=B$. The matrix elements of the unitary $\\mathcal U'$ belonging to $U'$ for vacuum, single photon operation, and the two-photon component\nread\n\\begin{eqnarray}\n  \\braket{0,0|\\mathcal U'|0,0} &=& 1 ,\\label{eqn:cphase:def1} \\\\\n  \\braket{1,0|\\mathcal U'|1,0} &=& A_{1, 1} = \\sin(\\vartheta), \\\\\n  \\braket{0,1|\\mathcal U'|0,1} &=& A_{2\\ic2} = \\sin(\\vartheta), \\\\\n  \\braket{1,1|\\mathcal U'|1,1} &=& \\mathrm{per}( A) = \\cos(2\\vartheta) = 1-2\\sin^2\\vartheta, \\label{eqn:cphase:def2}\n\\end{eqnarray}\nrespectively.\nSince we are restricted to $n\\le2$ modes, these four quantities determine the action of the core completely.\nWith the constraint\n\\begin{equation}\n  1-2\\sin^2(\\vartheta) = \\sin^2(\\vartheta)\n\\end{equation}\nthat ensures equal single- and two-photon amplitudes (equal probabilities for all dual-rail states),\nonly the two solutions $\\vartheta=\\pm\\arcsin(3^{-1\/2})$ are possible, giving rise to\n$\\varphi=0,\\pi$, respectively.\n\nHence, one finds that in this way, one \\emph{can} implement quantum phase gates, but only\ntwo different ones: One is not doing anything, and the other ones effect is a controlled\nphase of $\\pi$. This is exactly the gate of Refs.~\\cite{HT02,RLBW02}. In other words, \nwithout invoking at least a single additional mode, one can not go beyond\nthe known $\\pi$-phase in this fashion.\n\n\\subsection{Arbitrary phases}\nHowever, we can extend this scheme: \nThe restriction to unitary two-mode beam splitters can be relaxed.\nInstead of starting with $U\\in SU(2)$, we use an arbitrary matrix $A\\in\\mathbbm{C}^{2\\times2}$.\nThen we will embed the two-mode matrix into a higher dimensional unitary, such that an \nappropriately rescaled $A$ forms a principle submatrix of a larger unitary matrix $A'$. The optimal rescaling is simply\ndictated by the largest singular value of $A$. We will see that in the two-mode case only a single additional mode is already the most general extension, so the\nfull set-up would consist of a transformation on three modes involving \nat most three beam splitters. In this class of gates, for each $\\varphi$, \nthe one with the optimal probability of success $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$\ncan be identified.\n\n\\begin{figure}\n  \\includegraphics{phases_04.eps}\n  \\caption{Optimal success probability $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$ of phase gates with vacuum ancillas (one vacuum ancilla is already optimal)\n    {\\it vs.}\\ the phase $\\varphi$ (solid line). At $\\varphi=\\pi$ the result of Refs.~\\cite{HT02,RLBW02},\n    $\\ensuremath{p_{\\mathrm{s}}}(\\pi) = 1\/9$, is reproduced.\n    The intuitive assumption of a monotonous $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$ is not fulfilled: indeed, the success probability is worse than $1\/9$ in the interval $\\pi\/3 < \\varphi < \\pi$.\n    Due to phases $\\varphi<\\pi$ not being implementable with a single beam splitter, the additional unitary extension requires further measurements and therefore decreases the probability\n    of success near $\\varphi=\\pi$.\n    \\label{fig:phases} }\n\\end{figure}\n\n\\begin{theorem}[Optimal post-selected dual-rail controlled phase gate]\n  Consider linear optics, an arbitrary number of auxiliary vacuum modes and photon number resolving detectors. When post-selecting the state of the signal modes onto the computational sub-space\n  and the auxiliary modes onto the vacuum,\n  the optimal network on four modes implementing the gate represented in the computational basis of two dual-rail qubits by\n  $U = \\mathrm{diag} ( 1,1,1,\\mathrm{e}^{\\imath\\varphi} )$, $\\varphi\\in[0,\\pi]$,\n  has a success probability (shown in Fig.~\\ref{fig:phases}) of\n  \\begin{equation}\n    \\ensuremath{p_{\\mathrm{s}}}(\\varphi) = \\left(1+2\\left|\\sin\\frac{\\varphi}{2}\\right|+2^{3\/2}\\sin\\frac{\\pi-\\varphi}{4}\\sqrt{\\left|\\sin\\frac{\\varphi}{2}\\right|}\\right)^{-2} .\n  \\end{equation}\n\\end{theorem}\n\n{\\it Proof:}\nIn order to find $\\ensuremath{p_{\\mathrm{s}}}$---the same for all possible input states---we will first construct the linear transformation of the relevant creation \noperators and identify the optimal unitary extension afterwards.\n\\begin{itemize}\n\\item \n{\\it Two-mode transformation:}\nThe two-mode transformation resulting from solving the equations (\\ref{eqn:cphase:def1})--(\\ref{eqn:cphase:def2}) imposed by the gate we want to build is\n\\begin{equation}\\label{eqn:cphase:solution}\n  A = \\ensuremath{p_{\\mathrm{s}}}^{1\/4}\\left[ \\begin{array}{cc} x & \\left(\\mathrm{e}^{\\imath\\varphi} - 1\\right)x\/y \\\\ y\/x & 1\/x  \\end{array} \\right] .\n\\end{equation}\n$x$ and $y$ are free non-zero complex parameters. By writing\n\\begin{equation}\n  A = \\ensuremath{p_{\\mathrm{s}}}^{1\/4}\\,\\mathrm{diag} (a,a^{-1})\n  \\cdot\\left[ \\begin{array}{cc} 1&\\mathrm{e}^{\\imath\\varphi}-1 \\\\ 1&1 \\end{array}\\right]\\cdot\\mathrm{diag}(b,b^{-1}) ,\n\\end{equation}\nwith $a=xy^{-1\/2}$ and $b=y^{1\/2}$ we see that the singular values of $A$ only depend on $|a|^2$ and $|b|^2$, so not on the phases of $a$,$b$, and $x$ and $y$.\n\nThe general solution to the dual-rail problem is actually composed of the transformation $A$ together with appropriate damping of the by-standers:\nthe probability of success can not be enhanced by considering a full transformation on all four modes.\nThis can be seen by writing the polynomial system given by the dual-rail problem similar\nto (\\ref{eqn:cphase:def1})--(\\ref{eqn:cphase:def2}), consisting of $16$ quadratic equations\nin the matrix elements of $B\\in\\mathbbm{C}^{4\\times 4}$. It turns out that by permuting modes and appropriate variable substitutions all solutions can be brought into the form\n$B \\propto \\mathbbm{1}_2 \\otimes A$,\nconsisting of the two-mode core given in Eqn.~(\\ref{eqn:cphase:solution}) and two by-passed modes.\n\n\\item {\\it Optimal extension:}\nGiven the $2\\times2$ matrix $A$ that realises the transformation we are looking for,\nthe optimal unitary \nextensions can be identified. Let us extend the first and second row vectors (denoted by $A_1$ and $A_2$) to dimension $3$ by appending $A_{1\\ic3}$ and $A_{2\\ic3}$,\nrespectively, in such a way as to allow for unitarity of the extended matrix, $A'\\in SU(3)$.\n\n\nTo see why a dimension of three is already sufficient consider a linear transformation of the creation operators of $n$ modes, described by its a (not necessarily unitary) matrix $A$.\nBy using the singular value decomposition (SVD)\nit can be decomposed as $A=V\\cdot D\\cdot W^{-1}$ where $V$ and $W$ are unitary (and therefore have immediate interpretations as physical beam splitter matrices themselves),\nand $D=\\{d_1,\\ldots,d_n\\}$ is a diagonal matrix with real non-negative entries $d_1\\ge d_2\\ge\\ldots\\ge d_n$, the singular values.\nIn terms of linear optics $D$ can be interpreted as mixing each mode $k=1,\\ldots,n$ with an additional\nmode $n+k$ in the vacuum state which will be post-selected in the vacuum afterwards~\\cite{MLU+06,Kieling08}.\nThen, $d_k$ describes the transmittivity of the beam splitter used to couple modes $k$ and $n_k$.\nWithout loss of generality one can assume $d_1=1$, vacuum mixing for the first mode.\nThis can be achieved by rescaling with the inverse of the largest singular value, so $A \\mapsto d_1^{-1}A$, which implies $d_k\\mapsto d_1^{-1}d_k$.\nNote that such a ``global'' rescaling of $A$ does not change the post-selected action on the computational sub-space but only the success probability of it according to $\\ensuremath{p_{\\mathrm{s}}}\\mapsto d_1^{-2n}\\ensuremath{p_{\\mathrm{s}}}$.\nTherefore, in general, there are $n-1$ additional\nvacuum modes required to extend an $n$-mode linear transformation to a unitary, and thus physical, network. Also please note, that the constraint $d_1=1$\nhas to be taken into account for any optimisation of success probabilities of $A$. Here we will not explicitly use this decomposition further, but the constraint will be implemented\nimplicitly by requiring the $n-1$-mode extension to be unitary.\n\n\nIn our specific $3$-mode extension we choose\n$A_{1\\ic3}$ and $A_{2\\ic3}$ such that the new row vectors are orthogonal. By multiplying them by the root of the inverse of their respective norms, $|A'_1|$ and $|A'_2|$,\nthey will be normalised. Finding a third orthogonal vector to fill the unitary matrix can be done with the complex cross product $(A'_1\\times A'_2)^*$, or\nin general by choosing a vector at random and orthogonalising it with respect to the given ones.\n\nThe dependence of the success probability on the extension is \n\\begin{equation}\n\t\\ensuremath{p_{\\mathrm{s}}}=\\left(|A'_1||A'_2|\\right)^{-2}.\n\\end{equation}\t\nTherefore, the objective is to\n\\begin{eqnarray}\n  \\mbox{minimise}   && f = |A'_1|^2 |A'_2|^2 \\\\\n  \\mbox{subject to} && A'_1 (A'_2)^{\\dagger} = A_1 A_2^{\\dagger} + A_{1\\ic3}A^{*}_{2\\ic3} = 0 \\label{eqn:subject} .\n\\end{eqnarray}\nThe first observation is that the row-scaling by $x$ is already included in the norm of the row vectors, leaving us with one parameter less.\nBy using the phase of $y$, we can assure that $A_1 A_2^{\\dagger}$ is real and \npositive and also $\\arg(A_{1\\ic3}) - \\arg(A_{2\\ic3}) = \\pm\\pi$. This \nconstrained minimisation problem in $A_{1\\ic2}$ and $A_{1\\ic3}$ can indeed be solved\n(by using Lagrange's multiplier rule and showing constraint qualification)\nand  we find $|y|=\\left(2(1-\\cos\\varphi)\\right)^{1\/4}$.\nThen an optimal solution (phases chosen conveniently) is\n\\begin{equation}\n  A_{1\\ic3} = A^*_{2\\ic3} = \\mathrm{e}^{{\\imath\\pi}\/{2}}\\left(\\sqrt{2\\left|\\sin\\frac{\\varphi}{2}\\right|}\\sin\n  \\frac{\\pi-\\varphi}{4}\\right)^{1\/2}\n\\end{equation}\nwith the probability of success given by\n\\begin{eqnarray}\\label{eqn:cphase:ps}\n  \\ensuremath{p_{\\mathrm{s}}}(\\varphi) &=& \\left(1+|y|^2+|y|\\sqrt{2-|y|^2}\\right)^{-2} \\\\\n      &=& \\left(1+2\\left|\\sin\\frac{\\varphi}{2}\\right|+2^{3\/2}\\sin\\frac{\\pi-\\varphi}{4}\\sqrt{\\left|\\sin\\frac{\\varphi}{2}\\right|}\\right)^{-2} .\n      \\nonumber\n\\end{eqnarray}\nThe reflectivities of the compensating beam splitters in the by-passed modes have to be chosen such that the success probability is constant for all dual-rail states, i.e., \n\\begin{equation}\n\tr=\\ensuremath{p_{\\mathrm{s}}}^{1\/4}.\n\\end{equation}\t\n\\end{itemize}\n\n\n\\begin{figure}\n  \\includegraphics{cphase_01.eps}\n  \\caption{Basic spatial modes based set-up obtained from translating an arbitrary $2\\times 2$ core into the language of linear optics. \n   The core extension is provided by mixing\n    with a vacuum mode on the central beam splitter. This mode, in turn, has to be post-selected in the vacuum state afterwards. The upper and lower beam splitters\n    implement the appropriate compensation by damping the by-passed modes (which is the same for both modes for the optimal solution which we consider).\n\\\\\n    The labels at the beam splitters will be used to identify them with the respective optical elements in Figs.~\\ref{fig:pgate03}--\\ref{fig:pgate:ppbs}.\n    In general, the parameters ({i.e.}, reflectivity and phases) of these elements depend on the gate's phase $\\varphi$.\n    Further, the notation $\\ket{\\mathfrak{0}}_i$ and $\\ket{\\mathfrak{1}}_i$ is used for the logical $0\/1$-modes of the $i$-th qubit to avoid confusion with Fock states. \\label{fig:pgate01} }\n\\end{figure}\n\n\nThe success probability $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$ of this gate\nis shown in Fig.~\\ref{fig:phases}. \nInterestingly---and quite surprisingly---the \nworst success probability is not achieved for the sign-flip ($\\varphi=\\pi$), but for\n$\\varphi\\approx2.05$. This means, gates delivering a phase shift slightly\nsmaller than $\\pi$ and thereby generating less entanglement will not give rise to a larger, but to a smaller success probability.\nAs expected, the success probability for very small \nphases increases and reaches unity for $\\varphi=0$: one can always do nothing \nat all with unit success probability.\n\n\n\n\n\\subsection{Integrated quantum photonics realizations}\nSophisticated circuits such as the one shown in Fig.~\\ref{fig:pgate:ppbs} can be built from \nbulk optical elements (mirrors, beamsplitters, etc.) Such circuits often\nrequire implementation of Sagnac interferometers (e.g., Ref.~\\cite{NOO+07}), partially polarizing beam splitters (PPBSs)~\\cite{PPBS}, or\nbeam displacers~\\cite{OPW+03,Process1} to achieve interferometric stability. For the most complicated circuits a combination of these elements is\nrequired. Indeed a (non-optimal) implementation of a two-qubit controlled unitary gate used a combination of beam displacers and\nPPBSs~\\cite{LBA+09}. However, such circuits are extremely challenging to align, are limited in performance by the quality of that alignment, and are ultimately not scalable.\n\nAn alternative approach based on lithographically fabricated integrated waveguides on a chip has recently been developed~\\cite{Integrated}.\nThis approach has demonstrated better performance, in terms of alignment and stability, as well as miniaturization and scalability.\nThe monolithic nature of these devices enables interferometers to be fabricated with precise phase and stability, making the Mach-Zehnder\ninterferometer shown in Fig.~\\ref{fig:pgate01} directly implementable without the need to stabilize the optical phase (either actively,\nor using the Sagnac-type architecture of Fig.~\\ref{fig:pgate:ppbs}), greatly simplifying the task of making complicated circuits:\nEssentially the circuit one draws on the blackboard can be directly `written' into the circuit.\nIndeed, integrated photonics circuits have been used to implement a circuit of several logic gates on four photons, to implement a compiled\nversion of Shor's quantum factoring algorithm in this way~\\cite{PMO09}. In fact laser direct write techniques have been\nused to `write' circuits in an even more direct fashion than the lithographic approach.\n\nAnother key advantage of integrated quantum photonics for the circuits described here is that on-chip phase control can be directly\nintegrated with the circuit~\\cite{IntegratedReconfigurable}, which could allow measurment of the success probability\ncurve (Fig.~\\ref{fig:phases} or Fig.~\\ref{fig:toffoli:ps} for example)\nto be directly mapped by sweeping the applied voltage.\n\n\\subsection{Experimental issues in free space}\n\\begin{figure}\n  \\figspace{\\vspace{.25cm}}\n  \\includegraphics{cphase_03a.eps}\\figspace{\\hspace{.25cm}\\vspace{.5em}}\n  \\caption{Setup for a controlled phase gate on two polarisation encoded dual-rail qubits. The logical modes of the two qubits are separated and re-united by means of polarising beam splitters (PBS).\n    Replacing the left and right beam splitters in Fig.~\\ref{fig:pgate01} by wave plates $\\lambda_1$ and $\\lambda_2$,\n    they become easier to tune to different $\\varphi$, and provide better stability. The lower beam splitter, $\\gamma$,\n    implements compensation of both, $\\ket{\\mathfrak{0}}_1$ and $\\ket{\\mathfrak{0}}_2$, modes. \n    $\\lambda_V$ is taken care of by the partially polarising beam splitter (PPBS, allowing for \n    different reflectivities for the two polarisations, further explained in Fig.~\\ref{fig:ppbs}) in the centre.\n    Additionally, one of the qubits has to be flipped prior to and after the circuit, which here is done by acting with a wave plate on the second qubit. \\label{fig:pgate03} }\n\\end{figure}\n\nIn order to render the proposed gates experimentally more feasible in free space, some simplifications\nhave to be done -- tailored to the specific physical implementation at hand.\nWaveguide based setups would not need further simplifications because stablity of the interferometers would be ensured by the rigid substrate.\nImplementations more suitable for beam displacer based setups are known~\\cite{LBA+09}.\nInfluenced by the gate model, those gates include a controlled $\\pi$-phase gate at the core. Operating at lower success probabilities,\nespecially at phases $\\varphi$ approaching $0$, the probability of success does not converge to $1$.\n\nIn the following we will discuss simplifications which shall allow for easier free-space implementation of the networks introduced above\nwhile still preserving optimality with respect to $\\ensuremath{p_{\\mathrm{s}}}$.\n\\begin{figure}\n  \\includegraphics{ppbs1.eps}\n  \\caption{From left to right: (i) A PPBS implementing a beam splitter with polarisation dependent reflectivity. (ii) It is equivalent to an interferometer between two PBS\n    where the PPBS' reflectivities are incorporated by means of wave plates, $\\lambda_1$ and $\\lambda_2$. (iii) By identifying the two PBS, the interferometer collapses into a closed\n    loop (which is more compact and more robust in experimental implementations), leaving only one PBS.\n    In the second and third circuit, a polarisation flip of the second qubit before and after the circuit is added by using a wave plate.\\label{fig:ppbs} }\n\\end{figure}\nA straightforward set-up on dual-rail encoding that realises a three-mode unitary and \ncompensates the amplitudes in the remaining modes is shown in Fig.~\\ref{fig:pgate01}.\nIt includes one interferometer, but the whole gate would sit inside a double interferometer,\nbecause local unitaries on the input and output qubits would require classical interference.\nThus, the complexity of this gate is best described as a nested three-fold interferometer.\nIn this first stage, the parameters (reflectivities and phases) of all five beam splitters\ndepend on $\\varphi$.\n\nTo get rid of some of the interferometers, polarisation encoding is convenient. Two modes can be united in one spatial mode, resulting in inherent stability (neglecting\nbirefringence of the optical medium) of some interferometers. Rotations on these modes can be carried out easily using wave plates. Because the core acts on modes coming from different\ndual-rail qubits, they have to be combined into a single spatial mode before. This is achieved with a PBS, thus permuting $H$ and $V$ modes. Damping of both by-passed modes can be done simultaneously\nby a single polarisation insensitive beam splitter coupled to the vacuum. A straightforward translation of Fig.~\\ref{fig:pgate01}\ninto polarisation encoding, thereby collapsing the network into a single interferometer, is shown in Fig.~\\ref{fig:pgate03}.\n\nDue to the asymmetry of the core, still one PPBS is used, the reflectivity of one polarisation component of which actually depends on $\\varphi$, the other one being $1$. Fig.~\\ref{fig:ppbs} shows\nhow a tunable PPBS can be constructed, introducing another interferometer.\n\nIterating the ideas that led to the compact PPBS implementation once more yields a collapsed form of the phase gate based on only two PBS and a couple of wave plates. Due\nto the paths of light being very similar, this set-up should also be more robust.\nThis is illustrated by Fig.~\\ref{fig:pgate:ppbs} since it has an overall Mach-Zehnder interferometer structure to it, by feeding one displaced sagnac loop into another.\n\n\n\\subsection{Simple decomposition}\nHere we want to obtain a simple decomposition and explicit $\\varphi$-dependence of the involved elements of the full $3\\times3$ unitary transformation $U\\in SU(3)$.\nTo do so, we interpret the effective core, acting on the $\\ket{\\mathfrak{1}}_1$ and $\\ket{\\mathfrak{1}}_2$ modes in Fig.~\\ref{fig:pgate01}, in a different way:\nin between $U_{\\lambda_1}$ (the unitary beam splitter matrix of $\\lambda_1$)\nand $U_{\\lambda_2}$, there acts a diagonal mode transformation such that the first mode is unaffected and the amplitude of the second one is damped\n(due to $U_{\\lambda_V}$ coupling it with reflectivity $r_V$ to a vacuum mode which will be projected onto the vacuum).\n\nNow considering the singular value decomposition (SVD) of the matrix representing the core transformation, $A=V\\cdot\\Sigma\\cdot W^{\\dagger}$ we can identify $V=U_{\\lambda_2}$,\n$W^{\\dagger}=U_{\\lambda_1}$ and $\\Sigma=\\mathrm{diag}\\{1,r_V\\}$. As we have seen earlier, optimal extensions of $2\\times2$ cores only require global rescaling, which commutes with the unitaries involved.\nTherefore we can use the much simpler original form of $A$ in Eqn.~(\\ref{eqn:cphase:solution}), and we choose $x=1$ and $y=-\\sqrt{\\mathrm{e}^{\\imath\\varphi}-1}$.\n\nWe find the singular values of $A$\n\\begin{equation}\n  \\sigma_{\\pm} = \\sqrt{1+2\\sin\\frac{\\varphi}{2}\\pm2(2-2\\cos\\varphi)^{1\/4}\\cos\\frac{\\varphi+\\pi}{4}} .\n\\end{equation}\nGlobal rescaling amounts to fixing the largest singular value to unity, so the new singular values are $1$ and $r_V=\\sigma_-\/\\sigma_+$.\n\nDue to $\\det U_{\\lambda_1}=\\det U_{\\lambda_2}=1$\nwe need to attach a phase to one singular value as well in order to apply the identification of the matrices introduced above. \nThen the SVD of $A$ yields\n\\begin{equation}\n  U_{\\lambda_1} = \\frac{1}{\\sqrt{2}}\\left[ \\begin{array}{cc} -1 & 1 \\\\ -1 & -1  \\end{array} \\right]\n\\end{equation}\nand $U_{\\lambda_2} = U_{\\lambda_1}^{-1}\\cdot\\mathrm{e}^{\\imath\\phi_+\\sigma_z}$ with\n\\begin{equation}\n  \\phi_\\pm=\\mathrm{arccot}\\left[ \\cot\\frac{\\varphi+\\pi}{4}\\pm\\left((2-2\\cos\\varphi)^{1\/4}\\sin\\frac{\\varphi+\\pi}{4}\\right)^{-1}\\right]\n\\end{equation}\nwhere the order of rows and columns in the matrices is as in Fig.~\\ref{fig:pgate01} from top to bottom.\n\nThe ``complex singular values'' are $1$ (the first mode is not affected) and $\\sigma_-\/\\sigma_+\\exp{\\imath \\phi_++\\phi_-}$. The latter can be achieved by using the aforementioned\ncoupling to the vacuum with a reflectivity of $r_V$ and a phase upon reflection of $\\phi_++\\phi_-$. The further ingredients are the beam splitters required for the ``damping'' of the\nby-standers as discussed earlier.\nBy confirming $1\/\\sigma_+^4 = \\ensuremath{p_{\\mathrm{s}}}$ in the range $0\\le\\varphi\\le\\pi$, the optimality of this construction is assured.\n\n\n\\section{Event-ready gates}\nComing from post-selected gates, the next step towards scalable quantum computation would be to build\ngates not requiring measurements on the output modes. Intuitively it is clear that the construction of\na controlled phase gate in this class will be more demanding with respect to the resources (such as\nthe number of auxiliary modes and photons, size and complexity of the network) involved.\n\nEspecially the number of additional photons will change drastically: having had none in the post-selected case of a controlled $\\pi$-phase gate,\ntwo are required in the class of event-ready gates. We will use this example as a motivation for a detour to discussing a number\nof different methods that could be useful for handling linear optics state preparation.\n\nTo do so, we notice that a controlled $\\pi$-phase gate is more constrained than\na device that creates EPR pairs from single photons.\nThis is meant in the sense that it not only amounts to\na state transformation from two single photons to an EPR pair, but a full unitary transformation\non the entire computational state space in dual-rail encoding (and creating an EPR pair when applied to a\ncertain product input corresponding to the product state of two photons).\nIn the following two sections we will be concerned with different methods to describe linear optics state preparation\nand will apply them to the specific example at hand (i.e., heralded dual-rail EPR pair generations from single photons).\n\nIt will turn out, that the construction of an EPR pair out of single photons by means of linear optics, vacuum modes,\none additional photon, and detectors is not possible.\nOf course, directly solving the polynomial equations in the matrix elements of $A$ (generalisation of Eqns.~(\\ref{eqn:cphase:def1}) to~(\\ref{eqn:cphase:def2})) will yield the same result -- no solutions unless two\nadditional photons are involved.\nHaving excluded the cases of zero and one auxiliary photons, a set-up with two of them \nis possible, proven by the existence of such a scheme (EPR construction~\\cite{ZBL+06} as well\nas controlled-$Z$ gate~\\cite{KLM01}).\n\n\n\\subsection{State transformations}\nAn obvious way of looking at states of exactly $2$ photons in $m$ bosonic field modes is the following. \nSuch a state vector can be written as\n\\begin{eqnarray}\n  |\\psi_M\\rangle &=& P({\\bf a}^\\dagger)|\\mathrm{vac}\\rangle \\nonumber\\\\ &=& \\sum\\limits_{i,j=1}^{m} M_{i,j} \n  a_i^\\dagger a_j^\\dagger |\\mathrm{vac}\\rangle = ({\\bf a}^\\dagger )^T M {\\bf a}^\\dagger |\\mathrm{vac}\\rangle ,\n\\end{eqnarray}\nwhere $M$ is a symmetric $m\\times m$ matrix~\\footnote{The $m=4$ case was used already in \nRefs.~\\cite{LCS99,Calsamiglia02}. Thanks to D.~Uskov for pointing out the nice structure of this.}.\nThe application of a unitary mode-transformation $U$---representing a linear optical network---is reflected by\n\\begin{eqnarray}\n  |\\psi_M\\rangle \\mapsto |\\psi_{M'}\\rangle &=& (U{\\bf a}^\\dagger)^T M (U{\\bf a}^\\dagger) |\\mathrm{vac}\\rangle \\\\\n                  &=& ({\\bf a}^\\dagger)^T M' {\\bf a}^\\dagger |\\mathrm{vac}\\rangle\n\\end{eqnarray}\nwith $M'=U^T MU$ clearly again being symmetric.\nAs a special case of the singular value decomposition~\\cite{HornJohnson}, a diagonal $M'$ can be achieved,\ngiven an arbitrary input state vector $|\\psi_M\\rangle$.\n\nNow let us choose $U$ such that $M'$ is diagonal. Then, labelled by\n\\begin{equation}\n  \\nu' = \\mathrm{rank} (M'), \n\\end{equation}\t\nthere are $m$ different classes of states~\\cite{Kieling08,Kieling09}, in each of which is the states are composed by superpositions of $2$ photons in either of $\\nu'$ modes.\nThese classes are separated by linear optical mode transformations requiring additional modes.\nDecreasing the rank is possible by allowing for auxiliary vacuum modes. However, to increase the rank by $1$ one additional photon is required.\n\nFurther, the state matrix $M$ of two single photons on four modes has rank $\\nu=2$ while an EPR pair corresponds to a matrix with rank $\\nu=4$.\nTherefore, the desired state transformation requires at least two additional single photons.\n\n\\subsection{Polynomial factorisation}\nAn alternative approach is the following~\\cite{Kieling08}: The polynomial describing \nthe objective state vector $|\\psi\\rangle = P({\\bf a}^\\dagger)|\\mathrm{vac}\\rangle$\nwith\n\\begin{equation}\n  P({\\bf a}^\\dagger) = 2^{-1\/2}\\left(a^{\\dagger}_1 a^{\\dagger}_3 + a^{\\dagger}_2 a^{\\dagger}_4\\right)\n\\end{equation}\t\ndoes not factorise over $\\mathbbm{C}$.\nUsing Lemmata~1 and~2 from Ref.~\\cite{Ruppert99}, the property of factorisation of a bivariate polynomial \n\\begin{equation}\n  p(x,y)=\\sum_{i,j=0}^mp_{i,j}x^iy^j \n\\end{equation}\nover $\\mathbbm{C}$ can be tested by assessing the rank of a \ncomplex $2m(2m-1)\\times(m+1)(2m-1)$ matrix. Further, \napplying Lemma~7 of Ref.~\\cite{Kaltofen95}, this technique can be extended to multi-variate polynomials.\nNow, a state can be constructed from a product state using linear optical gate arrays \niff the corresponding polynomial is factorisable.\nIn the case mentioned before (dual-rail EPR pair, so four variables), one can use the resulting $12\\times 9$ matrix to confirm in the language of polynomials of creation operators that\nadditional resources are in fact required.\n\n\\begin{figure}\n  \\figspace{\\vspace{.25cm}}\\includegraphics{cphase_08.eps}\n  \\caption{Compact implementation of a controlled phase gate by using a single loop to implement the central PPBS and the compensation beam splitters simultaneously. Additionally, the two PBS are identified,\n     resulting in a second loop. All omitted modes are initialised in the vacuum and post-selected in the vacuum state (which will be achieved in practice by counting the photons in the other output).\n     \\label{fig:pgate:ppbs} }\n\\end{figure}\n\n\n\\begin{figure}\n \\includegraphics{toffoli.eps}\n  \\caption{Optimal success probabilities of generalised Toffoli gates.\n    The features exhibited by $\\ensuremath{p_{\\mathrm{s}}}(\\varphi)$ are similar to the ones observed at the controlled phase gate (Fig.~\\ref{fig:phases}):\n    There is a shallow dip between $\\varphi=\\pi\/2$ and $\\varphi=\\pi$ below $\\ensuremath{p_{\\mathrm{s}}}(\\pi)=\\ensuremath{p_{\\mathrm{s}}}(\\pi\/2)$ and a steep incline (more pronounced than for the controlled phase gate) for small phases towards $\\ensuremath{p_{\\mathrm{s}}}(0)=1$.\n    \\label{fig:toffoli:ps}}\n\\end{figure}\n\n\n\\section{Toffoli gates}\nIn the same way as above, we can consider a generalised Toffoli gate, the effect of which on the \ncomputational basis realized as dual-rail encodings can be described by the unitary \n\\begin{equation}\n\tU=\\mathrm{diag}(1,1,1,1,1,1,1,\\mathrm{e}^{\\imath \\varphi}).\n\\end{equation}\t\nThe solutions to the polynomial equations describing the action on the three-mode core---up to mode permutations---can be parametrised by $x,y\\in\\mathbbm{R}$ and are given by the matrix\n\\begin{equation}\n  A=\\ensuremath{p_{\\mathrm{s}}}^{1\/6} \\left[\\begin{array}{ccc} 1 & \\frac{\\mathrm{e}^{\\imath\\varphi}-1}{xy} & 0 \\\\ 0 & 1 & y \\\\ x & 0 & 1 \\end{array}\\right] .\n\\end{equation}\nThis has to be understood similar to the $2$-mode core used by the controlled phase gate. However, here we do not solve the full optimisation problem, but only consider global rescaling.\n\nFor a unitary extension all singular values of $A$ have to be at most $1$. In order to avoid to formulate \ncubic singular values explicitly, we use the following constraints. Let\n$p_{AA^{\\dagger}}(\\lambda)=\\det(A A^{\\dagger}-\\lambda\\mathbbm{1}_3)$ be the characteristic polynomial of $A A^{\\dagger}$, the roots $\\lambda_{1,2,3}$ of which are the squared singular values of $A$.\nBy requiring $p_{AA^{\\dagger}}(1)=0$, one of the singular values has to be $1$. For the roots of $p_{AA^{\\dagger}}$ are real-valued and non-negative,\nthe condition that all other singular values are not larger \nthan $1$ is equivalent with the condition\nthat all derivatives of $p_{AA^{\\dagger}}$ have the same sign at $\\lambda=1$. \nMore formally, this results in further constraints of the form\n\\begin{equation}\n  (-1)^{\\alpha} p^{(k)}_{AA^{\\dagger}}(1) = (-1)^{\\alpha} \\left.\\frac{\\mathrm{d}p_{AA^{\\dagger}}(\\lambda)}{\\mathrm{d}\\lambda}\\right|_{\\lambda=1} \\ge0\n\\end{equation}\nfor $1\\le k<n$ where $\\alpha=0,1$ is fixed by the condition of $k=n$.\nFor $\\varphi=\\pi$ the optimal $\\ensuremath{p_{\\mathrm{s}}}$ compliant with these conditions is\n\\begin{equation}\n  \\ensuremath{p_{\\mathrm{s}}}(\\pi) = 1 + 3\\left( 2^{1\/3}-2^{2\/3} \\right) \\approx 1\/57 .\n\\end{equation}\nSee Fig.~\\ref{fig:toffoli:ps} for the maximum success probability in the range $0\\le\\varphi\\le\\pi$.\nThe corresponding networks could be constructed in the same way as above. However, they would consist of $3$-mode cores\n(an interferometer composed of three partially polarising beam splitters)\ninside separate interferometers\nfor each of the $3$ qubits.\nFor free space experiments more appealing approaches for the specific choice of \n$\\varphi=\\pi$ are presented in Refs.~\\cite{RRG07,Fiurasek06}, but \nleading only to success probabilities of at most $1\/72$.\n\n\\section{Remarks on process tomography}\n\nProcess tomography amounts to characterizing (or sometimes certifying) \nan unknown physical process. In practice, the task is to identify that completely\npositive map that is closest to the date with respect to some meaningful figure of merit.\nTo accomplish this task, \none has to consider a tomographically complete set of inputs and look at Hilbert-Schmidt\nscalar products of the output with observables, to faithfully reconstruct the matrix form of the channel \n\\cite{Process1,Process3}. \nPractically, the closest physical process can then be found by solving\na convex optimization problem. \nFormally equivalently, and in instances in an experimentally simpler fashion, one can, instead of\nsending in a full set of input states, submit half of a single fixed maximally entangled state, and hence \nreconstruct the channel from the Choi matrix, then referrred to as entanglement-assisted \nprocess tomography.\n\nThe latter technique can clearly also be applied in case of a postselected quantum gate\nlike a phase gate. Yet, even without entangled inputs and including the actual measurement,\none can  reconstruct the resulting POVM elements, to which essentially postselected gates\namount to when one faithfully includes also the actual measurement in the black-box description\nof the process. If one has a well-characterized source at hand, then the statistics of\n\\begin{equation}\n\tp_{j,k}= \\text{tr}[\\rho_j A_k],\n\\end{equation}\nuniquely characterize the process,\nwhere $\\{\\rho_j\\}$ form a tomographically complete well-characterized\ninput set, and $A_1,\\dots, A_K\\geq $ constitute a POVM, i.e., \n\\begin{equation}\n\t\\sum_{k=1}^K A_k=\\mathbbm{1}\n\\end{equation}\n(for an experimental realization of such an approach, see Ref.\\ \\cite{Process2}). In practice,\none uses methods of convex optimization to identify\nthe closest physical process to the given data.\n\nFor the purposes of the present work, \none of the outcomes $k=1$, that is, a specific pattern $A_1$\nof detection, then gives rise to the actual postselected linear\noptical quantum gate. In this way, one can reconstruct\npostselected quantum gates, without using an ancilla-based approach,\neven faithfully including the final measurement as part of the process.\n\n\\section{Conclusion}\n\nWe have shown how to obtain the maximum probability of success of controlled phase- and Toffoli-like gates in \nthe class of post-selected linear optics dual-rail gates without additional photons.\nFurther, constructions of networks for the smaller gates suitable for experimental implementation have been \ngiven, and techniques elaborated upon that allow for the assessment of the possibility\nof certain linear optical schemes.\nFor further progress concerning the eventual full optimization of linear optical processes \nit would be interesting\nto investigate (i) optimal constructions with respect to a class of gates inherently incorporating such \nexperimental constraints or (ii) identify further\ndecomposition techniques from a given linear optics mode transformation\ninto suitable physical networks, respecting these constraints.\n\n\\section*{Acknowledgements}\n\nFor insightful discussions concerning current \nexperimental \nabilities we would like to thank\nM.~Du{\\v{s}}ek,\nW.~Wieczorek,\nC.~Schmid, and\nJ.~Matthews. \nKK was supported by Microsoft Research through the \nEuropean PhD Programme and the EU (MINOS), \nJE by the EU (QAP, COMPAS, MINOS) \nand the EURYI award.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminaries}\n\nIn  \\cite{P}, H. Poincar\\'e has constructed a flow on a torus \nwith an exceptional minimal set. \nA. Denjoy has constructed such a $C^1$-flow \\cite{D}. \nOn the other hand,  \nthe author has shown that \nthere are no exceptional minimal sets of $C^2$-flows on tori. \nThis result is generalized to the compact surface cases by Schwartz, A. J. \\cite{S}.\nOn the other hand, \nin \\cite{NZ}, \nthey have shown the characterization of \nthe non-wandering flows on compact surfaces with finitely many singularities. \nIn \\cite{Mar}, \none has given a description near orbits of the non-wandering flow with \nthe set of singularities which is totally disconnected. \nMoreover, \nin  \\cite{M}, \nit has shown that \nif a nontrivial recurrent point $x$ of a surface flow \nbelongs to the limit\nset of another nontrivial recurrent point $y$, \nthen \n$y$ \nbelongs to the limit set of $x$. \nWe sharpen these results to analysis surface flows. \nIn particular, \nit has shown that \nthe orbit class of each \nnontrivial weakly recurrent point is \nthe orbit closure in \nthe set of regular weakly recurrent points. \nIn this paper, \nwe show the non-existence of exceptional minimal sets of \ncontinuous non-wandering flows \nwith arbitrary singularities \non compact surfaces \nand obtain  \na topological characterization of \nnon-wandering flows \nand \na topological characterization of \nnon-periodic proper orbits \nof non-wandering flows.  \nMoreover, we give \na smoothability of non-wandering flows \nand \na characterization of topological transitivity \nfor continuous flows. \nIn addition, \nwe construct a smooth flow on $\\mathbb{T}^2$ \nand a foliation on an open manifold contained in $\\mathbb{R}^2$ \nsuch that \n$\\mathrm{LD}$ and $\\mathrm{P}$ are dense, \nwhere \n$\\mathrm{LD}$ is the union of locally dense orbits (resp. leaves) \nand \n$\\mathrm{P}$ is the union of non-closed proper orbits (resp. leaves). \nOn the other hand, \nif $\\mathop{\\mathrm{Sing}}(v)$ \nconsists of finitely many contractible connected components, \nthen $\\mathrm{LD}$ is open \nand $\\mathrm{P}$ \nconsists of finitely many orbits. \n\n\n\nBy flows, \nwe mean continuous $\\mathbb{R}$-actions on surfaces.\nLet \n$v: \\mathbb{R} \\times M \\to M$ be a flow on a compact surface $M$. \nPut $v_t( \\cdot ) := v(t, \\cdot )$ \nand $O_v( \\cdot ) := v(\\mathbb{R}, \\cdot )$.  \nA subset of $M$ is said to be saturated \nif \nit is a union of orbits. \nRecall that \na point $x$ of $M$ is \nsingular if \n$x = v_t(x)$ for any $t \\in \\R$, \nis regular if \n$x$ is not singular, \nand \nis periodic if \nthere is positive number $T > 0$\nsuch that \n$x = v_T(x)$ and  \n$x \\neq v_t(x)$ for any $t \\in (0, T)$. \nDenote by \n$\\mathop{\\mathrm{Sing}}(v)$ \n(resp. $\\mathop{\\mathrm{Per}}(v)$) \nthe set of singular (resp. periodic) points. \nA point $x$ is wandering if \nthere are a neighbourhood $U$ of $x$ and \na positive number $N$ \nsuch that \n$\\bigcup_{t > N} v_t(U) \\cap U = \\emptyset$, \nand is  \nnon-wandering if  \n$x$ is not wandering \n(i.e. for each neighbourhood $U$ of $x$ and \neach positive number $N$, \nthere is $t \\in \\mathbb{R}$ with $|t| > N$ such that \n$v_t(U) \\cap U \\neq \\emptyset$). \nAn orbit is non-wandering if \nit consists of non-wandering points \nand \nthe flow $v$ is non-wandering if \nevery point is non-wandering. \nFor a point $x \\in M$, \ndefine \nthe omega limit set $\\omega(x)$\nand \nthe alpha limit set $\\alpha(x)$ \nof $x$\nas follows: \n$\\omega(x) \n:= \\bigcap_{n\\in \\mathbb{R}}\\overline{\\{v_t(x) \\mid t > n\\}} \n$, \n$\\alpha(x) \n:= \\bigcap_{n\\in \\mathbb{R}}\\overline{\\{v_t(x) \\mid t < n\\}} \n$. \nA point $x$ of $M$ is \npositive recurrent (resp. negative recurrent) if \n$x \\in \\omega(x)$ \n(resp.  $x \\in \\alpha(x)$),  \nand that \n$x$ is recurrent (resp. weakly recurrent) \nif $x$ is positive and (resp. or) negative recurrent. \nA (weakly) recurrent orbit is an orbit of such a point. \nAn orbit is proper if \nit is embedded, \nlocally dense if \nthe closure of it has nonempty interior, \nand \nexceptional if \nit is neither proper nor locally dense.  \nA point is proper (resp. locally dense, exceptional) if \nso is its orbit. \nDenote by \n$\\mathrm{LD}$ \n(resp. $\\mathrm{E}$, $\\mathrm{P}$)\nthe union of locally dense orbits \n(resp. exceptional orbits, \nnon-closed proper orbits).  \nNote \n$\\mathrm{P}$ is the complement of \nthe set of weakly recurrent points. \nBy the definitions,  \nwe have a decomposition \n$\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{P}\n\\sqcup \\mathrm{LD} \n\\sqcup \\mathrm{E} = M$. \nA (weakly) recurrent orbit is nontrivial if \nit is not closed.  \nNote that \nthe union of nontrivial weakly recurrent orbits \ncorresponds with $\\mathrm{LD} \\sqcup \\mathrm{E}$.   \nA quasi-minimal set of $v$ is the closure of a nontrivial weakly recurrent orbit. \nIt's known that \nthe total number of quasi-minimal sets for $v$ \ncannot exceed $g$ if $M$ is an orientable surface of genus $g$ \\cite{M}, \nand $\\frac{p-1}{2}$ if $M$ is a non-orientable surface of genus $p$ \\cite{Ma}. \nTherefore \nthe closure \n$\\overline{\\mathrm{LD \\sqcup E}}$ consists of finitely many quasi-minimal sets. \nBy a limit cycle, we mean \na periodic orbit of $v$ which is the $\\alpha$-limit set \nor the $\\omega$-limit set of some point \nnot on the periodic orbit. \n\n\n\n\n\n\\section{A Topological characterization of \nnon-wandering surface flows with arbitrary singularities}\n\nLet $v$ be a continuous flow on a compact surface $M$. \nWe call \nthat a collar $A$ of an periodic orbit $O \\subset A$  is \nan annulus $A$ one of whose connected component of \n$\\partial A$ is $O$, \nwhere $\\partial A := \\overline{A} - \\mathrm{int} A$ \nis the topological boundary of $A$. \nFirst, we state a following easy observation. \n\n\\begin{lemma}\\label{lem00}\n$O \\subset \\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{P}$ \nfor an orbit $O$ with $\\overline{O} \\cap \\mathop{\\mathrm{Per}}(v) \\neq \\emptyset$.  \nMoreover \neach limit cycle is contained in \n$\\partial (\\mathrm{int} \\mathrm{P})$. \n\\end{lemma}\n\n\\begin{proof}\nLet $O$ be an non-periodic orbit with $\\overline{O} \\cap \\mathop{\\mathrm{Per}}(v) \\neq \\emptyset$. \nThen $\\overline{O} - O$ contains a limit cycle $\\gamma$. \nThe  flow box theorem (cf. Theorem 1.1, p.45\\cite{ABZ}) implies that \na limit cycle $\\gamma$ is covered by finitely many flow boxes $\\{ U_i' \\}$. \nSince $\\overline{O} \\cap \\gamma \\neq \\emptyset$,  \nthe one-sided holonomy of $\\gamma$ is contracting or expanding. \nFix a point $z$ of $O$ in a flow box $U_i'$. \nThen the point $z$ has an open neighbourhood $U \\subset U_i'$ \nwhich is a flow box such that \n$O \\cap U$ is one arc. \nThen $\\cup_{t \\in \\mathbb{R}} v_t(U)$ \nis an open neighbourhood of $O$ in \nwhich $O$ is closed.    \nHence $O \\subset \\mathrm{P}$.  \nLet $U' \\subset \\bigcup_i U_i'$ be \na sufficiently small collar of $\\gamma$ \nwhere the holonomy along $\\gamma$ is contracting or expanding.  \nThen the orbit closure of each point $y$ in $U' - \\gamma$ contains $\\gamma$ \nbut the orbit of $y$ is not closed. \nTherefore $y \\in \\mathrm{P}$\n and so $V := \\cup_{t \\in \\mathbb{R}} v_t(U') - \\gamma \\subseteq \\mathrm{P}$ \nis a saturated open subset with $\\gamma \\subseteq \\partial V \n\\subseteq \\partial (\\mathrm{int} \\mathrm{P})$. \nThis implies the second assertion. \n\\end{proof}\n\n\nRecall that \nthe orbit class $\\hat{O}$ of an orbit $O$ is \nthe union of points each of whose orbit closure \ncorresponds with $\\overline{O}$ \n(i.e.  $\\hat{O} := \\{ y \\in M \\mid \\overline{O} = \\overline{O_v(y)} \\}$). \nNow we show that \nthe orbit class of each \nnontrivial weakly recurrent point is \nthe orbit closure in \nthe set of regular weakly recurrent points, \nwhich \nrefine a Ma\\v \\i er-type result \\cite{M}.  \n\n\\begin{proposition}\\label{lem001}\nFor an \norbit $O \\subset \\mathrm{LD} \\sqcup \\mathrm{E}$,  \nwe have \n$\\hat{O} = \\overline{O} \n\\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}) \n\\subset \\mathrm{LD} \\sqcup \\mathrm{E}$. \n\\end{proposition}\n\n\\begin{proof}\nLet $Q$ be a quasi-minimal set which is the closure of $O$. \nBy Lemma \\ref{lem00}, \nwe have $Q \\cap \\mathop{\\mathrm{Per}}(v)  = \\emptyset$. \nNote that \nthe inverse image of \nproper (resp. locally dense, exceptional) orbits \nby any finite covering \nare also proper (resp. locally dense, exceptional). \nBy taking a double covering of $M$ and the doubling of $M$ \nif necessary, \nwe may assume that $v$ is transversally orientable \nand $M$ is closed and orientable. \nFor any point $y \\in \\hat{O} - O$, \nwe have \n$\\overline{O_v(y)} = \\overline{O}$ \nand so \n$O \\subseteq \\omega(y) \\cup \\alpha(y)$. \nSince $y \\in \\overline{O}  - O$, \nwe obtain $y \\in \\omega(y) \\cup \\alpha(y)$. \nThen $y$ is not proper. \nThe regularity of $y$ implies \n$y \\notin \\mathop{\\mathrm{Sing}}(v)$ \nand so \n$y \\in Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}) \n\\subseteq \\mathrm{LD} \\sqcup \\mathrm{E}$. \nThus \n$\\hat{O} \\subseteq Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P})$. \nOn the other hand, \nwe show that \n$\\hat{O} \\supseteq Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P})$. \nIndeed, \nif there is exactly one quasi-minimal set $Q$, \nthen \n$\\overline{O_v(x)} = Q$ for any \n$x \\in Q  \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}) \n\\subseteq \\mathrm{LD} \\sqcup \\mathrm{E}$. \nThus we may assume that \nthere are at least two quasi-minimal sets. \nThe above Ma\\v \\i er work \\cite{M} \n(cf. Remark 2 \\cite{AZ}) impies that \nthe genus of  $M$ is at least two.  \nNote \nCherry has proved that a quasi-minimal set contains \na continuum of nontrivially recurrent orbits each of  which is dense\nin the quasi-minimal set (Theorem VI\\cite{C}).  \nTherefore $Q$ contains \nnontrivially recurrent orbits. \nFix a recurrent point $x \\in (\\mathrm{LD} \\sqcup \\mathrm{E}) \\cap Q$ \nwhose orbit closure is $Q$. \nFor \nany point $y \\in Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P})$, \nwe have \n$y \\in \\mathrm{LD} \\sqcup \\mathrm{E}$  \nand so \n$y$ is weakly recurrent. \nBy the above Cherry result, \nthere is a recurrent point $z \\in \\overline{O_v(y)}$ \nwhose orbit closure is $\\overline{O_v(y)}$. \nBy another Ma\\v \\i er theorem (cf. Theorem 4.2 \\cite{AZ}) \nand its dual, \nwe obtain \n$\\omega(x) = \\omega(z)$ \nand \n$\\alpha(x) = \\alpha(z)$.  \nThus \n$\\overline{O} = \nQ = \n\\overline{O_v(x)} = \n\\overline{O_v(z)} = \n\\overline{O_v(y)}$. \nThis means  \n$\\hat{O} = Q \\setminus (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P})$.  \n\\end{proof}\n\nWe state a key lemma which is a relation between exceptional \nand proper orbits.\nRecall that a subset $S$ of a surface $M$ is essential if \nsome connected component of $S$ \nis neither null homotopic \nnor homotopic to a subset of the boundary $\\partial M$. \n\n\n\\begin{lemma}\\label{lem0bb}\n$\\overline{\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{LD}} \n\\cap \\mathrm{E}= \\emptyset$ \nand \n$\\overline{\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{E}} \n\\cap \\mathrm{LD}= \\emptyset$.  \nMoreover \n$\\mathrm{E}\\subset  \\mathrm{int} \\overline{\\mathrm{P}}$. \n\\end{lemma}\n\n\n\\begin{proof}\nBy taking a double covering of $M$ and the doubling of $M$ \nif necessary, \nwe may assume that $v$ is transversally orientable \nand $M$ is closed and orientable. \nBy the Ma\\v \\i er theorem, \n$\\overline{\\mathrm{E}}$\n(resp. $\\overline{\\mathrm{LD}}$) \nconsists of finitely many closures of \nexceptional (resp. locally dense) orbits.  \nBy Proposition \\ref{lem001}, \nwe have \n$\\overline{\\mathrm{LD}} \\cap \\mathrm{E} = \\emptyset$ \nand \n$\\mathrm{LD} \\cap \\overline{\\mathrm{E}} = \\emptyset$. \nRecall that \nthe flow box theorem \nimplies that \nthe orbits of $v$ on a surface $M - \\mathop{\\mathrm{Sing}}(v)$ \nform a foliation $\\mathcal{F}$. \nMoreover there is \na transverse foliation $\\mathcal{L}$ for this foliation $\\mathcal{F}$ by \nProposition 2.3.8 (p.18 \\cite{HH}). \nWe show \n$ \\overline{\\mathop{\\mathrm{Per}}(v)} \\cap (\\mathrm{E} \\sqcup \\mathrm{LD}) =  \\emptyset$.\nOtherwise\nthere is a sequence of periodic orbits $O_i$ \nsuch that \nthe closure $\\overline{\\cup_i O_i}$ \ncontains \na point $x \\in \\mathrm{E} \\sqcup \\mathrm{LD}$.  \nBy Lemma \\ref{lem00}, \nwe have  $\\overline{O_v(x)} \\cap \\mathop{\\mathrm{Per}}(v) = \\emptyset$. \nWe show that \nthere is $K \\in \\mathbb{Z}_{>0}$ such that \n$O_k$ is contractible in $M_K$ \nfor any $k > K$ \nwhere $M_K$ is the resulting closed surface \nof adding \n$2K$ disks to $M - (O_1 \\sqcup \\cdots \\sqcup O_K)$. \nIndeed,  \nwe may assume that \n$O_1$ is \nessential \nby renumbering. \nLet $M_1$ be \nthe resulting closed surface \nof adding two center disks to $M - O_1$. \nThen $g(M_1)< g(M)$, \nwhere $g(N)$ is the genus of a surface $N$. \nSince $M$ and $M_1$ are closed surfaces, \nby induction \nfor essential closed curves \nat most $g(M)$ times, \nthe assertion is followed. \nSince $M$ is normal, \nthere are open disjoint neighbourhoods $U_x$ and $V$ of \n$\\overline{O_v(x)}$ and $\\cup_{i \\leq K} O_i$ respectively.\nThen \nthere is \na transverse arc $\\gamma \\subset U_x \\cap l_x$ through $x$ \nwhich does not intersect $\\cup_{i \\leq K} O_i$, \nwhere $l_x \\in \\mathcal{L}$. \nSince $O_v(x)$ is exceptional or locally dense, \nwe have \n$x$ is a  weakly recurrent point and so \nthere is an arc $\\gamma'$ in $O_v(x)$ \nwhose boundaries are contained in $\\gamma$. \nSince $M$ is normal, \nthere is a neighbourhood of $\\gamma \\cup \\gamma'$ \nwhich does not intersect $\\cup_{i \\leq K} O_i$. \nBy the waterfall constraction (cf. Lemma 1.2, p.46\\cite{ABZ}), \nwe can construct \na closed transversal $T$ for $v$ through $x$ \nwhich does not intersect $\\cup_{i \\leq K} O_i$. \nLet $v_K$ be a resulting flow on $M_K$ by adding center disks. \nThen $T$ is also a closed transversal for $v_K$. \nHowever there is $k > K$ such that \n$T$ intersects $O_k$ which is contractible in $M_K$. \nThis is impossible. \nThe closedness of $\\mathop{\\mathrm{Sing}}(v)$ \nimplies the first assertion. \nSince $M = \n\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \n\\mathrm{LD}\\sqcup \\mathrm{P} \\sqcup \\mathrm{E}$, \nwe obtain that \n$\\mathrm{E}$ is contained in an open subset \n$M - \\overline{\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{LD}} \n\\subseteq \\mathrm{P} \\sqcup \\mathrm{E}$.  \nTherefore \n$\\mathrm{E} \\subseteq \\mathrm{int}(\\mathrm{P} \\sqcup \\mathrm{E})$. \nSince $\\overline{\\mathrm{E}}$ consists of finitely many closures of exceptional orbits, \nwe have that \n${\\mathrm{E}}$ is nowhere dense. \nThis implies \n$\\mathrm{E} \\subset \\overline{\\mathrm{P}}$ \nand so  \n$\\mathrm{E} \\subset \\mathrm{int}(\\mathrm{P} \\sqcup \\mathrm{E}) \n \\subset  \\mathrm{int} \\overline{\\mathrm{P}}$. \n\\end{proof}\n\n\n\n\n\n\n\nFrom now on, \nwe consider only non-wandering cases \nin this section. \n\n\n\\begin{lemma}\\label{lem0aa}\nLet $v$ a non-wandering flow on a compact surface $M$. \nThen \n$\\mathop{\\mathrm{Per}}(v)$ is open, \n$M = \n\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \n\\mathrm{LD}\\sqcup \\mathrm{P}$, \nand \n$\\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)}  \n\\supseteq M - \\mathop{\\mathrm{Sing}}(v)$. \n\\end{lemma}\n\n\n\n\n\n\n\\begin{proof}\nBy taking a double covering of $M$ if necessary, \nwe may assume that $v$ is transversally orientable. \nBy Theorem III.2.12, III.2.15 \\cite{BS}, \nthe set of recurrence points is dense in $M$. \nBy Lemma \\ref{lem00}, \nthere are no limit cycles. \nWrite $U := M - \\overline{\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{LD}}$.  \nBy Lemma \\ref{lem0bb}, \nthis $U \\subseteq \\mathrm{P} \\sqcup \\mathrm{E}$ is an open \\nbd of \n$\\mathrm{E}$. \nSince each point of $\\mathrm{P}$ is not weakly recurrent, \nwe have $\\overline{\\mathrm{E}} \\supseteq U$. \nSince $\\mathrm{E}$ is nowhere dense, \nwe have $U$ is empty and so is $\\mathrm{E}$. \nHence \n$M = \n\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathop{\\mathrm{Per}}(v) \n\\sqcup \n\\mathrm{LD}\\sqcup \\mathrm{P}$. \nSince there are no limit cycles, \nwe obtain \n$\\overline{\\mathrm{LD} \n} \\cap \\mathop{\\mathrm{Per}}(v) = \\emptyset$.  \nFix a periodic orbit $O$. \nThen there is an annular  \n\\nbd $V$ of $O$ \nwhich is a finite union of flow boxes \nsuch that \n$V \\cap \n(\\mathop{\\mathrm{Sing}}(v) \\sqcup \\overline{\\mathrm{LD}} )\n = \\emptyset$. \nHence \n$V \\subseteq \n\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{P}$.  \nSince the set of recurrence points is dense in $M$, \nwe have that \n$V \\cap \\mathop{\\mathrm{Per}}(v)$ \nis dense in $V$. \nTherefore \nthe holonomy of $O$ is \nidentical \nand so \n$V \\subseteq \\mathop{\\mathrm{Per}}(v)$. \nThis means that \n$\\mathop{\\mathrm{Per}}(v)$ is open. \nFor any $x \\in \\mathrm{P}$, \nsince the regular weakly recurrent points form \n$\\mathrm{Per}(v) \\sqcup \\mathrm{LD}$, \nby non-wandering property, \neach neighbourhood of $x$ meets \n$\\mathrm{Per}(v) \\sqcup \\mathrm{LD}$ \nand so \n$\\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)} \\supseteq \\mathrm{P}$.  \n\\end{proof}\n\n\n\n\n\n\n\n\nNow we state the characterization of non-wandering flows. \n\n\\begin{theorem}\\label{prop0c}\nLet $v$ be a continuous flow on a compact surface $M$. \nThen \n$v$ is non-wandering \nif and only if \n$\\overline{\\mathrm{LD} \\sqcup \\mathop{\\mathrm{Per}}(v)} \\cup \\mathop{\\mathrm{Sing}}(v) \n= M$.  \nIn particular, \nif $v$ is non-wandering, \nthen \n$\\mathop{\\mathrm{Per}}(v)$ \nis open \nand \nthere are no exceptional orbits. \n\\end{theorem}\n\n\n\n\n\\begin{proof}\nSuppose that $v$ is non-wandering. \nBy Lemma \\ref{lem0aa}, \nwe have \n$\\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)} \\cup \\mathop{\\mathrm{Sing}}(v) \n= M$.  \nConversely, \nsuppose that \n$\\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)} \\cup \\mathop{\\mathrm{Sing}}(v) = M$.  \nFor any regular point $x$ of $M$, \nwe have \n$x \\in \\overline{\\mathrm{LD}\\sqcup \\mathop{\\mathrm{Per}}(v)}$. \nThis shows that $v$ is non-wandering. \n\\end{proof}\n\n\n\nWe state the following charcterization of \nthe union \n$\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)$ of non-periodic proper orbits. \n\n\n\\begin{proposition}\\label{lem32}\nLet $v$ be a continuous non-wandering flow \non a compact surface $M$. \nThen $\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v) = \n\\{ x \\in M \\mid \\omega(x) \\cup \\alpha(x) \\subseteq \\mathop{\\mathrm{Sing}}(v) \\}$. \n\\end{proposition}\n\n\\begin{proof}\nWe may assume \n$M$ is connected. \nBy taking a double covering of $M$ and the doubling of $M$ \nif necessary, \nwe may assume that $v$ is transversally orientable \nand $M$ is closed and orientable. \nObviously \n$\\mathrm{P}  \\sqcup \\mathop{\\mathrm{Sing}}(v) \n\\supseteq \\{ x \\in M \\mid \\omega(x) \\cup \\alpha(x) \\subseteq \\mathop{\\mathrm{Sing}}(v) \\}$. \nTherefore \nit suffices to show this converse relation.\nFix a point $y \\in \\mathrm{P}$. \nBy Lemma \\ref{lem00}, \nthe non-wandering property implies  \n$\\overline{O_v(y)} \\cap \\mathop{\\mathrm{Per}}(v) = \\emptyset$. \nWe show \n$\\overline{O_v(y)} \\cap \\mathrm{LD} = \\emptyset$. \nOtherwise \n$\\overline{O_v(y)}$ contains a locally dense orbit $O$. \nSince $\\mathrm{int}\\overline{O}$ is an open \\nbd of $O$, \nwe have  \n$O_v(y) \\cap \\mathrm{int}\\overline{O} \\neq \\emptyset$. \nThe relation \n$y \\in \\overline{O} \\subseteq \\overline{O_v(y)}$ \nimplies that \n$y$ is not proper, which is a contradiction. \nSince $\\mathrm{E} = \\emptyset$, \nwe have \n$\\overline{O_v(y)} \\subset \\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)$. \nSuppose that \n$y \\in \\overline{\\mathrm{LD}}$. \nThus there is a recurrent point in $\\mathrm{LD}$ \nwhose orbit closure contains $O_v(y)$. \nRemoving the singular points, \nTheorem 3.1\\cite{Marz} implies \n$\\omega(y) \\cup \\alpha(y) \\subseteq \\mathop{\\mathrm{Sing}}(v)$. \nThis means \n$y \\in \n\\{ x \\in M \\mid \\omega(x) \\cup \\alpha(x) \\subseteq \\mathop{\\mathrm{Sing}}(v) \\}$. \nOtherwise \n$y \\notin \\overline{\\mathrm{LD}}$. \nSuppose that \neach periodic orbit is null homotopic (i.e. non-essential). \nFix a saturated \\nbd $W$ of $y$. \nThen the intersection   \n$W \\cap \\mathop{\\mathrm{Per}}(v)$ \nis open dense in $W$.  \nTo apply Theorem 3.1\\cite{Marz}, \nwe need replace a small flow box near $O_v(y)$. \nTake a flow box $B \\subset W$ \nwhich can be identified with $[-1, 1] \\times [-1, 1]$ \nsuch that \n$O_v(y) \\cap B = [-1, 1] \\times \\{ 0 \\}$ \nand \neach orbit in $B$ is $[-1, 1] \\times \\{ a \\}$ for some $a \\in [-1, 1]$. \nFix a small transverse arc $\\gamma$ through $y$ in $B$. \nLet $\\gamma_+$ be a connected component of $\\gamma \\setminus O$. \nSince each periodic orbit is null homotopic, \neach periodic orbit intersects $\\gamma_+$ at most one point. \nThis means that \neach point $ p:= (1, \\varepsilon)$ in $\\mathop{\\mathrm{Per}}(v) \\cap (\\{ 1 \\} \\times [-1, 1])$ \nintersecting $\\gamma_+$ \ngoes back to a point $(-1, \\varepsilon)$ in $B$ \n(i.e. $O_v(p) \\cap B = \\{ 1 \\} \\times \\{ \\varepsilon \\}$). \nTherefore \nthe flow $v|_B$ induces \na homeomorphism \nfrom $\\{-1 \\} \\times [-1, 1]$ \nto $\\{ 1 \\} \\times [-1, 1]$ \nwhich can be identified with the identity mapping $1_{[-1, 1]}$ on $[-1, 1]$. \nReplacing $1_{[-1, 1]}$ \nwith a homeomorphism $f$ which is contracting near $0$ \n(e.g. $f(x) = x^3$), \nwe obtain the resulting continuous flow $w$ \nsuch that \n$O_v(y) = O_w(y)$ and \n$\\mathop{\\mathrm{Sing}}(w) = \n\\mathop{\\mathrm{Sing}}(v)$, \nmodifying $v$ in $B$. \nWe identify $O_v(y) \\cap B = [-1, 1] \\times \\{ 0 \\}$ \nwith $\\mathrm{dom} (f) = [-1, 1]$. \nBy the Baire category theorem, \nthe countable intersection \n$\\bigcap_{n \\in \\Z} f^{n}([-1, 1] \\cap \\mathop{\\mathrm{Per}}(v))$ \nis dense in $[-1, 1]$. \nThus \nthere is a point $z \\in \\mathop{\\mathrm{Per}}(v)$ \nsuch that \n$O_w(z)$ is proper \nand \n$y \\in \\overline{O_w(z)} - O_w(z)$.  \nRemoving the singular points, \nTheorem 3.1\\cite{Marz} for $w$ implies \n$\\overline{O_w(y)} - O_w(y) = \n\\alpha_w(y) \\cup \\omega_w(y) \\subseteq \\mathop{\\mathrm{Sing}}(w)$. \nSince \n$O_v(y) = O_w(y)$ and \n$\\mathop{\\mathrm{Sing}}(w) = \n\\mathop{\\mathrm{Sing}}(v)$, \nwe have \n$\\omega(y) \\cup \\alpha(y) \\subseteq \\mathop{\\mathrm{Sing}}(v)$. \nSuppose that \nthere are essential periodic orbits. \nLet $O$ be an essential periodic orbit. \nCutting this periodic orbit $O$, \nthe remainder $M - O$ has \nnew two boundaries. \nAdding two center disks to the two boundaries of $M - O$,  \nwe obtain \na new closed surface $M'$ whose genus is less than one of $M$ \nand the resulting non-wandering flow $v'$ on $M'$ \nwith \n$\\mathop{\\mathrm{Sing}}(v') = \\mathop{\\mathrm{Sing}}(v)$\nsuch that \n$\\alpha(y) = \\alpha_{v'}(y)$ \nand \n$\\omega(y) = \\omega_{v'}(y)$, \nwhere \n$\\alpha_{v'}(y)$ (resp. $\\omega_{v'}(y)$) is \nthe alpha (resp. omega) limit set of $y$ with respect to $v'$. \nBy finite iterations, \nwe obtain a new closed surface $M^{\\dagger}$ \nand the resulting non-wandering flow $v^{\\dagger}$ on $M^{\\dagger}$ \nwith \n$\\mathop{\\mathrm{Sing}}(v^{\\dagger}) = \\mathop{\\mathrm{Sing}}(v)$\nsuch that \n$\\alpha(y) = \\alpha_{v^{\\dagger}}(y)$ \nand \n$\\omega(y) = \\omega_{v^{\\dagger}}(y)$ \nsuch that \neach periodic orbit of $v^{\\dagger}$ is null homotopic. \nThis implies \n$\\omega(y) \\cup \\alpha(y) \\subseteq \\mathop{\\mathrm{Sing}}(v)$. \n\\end{proof}\n\n\n\n\n\n\n\n\nTaking a suspension of \na non-wandering circle homeomorphism, \nwe have the following statement. \n\n\n\n\n\\begin{corollary}\\label{cor26}\nEach non-wandering continuous homeomorphism \non $\\S^1$  \nis topologically conjugate to \na rotation. \n\\end{corollary}\n\n\n\n\\begin{proof}\nLet $v$ be the suspension of \na non-wandering circle homeomorphism $f$. \nThen $v$ is a non-wandering continuous flow on $\\T^2$.\nSince $v$ is a suspension, \n there are no singular points. \n By Proposition \\ref{lem32}, \n we have \n $\\mathrm{P} = \\emptyset$.  \nTheorem \\ref{prop0c} implies \n $\\mathop{\\mathrm{Per}}(v) \\sqcup \\mathrm{LD} = \\T^2$. \n Since $\\mathop{\\mathrm{Per}}(v)$ \nis open, \nthe complement \n$\\mathrm{LD} = \\T^2 - \\mathop{\\mathrm{Per}}(v)$ \nis closed. \nBy Lemma \\ref{lem0bb}, \nwe obtain \n$\\overline{\\mathop{\\mathrm{Per}}(v)} \n\\cap \\overline{\\mathrm{LD}} \n= \\overline{\\mathop{\\mathrm{Per}}(v)} \n\\cap {\\mathrm{LD}}  \n= \\emptyset$. \nThis implies that \nthe both of \n$\\mathop{\\mathrm{Per}}(v)$ \nand $\\mathrm{LD}$ \nare open and closed. \nSince $\\T^2$ is connected, \nwe have that \n$v$ is pointwise periodic or \nminimal. \nThen $f$ is topologically conjugate to periodic or \nminimal. \nThis means that \n$f$ is topologically conjugate to \na rotation. \n\\end{proof}\n\n\n\n\nBy the smoothing result \\cite{G}, \nwe have the following result.\n\n\\begin{corollary}\\label{cor21}\nEach non-wandering continuous flow on a compact surface $M$ \nis topologically equivalent to a $C^{\\infty}$ flow. \n\\end{corollary}\n\n\n\\begin{proof}\nWe may assume that \n$M$ is connected. \nBy Theorem \\ref{prop0c}, \nthere are no exceptional orbits. \nBy Proposition \\ref{lem32}, \nthe closure of each orbit in $\\mathrm{P}$ contains \nsingular points.  \nWe show that \nif the closure of locally dense orbit is minimal \nthen it is the whole surface which is $\\mathbb{T}^2$. \nIndeed, \nlet $O$ be a locally dense orbit whose closure is minimal. \nThen $\\overline{O}$ contains \nneither singular points\nnor periodic points. \nSince the closure of an orbit in $\\mathrm{P}$ contains \nsingular points, \nthe minimal set $\\overline{O}$ consists of \nlocally dense orbits. \nSince each point of $\\overline{O}$ is contained \nin the interior of $\\overline{O}$, \nthis minimal set $\\overline{O}$ is closed and open, and so  \nis the whole surface  $M$ which is homeomorphic to $\\mathbb{T}^2$.\nTherefore each minimal set is \neither a closed orbit or $\\mathbb{T}^2$. \nBy the smoothing theorem \\cite{G}, \nthis continuous flow is topologically equivalent to a $C^{\\infty}$ flow. \n\\end{proof}\n\n\n\n\nWe state uniformity of $\\mathop{\\mathrm{Per}}(v)$ \nof a non-wandering continuous flow $v$. \n\n\\begin{corollary}\\label{cor29}\nLet $v$ be a non-wandering continuous flow \non a compact surface $M$.  \nThen \neach connected component of $\\mathop{\\mathrm{Per}}(v)$ \nis either \na connected component of $M$,   \nan open annulus, or \nan open M\\\"obius band. \nIf \n$\\mathop{\\mathrm{Sing}}(v)$ \nconsists of finitely many contractible connected components, \nthen $\\mathrm{P}$ consists of finitely many separatrices,  \n$\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}$ is closed, \nand \n$\\mathrm{LD}$ is open. \n\\end{corollary}\n\n\n\\begin{proof} \nSince $\\mathop{\\mathrm{Per}}(v)$ is open, \nthe flow box theorem \nimplies that \neach periodic orbit has a saturated \\nbd \nwhich is either an annulus or a M\\\"obius band. \nNotice that  \na union of one saturated M\\\"obius band and one saturated annulus with one intersection \nis a M\\\"obius band \nand that \na union of two saturated annuli (resp. M\\\"obius bands) with one intersection \nis an annulus (resp. a Klein bottle).\nFix a connected component $U$ of $\\mathop{\\mathrm{Per}}(v)$.  \nIf $\\partial U = \\emptyset$, \nthen $U$ is a connected component of $M$. \nIf $\\partial U \\neq \\emptyset$, \nthen $U$ is \neither an annulus \nor a M\\\"obius band. \nSuppose that $\\mathop{\\mathrm{Sing}}(v)$ \nconsists of finitely many contractible connected components. \nCollapsing each connected component of \n$\\mathop{\\mathrm{Sing}}(v)$ into a point, \nby Theorem 1\\cite{RS}, \nthe resulting space is homeomorphic to \nthe original space $M$ \nand the resulting flow is non-wandering. \nThus we may assume that \n$\\mathop{\\mathrm{Sing}}(v)$ is finite.  \nBy Theorem 3\\cite{CGL}, \neach singular point is \neither a center or a multi-saddle. \nBy Proposition \\ref{lem32}, \neach orbit in $\\mathrm{P}$ is \na separatrix. \nTherefore \n$\\mathrm{P}$ consists of finitely many separatrices and so   \n$\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P}$ is closed.   \nSince $\\overline{\\mathop{\\mathrm{Per}}(v)} \\cap \\mathrm{LD} = \\emptyset$, \nthe complement \n$\\mathrm{LD} = M - (\\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{P} \\sqcup \\mathop{\\mathrm{Per}}(v))$ is open. \n\\end{proof} \n\n\n\nWe show that \n$\\mathrm{LD}$ is not open in general.  \nIn fact, \nwe construct \na non-wandering flow on $\\T^2$ \nsuch that \n$\\mathrm{P}$ and \n$\\mathrm{LD}$ are dense as follows. \n\n\n\n\n\\begin{example}\\label{ex1}\nConsider an irrational rotation $v$ on $\\T^2$. \nFix any points $p \\in \\T^2$. \nLet  $( t_i )_{i \\in \\Z}$ of $\\R$ be \na sequence such that \n$\\lim_{i \\to \\infty} t_i = \\infty$, \n$\\lim_{i \\to - \\infty} t_i = - \\infty$, \nand \n$\\lim_{i \\to \\infty} p_i \n= \\lim_{i \\to - \\infty} p_i \n= p_0$, \nwhere $p_i := v(t_i, p_0) \\in O_v(p)$.  \nUsing dump functions, \nreplace $O_v(p)$ \nwith a union of countably many singular points $p_i$ ($i \\in \\Z$) \nand countably many proper orbits. \nLet $v'$ be the resulting vector field. \nFor any point $x \\in \\T^2 - O(p)$, \nwe have $O_v(x) = O_{v'}(x)$. \nMoreover $O_v(p) - \\mathop{\\mathrm{Sing}}(v') = \\mathrm{P}(v')$.  \nBy construction, \n$\\mathrm{LD}$ is not open but \n$\\overline{\\mathrm{P}} = \\overline{\\mathrm{LD}} = \\T^2$. \nThis example also shows that \nthe finiteness condition in \nCorollary \\ref{cor29} \nis necessary. \n\\end{example}\n\nIn \\cite{H2}, \na foliation $\\F$  on a manifold $M$ is \nsaid to be ``rare species'' \nif \neither \nall leaves are exceptional \nor \n$\\F$ has at least two of three type (i.e. proper, locally dense, exceptional) \nand the union of leaves of each type is dense. \nThe author \nhave constructed \nsome kind of   \ncodimension one ``rare species''  foliations on compact $3$-manifolds.\nAnalogically we call that \na flow $v$ on $M$ is \n``rare species''\nif \neither \n$\\mathrm{E} = M$ \nor \n$v$ has at least two of three type (i.e. proper, locally dense, exceptional) \nand the union of orbits of each type is dense. \nNote \nthat ``rare species'' surface flows \nare non-wandering and so \nhave no exceptional orbits. \nIn contrast to \nfoliations, \nthis implies \nthere is only one possible kind of \n``rare species'' flows on compact surfaces.  \nNotice that \nthe above example is \na ``rare species'' flows on $\\mathbb{T}^2$. \nTherefore, \nwe obtain the following statement. \n\n\n\n\n\\begin{proposition}\\label{prop}\nThere are smooth ``rare species'' flows on $\\mathbb{T}^2$.   \nOn the other hand, \nfor each ``rare species'' flow $v$ on a compact surface $M$, we have \n$M = \\mathop{\\mathrm{Sing}}(v) \\sqcup \\mathrm{LD} \\sqcup \\mathrm{P}$ \nand \n$\\overline{\\mathrm{LD}} = \n\\overline{\\mathrm{P}} = M$.   \n\\end{proposition}\n\n\nNote that \nall known codimension one foliations on \ncompact manifolds are not $C^1$ but continuous. \nIn contrast to codimension one foliations on \ncompact manifolds,  \neach ``rare species'' flow $v$ on a compact surface \nis topologically equivalent \nto a $C^{\\infty}$ flow. \nMoreover, \nwe obtain following examples. \n\n\\begin{corollary}\\label{cor}\nThere are codimension one ``rare species'' foliations \non open surfaces contained in $\\R^2$.  \nIn particular, \nthe union of locally dense leaves \nis not open. \n\\end{corollary}\n\n\\begin{proof}\nFix a minimal codimension one foliation $\\F$ \non open surfaces contained in $\\R^2$  \n(e.g. a foliation in Theorem \\cite{F}). \nReplace a locally dense leaf $L$ \ninto countably many singular points and \nproper leaves connecting two singular points,  \nand  \nremove the singular points as above construction for flows. \nThen we can obtain a desired foliation. \n\\end{proof}\n\n\n\n\\section{Applications of this characterization}\n\n\n\n\n\n\n\n\n\n\n\nRecall that \n$v$ is topologically transitive if \nit has a dense orbit.  \nClosed orbits are singular points or periodic orbits.\nA subset is said to be co-connected if \nthe complement of it is connected. \nWe have a following characterization of \ntransitivity for surface flows. \n\n\n\n\\begin{theorem}\\label{th0}\nLet $v$ be a continuous flow on a compact surface $M$. \nThen \nthe following are equivalent: \n\\\\\n1. \n$v$ is topologically transitive. \n\\\\\n2.  \n$v$ is non-wandering \nsuch that \n$\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)$ is co-connected \nand \n$ \\mathop{\\mathrm{int}}(\\mathop{\\mathrm{Per}}(v) \n\\sqcup \\mathop{\\mathrm{Sing}}(v)) = \\emptyset$.  \n\\\\\n3. \n$v$ is non-wandering \nsuch that \nthe set of regular weakly recurrent points is connected \nand \nthe interior of \nthe union of closed orbits is empty.  \n\\\\\n4. \n$\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)$ is co-connected \nand \n$\\mathop{\\mathrm{int}} \\mathop{\\mathrm{Sing}}(v) = \n\\mathop{\\mathrm{int}} \\mathop{\\mathrm{Per}}(v) = \n\\mathop{\\mathrm{int}} \\mathrm{P} = \n\\emptyset$.  \n\nIn each case, \n$M \n= \\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathrm{P} \\sqcup \\mathrm{LD} \n= \\overline{\\mathrm{LD}}$ \nand \neach locally dense orbit is dense. \n\\end{theorem}\n\n\n\\begin{proof}\nObviously, \nthe conditions 2 and 3 are equivalent. \nSince $\\mathrm{P}$ is the complement of \nthe set of the weakly recurrent points, \n$v$ is non-wandering if \nand only if $\\mathop{\\mathrm{int}} \\mathrm{P} = \\emptyset$.  \nLemma \\ref{lem0aa} implies that \nthe conditions 2 and 4 are equivalent. \nSuppose that \n$v$ is topologically transitive. \nThen $v$ is non-wandering. \nThe openness of  $\\mathop{\\mathrm{Per}}(v)$ \nimplies $\\mathop{\\mathrm{Per}}(v) = \\emptyset$. \nBy transitivity, \nthere is a dense orbit $O$ \nand so $\\overline{\\mathrm{LD}} = M$. \nSince $O$ is connected \nand $O \\subseteq \\mathrm{LD} \n\\subseteq \\overline{O} = M$, \nwe have that \n$M - (\\mathrm{P} \\sqcup \\mathop{\\mathrm{Sing}}(v)) = \\mathrm{LD}$ is connected. \nConversely, \nsuppose \n$v$ is non-wandering \nsuch that \n$ \\mathop{\\mathrm{int}}(\\mathop{\\mathrm{Per}}(v) \n\\sqcup \\mathop{\\mathrm{Sing}}(v)) = \\emptyset$ \nand \n$M - (\\mathrm{P} \n\\sqcup \\mathop{\\mathrm{Sing}}(v))$ is connected. \nPut $U = M - (\\mathrm{P} \n\\sqcup \\mathop{\\mathrm{Sing}}(v))$. \nSince $\\mathop{\\mathrm{Per}}(v)$ is open, \nwe have \n$\\mathop{\\mathrm{Per}}(v)  = \\emptyset$ \nand so \n$\\mathop{\\mathrm{int}}\\mathop{\\mathrm{Sing}}(v) = \\emptyset$.  \nSince $v$ is non-wandering, \nwe have \n$\\overline{\\mathrm{LD}} = M = \n\\mathop{\\mathrm{Sing}}(v) \\sqcup \n\\mathrm{LD} \\sqcup \\mathrm{P}$. \nThus $\\mathrm{LD} = U$ is connected. \nFix an locally dense orbit $O$. \nFor any $z \\in \\hat{O}$, \nProposition \\ref{lem32} implies \n$z \\in \\mathrm{LD}$ \nand \nso \n$z \\in \\mathrm{int}\\overline{O_v(z)} = \\mathrm{int}\\overline{O}$. \nHence \n$\\hat{O} \\subseteq \\mathrm{int} \\overline{O}$ \nand so \n$\\hat{O} = \n\\overline{O} \\cap \\mathrm{LD} = \n\\mathrm{int} \\overline{O} \\cap \\mathrm{LD}$. \nThis means that \n$\\hat{O}$ is closed and open in $\\mathrm{LD}$ \nand so \n$\\overline{O} \\cap \\mathrm{LD} = \\mathrm{LD}$. \nTherefore  \n$M = \n\\overline{\\mathrm{LD}} = \n\\overline{O}$. \n\\end{proof}\n\n\nSince an area-preserving flow is non-wandering, \nProposition \\ref{lem32} implies \nthe following statement which is a generalization of Theorem A \\cite{MS}. \n\n\n\\begin{corollary}\\label{cor}\nLet $v$ be a continuous flow on a compact surface $M$. \nThen \n$v$ is topologically transitive \nif and only if \n$v$ is non-wandering \nsuch that \n$ \\mathop{\\mathrm{int}}(\\mathop{\\mathrm{Per}}(v) \n\\sqcup \\mathop{\\mathrm{Sing}}(v)) = \\emptyset$ \nand \n$\\{ x \\in M \\mid \\omega(x) \\cup \\alpha(x) \\subseteq \\mathop{\\mathrm{Sing}}(v) \\}$ is co-connected. \n\\end{corollary}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}