Fine compactified Jacobians of reduced curves
aa r X i v : . [ m a t h . AG ] S e p FINE COMPACTIFIED JACOBIANS OF REDUCED CURVES
MARGARIDA MELO, ANTONIO RAPAGNETTA, FILIPPO VIVIANI
Abstract.
To every singular reduced projective curve X one can associate many fine compactifiedJacobians, depending on the choice of a polarization on X , each of which yields a modular compactifi-cation of a disjoint union of a finite number of copies of the generalized Jacobian of X . We investigatethe geometric properties of fine compactified Jacobians focusing on curves having locally planar sin-gularities. We give examples of nodal curves admitting non isomorphic (and even non homeomorphicover the field of complex numbers) fine compactified Jacobians. We study universal fine compactifiedJacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space ofthe curve X . Finally, we investigate the existence of twisted Abel maps with values in suitable finecompactified Jacobians. Contents
1. Introduction 12. Fine Compactified Jacobians 73. Varying the polarization 134. Deformation theory 205. Universal fine compactified Jacobians 246. Abel maps 347. Examples: Locally planar curves of arithmetic genus 1 40References 431.
Introduction
Aim and motivation.
The aim of this paper is to study fine compactified Jacobians of a reduced projective connected curve X over an algebraically closed field k (of arbitrary characteristic), withspecial emphasis in the case where X has locally planar singularities.Recall that given such a curve X , the generalized Jacobian J ( X ) of X , defined to be the connectedcomponent of the Picard variety of X containing the identity, parametrizes line bundles on X that havemultidegree zero, i.e. degree zero on each irreducible component of X . It turns out that J ( X ) is asmooth irreducible algebraic group of dimension equal to the arithmetic genus p a ( X ) of X . However, if X is a singular curve, the generalized Jacobian J ( X ) is rarely complete. The problem of compactifyingit, i.e. of constructing a projective variety (called a compactified Jacobian) containing J ( X ) as an opensubset, is very natural and it has attracted the attention of many mathematicians, starting from thepioneering work of Mayer-Mumford and of Igusa in the 50’s, until the more recent works of D’Souza,Oda-Seshadri, Altmann-Kleiman, Caporaso, Pandharipande, Simpson, Jarvis, Esteves, etc... (we referto the introduction of [Est01] for an account of the different approaches).In each of the above constructions, compactified Jacobians parametrize (equivalence classes of) certainrank-1, torsion free sheaves on X that are assumed to be semistable with respect to a certain polarization.If the polarization is general (see below for the precise meaning of general), then all the semistable sheaveswill also be stable. In this case, the associated compactified Jacobians will carry a universal sheaf andtherefore we will speak of fine compactified Jacobians (see [Est01]).The main motivation of this work, and of its sequels [MRV1] and [MRV2], comes from the Hitchinfibration for the moduli space of Higgs vector bundles on a fixed smooth and projective curve C (see[Hit86], [Nit91]), whose fibers are compactified Jacobians of certain singular covers of C , called spectralcurves (see [BNR89], [Sch98] and the Appendix of [MRV1]). The spectral curves have always locallyplanar singularities (since they are contained in a smooth surface by construction), although they are not Key words and phrases.
Compactified Jacobians, locally planar singularities, Abel map. ecessarily reduced nor irreducible. It is worth noticing that, in the case of reduced but not irreduciblespectral curves, the compactified Jacobians appearing as fibers of the Hitchin fibration turn out tobe fine compactified Jacobians under the assumption that the degree d and the rank r of the Higgsbundles are coprime. However, in the general case, Chaudouard-Laumon in their work [CL10] and[CL12] on the weighted fundamental Lemma (where they generalize the work of Ngo on the fundamentalLemma, see [Ngo06] and [Ngo10]) have introduced a modified Hitchin fibration for which all fibers arefine compactified Jacobians.According to Donagi-Pantev [DP12], the conjectural geometric Langlands correspondence should in-duce, by passing to the semiclassical limit and taking into account that the general linear group GL r is equal to its Langlands dual group, an autoequivalence of the derived category of the moduli spaceof Higgs bundles, which should intertwine the action of the classical limit tensorization functors withthe action of the classical limit Hecke functors (see [DP12, Conj. 2.5] for a precise formulation). Inparticular, such an autoequivalence should preserve the Hitchin fibration, thus inducing fiberwise anautoequivalence of the compactified Jacobians of the spectral curves. This conjecture is verified in loc.cit. over the open locus of smooth spectral curves, where the desired fiberwise autoequivalence reducesto the classical Fourier-Mukai autoequivalence for Jacobians of smooth curves, established by Mukaiin [Muk81]. This autoequivalence was extended by D. Arinkin to compactified Jacobians of integralspectral curves in [Ari11] and [Ari13]. In the two sequels [MRV1] and [MRV2] to this work, which arestrongly based on the present manuscript, we will extend the Fourier-Mukai autoequivalence to any finecompactified Jacobian of a reduced curve with locally planar singularities. Our results.
In order to state our main results, we need to review the definition of fine compactifiedJacobians of a reduced curve X , following the approach of Esteves [Est01] (referring the reader to § J X ,locally of finite type over k , parametrizing simple, rank-1, torsion-free sheaves on X , which, moreover,satisfies the existence part of the valuative criterion of properness (see Fact 2.2). Clearly, J X admitsa decomposition into a disjoint union J X = ` χ ∈ Z J χX , where J χX is the open and closed subset of J X parametrizing sheaves I of Euler-Poincar´e characteristic χ ( I ) := h ( X, I ) − h ( X, I ) equal to χ . As soonas X is not irreducible, J χX is not separated nor of finite type over k . Esteves [Est01] showed that each J χX can be covered by open and projective subschemes, the fine compactified Jacobians of X , dependingon the choice of a generic polarization in the following way.A polarization on X is a collection of rational numbers q = { q C i } , one for each irreducible component C i of X , such that | q | := P i q C i ∈ Z . A torsion-free rank-1 sheaf I on X of Euler characteristic χ ( I )equal to | q | is called q -semistable (resp. q -stable ) if for every proper subcurve Y ⊂ X , we have that χ ( I Y ) ≥ q Y := X C i ⊆ Y q C i (resp. > ) , where I Y is the biggest torsion-free quotient of the restriction I | Y of I to the subcurve Y . A polarization q is called general if q Y Z for any proper subcurve Y ⊂ X such that Y and Y c are connected. If q is general, there are no strictly q -semistable sheaves, i.e. if every q -semistable sheaf is also q -stable(see Lemma 2.18); the converse being true for curves with locally planar singularities (see Lemma 5.15).For every general polarization q , the subset J X ( q ) ⊆ J X parametrizing q -stable (or, equivalently, q -semistable) sheaves is an open and projective subscheme (see Fact 2.19), that we call the fine compactifiedJacobian with respect to the polarization q . The name “fine” comes from the fact that there exists anuniversal sheaf I on X × J X , unique up to tensor product with the pull-back of a line bundle from J X ,which restricts to a universal sheaf on X × J X ( q ) (see Fact 2.2).Our first main result concerns the properties of fine compactified Jacobians under the assumptionthat X has locally planar singularities. Theorem A.
Let X be a reduced projective connected curve of arithmetic genus g and assume that X has locally planar singularities. Then every fine compactified Jacobian J X ( q ) satisfies the followingproperties:(i) J X ( q ) is a reduced scheme with locally complete intersection singularities and embedded dimensionat most g at every point;(ii) The smooth locus of J X ( q ) coincides with the open subset J X ( q ) ⊆ J X ( q ) parametrizing line bun-dles; in particular J X ( q ) is dense in J X ( q ) and J X ( q ) is of pure dimension equal to p a ( X ) ; iii) J X ( q ) is connected;(iv) J X ( q ) has trivial dualizing sheaf;(v) J X ( q ) is the disjoint union of a number of copies of J ( X ) equal to the complexity c ( X ) of thecurve X (in the sense of Definition 5.12); in particular, J X ( q ) has c ( X ) irreducible components,independently of the chosen polarization q (see Corollary 5.14). Part (i) and part (ii) of the above Theorem are deduced in Corollary 2.20 from the analogous state-ments about the scheme J X , which are in turn deduced, via the Abel map, from similar statementson the Hilbert scheme Hilb n ( X ) of zero-dimensional subschemes of X of length n (see Theorem 2.3).Part (iii) and part (iv) are proved in Section 5 (see Corollaries 5.6 and 5.7), where we used in a crucialway the properties of the universal fine compactified Jacobians (see the discussion below). Finally, part(v) is deduced in Corollary 5.14 from a result of J. L. Kass [Kas13] (generalizing previous results ofS. Busonero (unpublished) and Melo-Viviani [MV12] for nodal curves) that says that any relative finecompactified Jacobian associated to a 1-parameter regular smoothing of X (in the sense of Definition 5.9)is a compactification of the N´eron model of the Jacobian of the generic fiber (see Fact 5.11), togetherwith a result of Raynaud [Ray70] that describes the connected component of the central fiber of theabove N´eron model (see Fact 5.13). In the proof of all the statements of the above Theorem, we use inan essential way the fact that the curve has locally planar singularities and indeed we expect that manyof the above properties are false without this assumption (see also Remark 2.7).Notice that the above Theorem A implies that any two fine compactified Jacobians of a curve X withlocally planar singularities are birational (singular) Calabi-Yau varieties. However, for a reducible curve,fine compactified Jacobians are not necessarily isomorphic (and not even homeomorphic if k = C ). Theorem B.
Let X be a reduced projective connected curve.(i) There is a finite number of isomorphism classes of fine compactified Jacobians of X .(ii) The number of isomorphism classes of fine compactified Jacobians of a given curve X can be arbi-trarily large as X varies, even among the class of nodal curves of genus .(iii) If k = C then the number of homeomorphism classes of fine compactified Jacobians of a given curve X can be arbitrarily large as X varies, even among the class of nodal curves of genus . Part (i) of the above Theorem follows by Proposition 3.2, which says that there is a finite number offine compactified Jacobians of a given curve X from which all the others can be obtained via tensorizationwith some line bundle. Parts (ii) and (iii) are proved by analyzing the poset of orbits for the naturalaction of the generalized Jacobian on a given fine compactified Jacobian of a nodal curve. Proposition 3.4says that the poset of orbits is an invariant of the fine compactified Jacobian (i.e. it does not depend onthe action of the generalized Jacobian) while Proposition 3.5 says that over k = C the poset of orbits is atopological invariant. Moreover, from the work of Oda-Seshadri [OS79], it follows that the poset of orbitsof a fine compactified Jacobian of a nodal curve X is isomorphic to the poset of regions of a suitablesimple toric arrangement of hyperplanes (see Fact 3.8). In Example 3.11, we construct a family of nodalcurves of genus 2 for which the number of simple toric arrangements with pairwise non isomorphic posetof regions grows to infinity, which concludes the proof of parts (ii) and (iii).We mention that, even though if fine compactified Jacobians of a given curve X can be non isomorphic,they nevertheless share many geometric properties. For example, the authors proved in [MRV2] thatany two fine compactified Jacobians of a reduced X with locally planar singularities are derived equiv-alent under the Fourier-Mukai transform with kernel given by a natural Poincar´e sheaf on the product.This result seems to suggest an extension to (mildly) singular varieties of the conjecture of Kawamata[Kaw02], which predicts that birational Calabi-Yau smooth projective varieties should be derived equiv-alent. Moreover, the third author, together with L. Migliorini and V. Schende, proved in [MSV] that anytwo fine compactified Jacobians of X (under the same assumptions on X ) have the same Betti numbersif k = C , which again seems to suggest an extension to (mildly) singular varieties of the result of Batyrev[Bat99] which says that birational Calabi-Yau smooth projective varieties have the same Hodge numbers.As briefly mentioned above, in the proof of parts (iii) and (iv) of Theorem A, an essential role isplayed by the properties of the universal fine compactified Jacobians , which are defined as follows.Consider the effective semiuniversal deformation π : X →
Spec R X of X (see § s ∈ Spec R X , we denote by X s := π − ( s ) the fiber of π over s and by X s := X s × k ( s ) k ( s )the geometric fiber over s . By definition, X = X o = X o where o = [ m X ] ∈ Spec R X is the unique closedpoint o ∈ Spec R X corresponding to the maximal ideal m X of the complete local k -algebra R X . A olarization q on X induces in a natural way a polarization q s on X s for every s ∈ Spec R X which,moreover, will be general if we start from a general polarization q (see Lemma-Definition 5.3). Theorem C.
Let q be a general polarization on a reduced projective connected curve X . There existsa scheme u : J X ( q ) → Spec R X parametrizing coherent sheaves I on X , flat over Spec R X , whosegeometric fiber I s over any s ∈ Spec R X is a q s -semistable (or, equivalently, q s -stable) sheaf on X s . Themorphism u is projective and its geometric fiber over any point s ∈ Spec R X is isomorphic to J X s ( q s ) .In particular, the fiber of J X ( q ) → Spec R X over the closed point o = [ m X ] ∈ Spec R X is isomorphic to J X ( q ) .Moreover, if X has locally planar singularities then we have:(i) The scheme J X ( q ) is regular and irreducible.(ii) The map u : J X ( q ) → Spec R X is flat of relative dimension p a ( X ) and it has trivial relativedualizing sheaf.(iii) The smooth locus of u is the open subset J X ( q ) ⊆ J X ( q ) parametrizing line bundles on X . The first statement of the above Theorem is obtained in Theorem 5.4 by applying to the family π : X →
Spec R X a result of Esteves [Est01] on the existence of relative fine compactified Jacobians.In order to prove the second part of the above Theorem, the crucial step is to identify the completedlocal ring of J X ( q ) at a point I of the central fiber u − ( o ) = J X ( q ) with the semiuniversal deformationring for the deformation functor Def ( X,I ) of the pair ( X, I ) (see Theorem 4.5). Then, we deduce theregularity of J X ( q ) from a result of Fantechi-G¨ottsche-van Straten [FGvS99] which says that, if X haslocally planar singularities, then the deformation functor Def ( X,I ) is smooth. The other properties statedin the second part of Theorem C, which are proved in Theorem 5.5 and Corollary 5.7, follow from theregularity of J X ( q ) together with the properties of the geometric fibers of the morphism u .Our final result concerns the existence of (twisted) Abel maps of degree one into fine compactifiedJacobians, a topic which has been extensively studied (see e.g. [AK80], [EGK00], [EGK02], [EK05],[CE07], [CCE08], [CP10]). To this aim, we restrict ourselves to connected and projective reduced curves X satisfying the following Condition ( † ) : Every separating point is a node,where a separating point of X is a singular point p of X for which there exists a subcurve Z of X such that p is the scheme-theoretic intersection of Z and its complementary subcurve Z c := X \ Z .For example, every Gorenstein curve satisfies condition ( † ) by [Cat82, Prop. 1.10]. Fix now a curve X satisfying condition ( † ) and let { n , . . . , n r − } be its separating points, which are nodes. Denote by e X the partial normalization of X at the set { n , . . . , n r − } . Since each n i is a node, the curve e X is adisjoint union of r connected reduced curves { Y , . . . , Y r } such that each Y i does not have separatingpoints. We have a natural morphism τ : e X = a i Y i → X. We can naturally identify each Y i with a subcurve of X in such a way that their union is X and thatthey do not have common irreducible components. We call the components Y i (or their image in X ) the separating blocks of X . Theorem D.
Let X be a reduced projective connected curve satisfying condition ( † ) .(i) The pull-back map τ ∗ : J X −→ r Y i =1 J Y i I ( I | Y , . . . , I | Y r ) , is an isomorphism. Moreover, given any fine compactified Jacobians J Y i ( q i ) on Y i , i = 1 , . . . , r ,there exists a (uniquely determined) fine compactified Jacobian J X ( q ) on X such that τ ∗ : J X ( q ) ∼ = −→ Y i J Y i ( q i ) , and every fine compactified Jacobian on X is obtained in this way. ii) For every L ∈ Pic( X ) , there exists a unique morphism A L : X → J χ ( L ) − X such that for every ≤ i ≤ r and every p ∈ Y i it holds that τ ∗ ( A L ( p )) = ( M i , . . . , M ii − , m p ⊗ L | Y i , M ii +1 , . . . , M ir ) for some (uniquely determined) line bundle M ij on Y j for any j = i , where m p is the ideal of thepoint p in Y i .(iii) If, moreover, X is Gorenstein, then there exists a general polarization q with | q | = χ ( L ) − suchthat Im A L ⊆ J X ( q ) .(iv) For every L ∈ Pic( X ) , the morphism A L is an embedding away from the rational separating blocks(which are isomorphic to P ) while it contracts each rational separating block Y i ∼ = P into a semi-normal point of A L ( X ) , i.e. an ordinary singularity with linearly independent tangent directions. Some comments on the above Theorem are in order.Part (i), which follows from Proposition 6.6, says that all fine compactified Jacobians of a curvesatisfying assumption ( † ) decompose uniquely as a product of fine compactified Jacobians of its separatingblocks. This allows one to reduce many properties of fine compactified Jacobians of X to properties ofthe fine compactified Jacobians of its separating blocks Y i , which have the advantage of not havingseparating points. Indeed, the first statement of part (i) is due to Esteves [Est09, Prop. 3.2].The map A L of part (ii), which is constructed in Proposition 6.7, is called the L -twisted Abel map.For a curve X without separating points, e.g. the separating blocks Y i , the map A L : X → J X is thenatural map sending p to m p ⊗ L . However, if X has a separating point p , the ideal sheaf m p is not simpleand therefore the above definition is ill-behaved. Part (ii) is saying that we can put together the naturalAbel maps A L | Yi : Y i → J Y i on each separating block Y i in order to have a map A L whose restrictionto Y i has i -th component equal to A L | Yi and it is constant on the j -th components with j = i . Notethat special cases of the Abel map A L (with L = O X or L = O X ( p ) for some smooth point p ∈ X ) inthe presence of separating points have been considered before by Caporaso-Esteves in [CE07, Sec. 4 andSec. 5] for nodal curves, by Caporaso-Coelho-Esteves in [CCE08, Sec. 4 and 5] for Gorenstein curvesand by Coelho-Pacini in [CP10, Sec. 2] for curves of compact type.Part (iii) says that if X is Gorenstein then the image of each twisted Abel map A L is contained in a(non unique) fine compactified Jacobian. Any fine compactified Jacobian which contains the image of atwisted Abel map is said to admit an Abel map . Therefore, part (iii) says that any Gorenstein curve hassome fine compactified Jacobian admitting an Abel map. However, we show that, in general, not everyfine compactified Jacobian admits an Abel map: see Propositions 7.4 and 7.5 for some examples.Part (iv) is proved by Caporaso-Coelho-Esteves [CCE08, Thm. 6.3] for Gorenstein curves, but theirproof extends verbatim to our (more general) case. Outline of the paper.
The paper is organized as follows.Section 2 is devoted to collecting several facts on fine compactified Jacobians of reduced curves: in § J X parametrizing all simple torsion-free rank-1 sheaves on a curve X (seeFact 2.2) and we investigate its properties under the assumption that X has locally planar singularities(see Theorem 2.3); in § X (see Fact 2.19) and study themunder the assumption that X has locally planar singularities (see Corollary 2.20).In Section 3 we prove that there is a finite number of isomorphism classes of fine compactified Jacobiansof a given curve (see Proposition 3.2) although this number can be arbitrarily large even for nodal curves(see Corollary 3.10 and Example 3.11). In order to establish this second result, we study in detail in § X of a curve X (see § ( X,I ) of a pair ( X, I )consisting of a curve X together with a torsion-free, rank-1 sheaf I on X (see § § X and for a pair ( X, I ) as above.In Section 5, we introduce the universal fine compactified Jacobians relative to the semiuniversaldeformation of a curve X (see Theorem 5.4) and we study its properties under the assumption that X has locally planar singularities (see Theorem 5.5). We then deduce some interesting consequencesof our results for fine compactified Jacobians (see Corollaries 5.6 and 5.7). In § n Section 6, we introduce Abel maps: first for curves that do not have separating points (see § § Notations.
The following notations will be used throughout the paper. k will denote an algebraically closed field (of arbitrary characteristic), unless otherwise stated. All schemes are k -schemes, and all morphisms are implicitly assumed to respect the k -structure. A curve is a reduced projective scheme over k of pure dimension 1.Given a curve X , we denote by X sm the smooth locus of X , by X sing its singular locus and by ν : X ν → X the normalization morphism. We denote by γ ( X ), or simply by γ where there is no dangerof confusion, the number of irreducible components of X .We denote by p a ( X ) the arithmetic genus of X , i.e. p a ( X ) := 1 − χ ( O X ) = 1 − h ( X, O X )+ h ( X, O X ).We denote by g ν ( X ) the geometric genus of X , i.e. the sum of the genera of the connected componentsof the normalization X ν . A subcurve Z of a curve X is a closed k -subscheme Z ⊆ X that is reduced and of pure dimension1. We say that a subcurve Z ⊆ X is proper if Z = ∅ , X .Given two subcurves Z and W of X without common irreducible components, we denote by Z ∩ W the 0-dimensional subscheme of X that is obtained as the scheme-theoretic intersection of Z and W andwe denote by | Z ∩ W | its length.Given a subcurve Z ⊆ X , we denote by Z c := X \ Z the complementary subcurve of Z and weset δ Z = δ Z c := | Z ∩ Z c | . A curve X is called Gorenstein if its dualizing sheaf ω X is a line bundle. A curve X has locally complete intersection (l.c.i.) singularities at p ∈ X if the completion b O X,p of the local ring of X at p can be written as b O X,p = k [[ x , . . . , x r ]] / ( f , . . . , f r − ) , for some r ≥ f i ∈ k [[ x , . . . , x r ]]. A curve X has locally complete intersection (l.c.i.)singularities if X is l.c.i. at every p ∈ X . It is well know that a curve with l.c.i. singularities isGorenstein. A curve X has locally planar singularities at p ∈ X if the completion b O X,p of the local ring of X at p has embedded dimension two, or equivalently if it can be written as b O X,p = k [[ x, y ]] / ( f ) , for a reduced series f = f ( x, y ) ∈ k [[ x, y ]]. A curve X has locally planar singularities if X has locallyplanar singularities at every p ∈ X . Clearly, a curve with locally planar singularities has l.c.i. singular-ities, hence it is Gorenstein. A (reduced) curve has locally planar singularities if and only if it can beembedded in a smooth surface (see [AK79a]). A curve X has a node at p ∈ X if the completion b O X,p of the local ring of X at p is isomorphic to b O X,p = k [[ x, y ]] / ( xy ) . A separating point of a curve X is a geometric point n ∈ X for which there exists a subcurve Z ⊂ X such that δ Z = 1 and Z ∩ Z c = { n } . If X is Gorenstein, then a separating point n of X is a nodeof X , i.e. b O X,n = k [[ x, y ]] / ( xy ) (see Fact 6.4). However this is false in general without the Gorensteinassumption (see Example 6.5). Given a curve X , the generalized Jacobian of X , denoted by J ( X ) or by Pic ( X ), is the algebraicgroup whose group of k -valued points is the group of line bundles on X of multidegree 0 (i.e. having degree0 on each irreducible component of X ) together with the multiplication given by the tensor product.The generalized Jacobian of X is a connected commutative smooth algebraic group of dimension equalto h ( X, O X ). . Fine Compactified Jacobians
The aim of this section is to collect several facts about compactified Jacobians of connected reducedcurves, with special emphasis on connected reduced curves with locally planar singularities. Many ofthese facts are well-known to the experts but for many of them we could not find satisfactory referencesin the existing literature, at least at the level of generality we need, e.g. for reducible curves. Throughoutthis section, we fix a connected reduced curve X .2.1. Simple rank-1 torsion-free sheaves.
We start by defining the sheaves on the connected curve X we will be working with. Definition 2.1.
A coherent sheaf I on a connected curve X is said to be:(i) rank-1 if I has generic rank 1 on every irreducible component of X ;(ii) torsion-free (or pure, or S ) if Supp( I ) = X and every non-zero subsheaf J ⊆ I is such thatdim Supp( J ) = 1;(iii) simple if End k ( I ) = k .Note that any line bundle on X is a simple rank-1 torsion-free sheaf.Consider the functor(2.1) J ∗ X : { Schemes /k } → { Sets } which associates to a k -scheme T the set of isomorphism classes of T -flat, coherent sheaves on X × k T whose fibers over T are simple rank-1 torsion-free sheaves (this definition agrees with the one in [AK80,Def. 5.1] by virtue of [AK80, Cor. 5.3]). The functor J ∗ X contains the open subfunctor(2.2) J ∗ X : { Schemes /k } → { Sets } which associates to a k -scheme T the set of isomorphism classes of line bundles on X × k T . Fact 2.2 (Murre-Oort, Altman-Kleiman, Esteves) . Let X be a connected reduced curve. Then(i) The Zariski (equiv. ´etale, equiv. fppf ) sheafification of J ∗ X is represented by a k -scheme Pic( X ) = J X , locally of finite type over k . Moreover, J X is formally smooth over k .(ii) The Zariski (equiv. ´etale, equiv. fppf ) sheafification of J ∗ X is represented by a k -scheme J X , locally offinite type over k . Moreover, J X is an open subset of J X and J X satisfies the valuative criterion foruniversally closedness or, equivalently, the existence part of the valuative criterion for properness .(iii) There exists a sheaf I on X × J X such that for every F ∈ J ∗ X ( T ) there exists a unique map α F : T → J X with the property that F = (id X × α F ) ∗ ( I ) ⊗ π ∗ ( N ) for some N ∈ Pic( T ) , where π : X × T → T is the projection onto the second factor. The sheaf I is uniquely determined up totensor product with the pullback of an invertible sheaf on J X and it is called a universal sheaf .Proof. Part (i): the representability of the fppf sheafification of J ∗ X follows from a result of Murre-Oort(see [BLR90, Sec. 8.2, Thm. 3] and the references therein). However, since X admits a k -rational point(because k is assumed to be algebraically closed), the fppf sheafification of J ∗ X coincides with its ´etale(resp. Zariski) sheafification (see [FGA05, Thm. 9.2.5(2)]). The formal smoothness of J X follows from[BLR90, Sec. 8.4, Prop. 2].Part (ii): the representability of the ´etale sheafification (and hence of the fppf sheafification) of J ∗ X by an algebraic space J X locally of finite type over k follows from a general result of Altmann-Kleiman([AK80, Thm. 7.4]). Indeed, in [AK80, Thm. 7.4] the authors state the result for the moduli functor ofsimple sheaves; however, since the condition of being torsion-free and rank-1 is an open condition (seee.g. the proof of [AK80, Prop 5.12(ii)(a)]), we also get the representability of J ∗ X . The fact that J X is ascheme follows from a general result of Esteves ([Est01, Thm. B]), using the fact that each irreduciblecomponent of X has a k -point (recall that k is assumed to be algebraically closed). Moreover, since X admits a smooth k -rational point, the ´etale sheafification of J ∗ X coincides with the Zariski sheafificationby [AK79b, Thm. 3.4(iii)]. Since J ∗ X is an open subfunctor of J ∗ X then J X is an open subscheme of J X .Finally, the fact that J X satisfies the existence condition of the valuative criterion for properness followsfrom [Est01, Thm. 32].Part (iii) is an immediate consequence of the fact that J X represents the Zariski sheafification of J ∗ X (see also [AK79b, Thm. 3.4]). (cid:3) Notice however that the scheme J X fails to be universally closed because it is not quasi-compact. ince the Euler-Poincar´e characteristic χ ( I ) := h ( X, I ) − h ( X, I ) of a sheaf I on X is constant underdeformations, we get a decomposition(2.3) J X = a χ ∈ Z J χX , J X = a χ ∈ Z J χX , where J χX (resp. J χX ) denotes the open and closed subscheme of J X (resp. J X ) parametrizing simplerank-1 torsion-free sheaves I (resp. line bundles L ) such that χ ( I ) = χ (resp. χ ( L ) = χ ).If X has locally planar singularities, then J X has the following properties. Theorem 2.3.
Let X be a connected reduced curve with locally planar singularities. Then(i) J X is a reduced scheme with locally complete intersection singularities and embedded dimension atmost p a ( X ) at every point.(ii) J X is dense in J X .(iii) J X is the smooth locus of J X . The required properties of J X will be deduced from the analogous properties of the punctual Hilbertscheme (i.e. the Hilbert scheme of 0-dimensional subschemes) of X via the Abel map.Let us first review the needed properties of the punctual Hilbert scheme. Denote by Hilb d ( X ) theHilbert scheme parametrizing subschemes D of X of finite length d ≥
0, or equivalently ideal sheaves I ⊂ O X such that O X /I is a finite scheme of length d . Given D ∈ Hilb d X , we will denote by I D itsideal sheaf. We introduce the following subschemes of Hilb d ( X ): ( Hilb d ( X ) s := { D ∈ Hilb d ( X ) : I D is simple } , Hilb d ( X ) l := { D ∈ Hilb d ( X ) : I D is a line bundle } . By combining the results of [AK80, Prop. 5.2 and Prop 5.13(i)], we get that the natural inclusionsHilb d ( X ) l ⊆ Hilb d ( X ) s ⊆ Hilb d ( X )are open inclusions. Fact 2.4. If X is a reduced curve with locally planar singularities, then the Hilbert scheme Hilb d ( X ) hasthe following properties:(a) Hilb d ( X ) is reduced with locally complete intersection singularities and embedded dimension at most d at every point.(b) Hilb d ( X ) l is dense in Hilb d ( X ) .(c) Hilb d ( X ) l is the smooth locus of Hilb d ( X ) .The above properties do hold true if Hilb d ( X ) is replaced by Hilb d ( X ) s .Proof. Part (a) follows from [AIK76, Cor. 7] (see also [BGS81, Prop. 1.4]), part (b) follows from [AIK76,Thm. 8] (see also [BGS81, Prop. 1.4]) and part (c) follows from [BGS81, Prop. 2.3].The above properties do remain true for Hilb d ( X ) s since Hilb d ( X ) s is an open subset of Hilb d ( X )containing Hilb d ( X ) l . (cid:3) The punctual Hilbert scheme of X and the moduli space J X are related via the Abel map, which isdefined as follows. Given a line bundle M on X , we define the M -twisted Abel map of degree d by(2.4) A dM : s Hilb dX −→ J X ,D I D ⊗ M. Note that, by definition, it follows that(2.5) ( A dM ) − ( J X ) = Hilb d ( X ) l . The following result (whose proof was kindly suggested to us by J.L. Kass) shows that, locally on thecodomain, the M -twisted Abel map of degree p a ( X ) is smooth and surjective (for a suitable choice of M ∈ Pic( X )), at least if X is Gorenstein. Proposition 2.5.
Let X be a (connected and reduced) Gorenstein curve of arithmetic genus g := p a ( X ) .There exists a cover of J X by k -finite type open subsets { U β } such that, for each such U β , there exists M β ∈ Pic( X ) with the property that s Hilb gX ⊇ V β := ( A gM β ) − ( U β ) A gMβ −→ U β is smooth and surjective. roof. Observe that, given I ∈ J χX and M ∈ Pic( X ), we have:(i) I belongs to the image of A χ ( M ) − χM if (and only if) there exists an injective homomorphism I → M ;(ii) A χ ( M ) − χM is smooth along ( A χ ( M ) − χM ) − ( I ) provided that Ext ( I, M ) = 0.Indeed, if there exists an injective homomorphism I → M , then its cokernel is the structure sheaf of a0-dimensional subscheme D ⊂ X of length equal to χ ( M ) − χ ( I ) = χ ( M ) − χ with the property that I D = I ⊗ M − . Therefore A χ ( M ) − χM ( D ) = I D ⊗ M = ( I ⊗ M − ) ⊗ M = I, which implies part (i). Part(ii) follows from [AK80, Thm. 5.18(ii)] .Fixing M ∈ Pic( X ), the conditions (i) and (ii) are clearly open conditions on J χX ; hence the proof ofthe Proposition follows from the case n = g of the followingClaim: For any I ∈ J χX and any n ≥ g , there exists M n ∈ Pic( X ) with χ ( M ) = n + χ such that(a) there exists an injective homomorphism I → M n ;(b) Ext ( I, M n ) = 0.First of all, observe that, for any I ∈ J χX and any line bundle N , the local-to-global spectral sequence H p ( X, E xt q ( I, N )) ⇒ Ext p + q ( I, N ) gives that H ( X, H om ( I, N )) = Ext ( I, N ) , → H ( X, H om ( I, N )) → Ext ( I, N ) → H ( X, E xt ( I, N )) . Moreover, the sheaf E xt ( I, N ) = E xt ( I, O X ) ⊗ N vanishes by [Har94, Prop. 1.6], so that we get(2.6) Ext i ( I, N ) = H i ( X, H om ( I, N )) = H i ( X, I ∗ ⊗ N ) for i = 0 , I ∗ := H om ( I, O X ) ∈ J X . From (2.6) and Riemann-Roch, we get(2.7)dim Ext ( I, N ) − dim Ext ( I, N ) = χ ( I ∗ ⊗ N ) = deg N + χ ( I ∗ ) = deg N +2(1 − g ) − χ ( I ) = χ ( N ) − χ +1 − g. We will now prove the claim by decreasing induction on n . The claim is true (using (2.6)) if n ≫ M n is chosen to be a sufficiently high power of a very ample line bundle on X . Suppose now that wehave a line bundle M n +1 ∈ Pic n +1 ( X ) with χ ( M n +1 ) = n + 1 + χ (for a certain n ≥ g ) which satisfies theproperties of the Claim. We are going to show that, for a generic smooth point p ∈ X , the line bundle M n := M n +1 ⊗ O X ( − p ) ∈ Pic( X ) also satisfies the properties of the Claim.Using (2.7) and the properties of M n +1 , it is enough to show that M n := M n +1 ⊗ O X ( − p ), for p ∈ X generic, satisfies(*) dim Hom( I, M n ) = dim Hom( I, M n +1 ) − , (**) the generic element [ I → M n ] ∈ Hom(
I, M n ) is injective.Tensoring the exact sequence 0 → O X ( − p ) → O X → O p → I ∗ ⊗ M n +1 and taking cohomology, we get the exact sequence0 → Hom(
I, M n ) = H ( X, I ∗ ⊗ M n ) → Hom(
I, M n +1 ) = H ( X, I ∗ ⊗ M n +1 ) e −→ k p , where e is the evaluation of sections at p ∈ X . By the assumptions on M n +1 and (2.7), we have thatdim Hom( I, M n +1 ) = χ ( M n +1 ) − χ + 1 − g = n + 1 + 1 − g ≥ , and, moreover, that the generic element [ I → M n +1 ] ∈ Hom(
I, M n +1 ) is injective. By choosing a point p ∈ X for which there exists a section s ∈ H ( X, I ∗ ⊗ M n +1 ) which does not vanish in p , we get that (*)and (**) holds true for M n = M n +1 ⊗ O X ( − p ), q.e.d. (cid:3) Remark . From the proof of the second statement of Theorem 2.3(i) and Remark 2.7(iii) below, itwill follow that the above Proposition is, in general, false if g is replaced by any smaller integer.With the above preliminaries results, we can now give a proof of Theorem 2.3. Note that in loc.cit., this is stated under the assumption that X is integral. However, a close inspection of the proofreveals that this continues to hold true under the assumption that X is only reduced. The irreducibility is only used inpart (i) of [AK80, Thm. 5.18]. roof of Theorem 2.3. Observe that each of the three statements of the theorem is local in J X , i.e. it issufficient to check it on an open cover of J X . Consider the open cover { U β } given by Proposition 2.5.Part (i): from Fact 2.4(a), it follows that V β ⊂ s Hilb gX is reduced with locally complete intersectionsingularities and embedded dimension at most 2 g = 2 p a ( X ). Since ( A gM ) | V β is smooth and surjectiveinto U β , also U β inherits the same properties.Part (ii): from Fact 2.4(b), it follows that Hilb g ( X ) l ∩ V β is dense in V β . From the surjectivity of( A gM ) | V β together with (2.5), it follows that A gM ( V β ∩ Hilb g ( X ) l ) = U β ∩ J X is dense in A gM ( V β ) = U β .Part (iii): from Fact 2.4(c), it follows that Hilb g ( X ) l ∩ V β is the smooth locus of V β . Since ( A gM ) | V β issmooth and surjective and (2.5) holds, we infer that A gM ( V β ∩ Hilb g ( X ) l ) = U β ∩ J X is the smooth locusof A gM ( V β ) = U β . (cid:3) Remark . (i) Theorem 2.3 is well-known (except perhaps the statement about the embedded dimension) in thecase where X is irreducible (and hence integral): the first assertion in part (i) and part (ii) are dueto Altman-Iarrobino-Kleiman [AIK76, Thm. 9]; part (iii) is due to Kleppe [Kle81] (unpublished,for a proof see [Kas09, Prop. 6.4]). Note that, for X irreducible, part (ii) is equivalent to theirreducibility of J dX for a certain d ∈ Z (hence for all d ∈ Z ).(ii) The hypothesis that X has locally planar singularities is crucial in the above Theorem 2.3: • Altman-Iarrobino-Kleiman constructed in [AIK76, Exa. (13)] an integral curve without locallyplanar singularities (indeed, a curve which is a complete intersection in P ) for which J dX (forany d ∈ Z ) is not irreducible (equivalently, J dX is not dense in J dX ). Later, Rego ([Reg80, Thm.A]) and Kleppe-Kleiman ([KK81, Thm. 1]) showed that, for X irreducible, J dX is irreducibleif and only if X has locally planar singularities. • Kass proved in [Kas12, Thm. 2.7] that if X is an integral curve with a unique non-Gorensteinsingularity, then its compactified Jacobian J dX (for any d ∈ Z ) contains an irreducible compo-nent D d which does not meet the open subset J dX ⊂ J dX of line bundles and it is genericallysmooth of dimension p a ( X ). In particular, the smooth locus of J dX is bigger then the locus J dX of line bundles. • Kass constructed in [Kas15] an integral rational space curve X of arithmetic genus 4 for which J X is non-reduced.(iii) The statement about the embedded dimension in Theorem 2.3 is sharp: if X is a rational nodalcurve with g nodes and I ∈ J dX is a sheaf that is not locally free at any of the g nodes (and any J dX contains a sheaf with these properties), then it is proved in [CMK12, Prop. 2.7] that J dX isisomorphic formal locally at I to the product of g nodes, hence it has embedded dimension at I equal to 2 g .2.2. Fine compactified Jacobians.
For any χ ∈ Z , the scheme J χX is neither of finite type nor sepa-rated over k (and similarly for J χX ) if X is reducible. However, they can be covered by open subsets thatare proper (and even projective) over k : the fine compactified Jacobians of X . The fine compactifiedJacobians depend on the choice of a polarization, whose definition is as follows. Definition 2.8. A polarization on a connected curve X is a tuple of rational numbers q = { q C i } , onefor each irreducible component C i of X , such that | q | := P i q C i ∈ Z . We call | q | the total degree of q .Given any subcurve Y ⊆ X , we set q Y := P j q C j where the sum runs over all the irreduciblecomponents C j of Y . Note that giving a polarization q is the same as giving an assignment ( Y ⊆ X ) q Y such that q X ∈ Z and which is additive on Y , i.e. such that if Y , Y ⊆ X are two subcurves of X withoutcommon irreducible components, then q Y ∪ Y = q Y + q Y . Definition 2.9.
A polarization q is called integral at a subcurve Y ⊆ X if q Z ∈ Z for any connectedcomponent Z of Y and of Y c .A polarization is called general if it is not integral at any proper subcurve Y ⊂ X . Remark . It is easily seen that q is general if and only if q Y Z for any proper subcurve Y ⊂ X such that Y and Y c are connected. or each subcurve Y of X and each torsion-free sheaf I on X , the restriction I | Y of I to Y is notnecessarily a torsion-free sheaf on Y . However, I | Y contains a biggest subsheaf, call it temporarily J ,whose support has dimension zero, or in other words such that J is a torsion sheaf. We denote by I Y the quotient of I | Y by J . It is easily seen that I Y is torsion-free on Y and it is the biggest torsion-freequotient of I | Y : it is actually the unique torsion-free quotient of I whose support is equal to Y . Moreover,if I is torsion-free rank-1 then I Y is torsion-free rank-1. We let deg Y ( I ) denote the degree of I Y on Y ,that is, deg Y ( I ) := χ ( I Y ) − χ ( O Y ). Definition 2.11.
Let q be a polarization on X . Let I be a torsion-free rank-1 sheaf on X (not necessarilysimple) such that χ ( I ) = | q | .(i) We say that I is semistable with respect to q (or q -semistable) if for every proper subcurve Y ⊂ X ,we have that(2.8) χ ( I Y ) ≥ q Y . (ii) We say that I is stable with respect to q (or q -stable) if it is semistable with respect to q and if theinequality (2.8) is always strict. Remark . It is easily seen that a torsion-free rank-1 sheaf I is q -semistable (resp. q -stable) if andonly if (2.8) is satisfied (resp. is satisfied with strict inequality) for any subcurve Y ⊂ X such that Y and Y c are connected. Remark . Let q be a polarization on X and let q ′ be a general polarization on X that is obtainedby slightly perturbing q . Then, for a torsion-free rank-1 sheaf I on X , we have the following chain ofimplications: I is q -stable ⇒ I is q ′ -stable ⇒ I is q ′ -semistable ⇒ I is q -semistable . Remark . A line bundle L on X is q -semistable if and only if(2.9) χ ( L | Y ) ≤ q Y + | Y ∩ Y c | for any subcurve Y ⊆ X . Indeed, tensoring with L the exact sequence0 → O X → O Y ⊕ O Y c → O Y ∩ Y c → , and taking Euler-Poincar´e characteristics, we find that χ ( L | Y ) + χ ( L | Y c ) = χ ( L ) + | Y ∩ Y c | . Using this equality, we get that χ ( L | Y c ) ≥ q Y c ⇐⇒ χ ( L | Y ) = χ ( L ) − χ ( L | Y c ) + | Z ∩ Z c | ≤ | q | − q Y c + | Z ∩ Z c | = q Y + | Z ∩ Z c | , which gives that (2.8) for Y c is equivalent to (2.9) for Y . Remark . If X is Gorenstein, we can write the inequality (2.8) in terms of the degree of I Y as follows(2.10) deg Y ( I ) ≥ q Y − χ ( O Y ) = q Y + deg Y ( ω X )2 − δ Y , where we used the adjunction formula (see [Cat82, Lemma 1.12])deg Y ( ω X ) = 2 p a ( Y ) − δ Y = − χ ( O Y ) + δ Y . The inequality (2.10) was used to define stable rank-1 torsion-free sheaves on nodal curves in [MV12]; inparticular, there is a change of notation between this paper where q -(semi)stability is defined by meansof the inequality (2.8) and the notation of loc. cit. where q -(semi)stability is defined by means of theinequality (2.10).Polarizations on X can be constructed from vector bundles on X , as we now indicate. Remark . Given a vector bundle E on X , we define the polarization q E on X by setting(2.11) q EY := − deg( E | Y )rk( E ) , for each subcurve Y (or equivalently for each irreducible component C i ) of X . Then a torsion-free rank-1sheaf I on X is stable (resp. semistable) with respect to q E in the sense of Definition 2.11 if and only if χ ( I Y ) > ( ≥ ) q EY = − deg( E | Y )rk( E ) , .e. if I is stable (resp. semistable) with respect to E in the sense of [Est01, Sec. 1.2].Moreover, every polarization q on X is of the form q E for some (non-unique) vector bundle E . Indeed,take r > rq Y ∈ Z for every subcurve Y ⊆ X . Considera vector bundle E on X of rank r such that, for every subcurve Y ⊆ X (or equivalently for everyirreducible component C i of X ), the degree of E restricted to Y is equal to(2.12) − deg( E | Y ) = rq Y . Then, comparing (2.11) and (2.12), we deduce that q E = q .Finally, for completeness, we mention that the usual slope (semi)stability with respect to some ampleline bundle on X is a special case of the above (semi)stability. Remark . Given an ample line bundle L on X and an integer χ ∈ Z , the slope (semi)stabilityfor rank-1 torsion-free sheaves on X of Euler-Poincar´e characteristic equal to χ is equal to the above(semi)stability with respect to the polarization L q defined by setting(2.13) L q Y := deg( L | Y )deg L χ, for any subcurve Y of X . The proof of the above equivalence in the nodal case can be found in [Ale04,Sec. 1] (see also [CMKV15, Fact 2.8]); the same proof extends verbatim to arbitrary reduced curves.Notice that, as observed already in [MV12, Rmk. 2.12(iv)], slope semistability with respect to someample line bundle L is much more restrictive than q -semistability: the extreme case being when χ = 0,in which case there is a unique slope semistability (independent on the chosen line bundle L ) while thereare plenty of q -semistability conditions!The geometric implications of having a general polarization are clarified by the following result. Lemma 2.18.
Let I be a rank- torsion-free sheaf on X which is semistable with respect to a polarization q on X .(i) If q is general then I is also q -stable.(ii) If I is q -stable, then I is simple.Proof. Let us first prove (i). Since q is general, from Remark 2.10 it follows that if Y ⊂ X is a subcurveof X such that Y and Y c are connected then q Y Z . Therefore, the right hand side of (2.8) is not aninteger for such subcurves, hence the inequality is a fortiori always strict. This is enough to guaranteethat a torsion-free rank-1 sheaf that is q -semistable is also q -stable, by Remark 2.12.Let us now prove part (ii). By contradiction, suppose that I is q -stable and not simple. Since I is notsimple, we can find, according to [Est01, Prop. 1], a proper subcurve Y ⊂ X such that the natural map I → I Y ⊕ I Y c is an isomorphism, which implies that χ ( I ) = χ ( I Y ) + χ ( I Y c ). Since I is q -stable, we getfrom (2.8) the two inequalities ( χ ( I Y ) > q Y ,χ ( I Y c ) > q Y c . Summing up the above inequalities, we get χ ( I ) = χ ( I Y ) + χ ( I Y c ) > q Y + q Y c = | q | , which is acontradiction since χ ( I ) = | q | by definition of q -stability. (cid:3) Later on (see Lemma 5.15), we will see that the property stated in Lemma 2.18(i) characterizes thepolarizations that are general, at least for curves with locally planar singularities.For a polarization q on X , we will denote by J ssX ( q ) (resp. J sX ( q )) the subscheme of J X parametrizingsimple rank-1 torsion-free sheaves I on X which are q -semistable (resp. q -stable). If q = q E for somevector bundle E on X , then it follows from Remark 2.16 that the subscheme J sX ( q E ) (resp. J ssX ( q E ))coincides with the subscheme J sE (resp. J ssE ) in Esteves’s notation (see [Est01, Sec. 4]). By [Est01, Prop.34], the inclusions J sX ( q ) ⊆ J ssX ( q ) ⊂ J X are open. Fact 2.19 (Esteves) . Let X be a connected curve.(i) J sX ( q ) is a quasi-projective scheme over k (not necessarily reduced). In particular, J sX ( q ) is ascheme of finite type and separated over k .(ii) J ssX ( q ) is a k -scheme of finite type and universally closed over k . iii) If q is general then J ssX ( q ) = J sX ( q ) is a projective scheme over k (not necessarily reduced).(iv) J X = [ q general J sX ( q ) . Proof.
Part (i) follows from [Est01, Thm. A(1) and Thm. C(4)].Part (ii) follows from [Est01, Thm. A(1)].Part (iii): the fact that J ssX ( q ) = J sX ( q ) follows from Lemma 2.18. Its projectivity follows from (i) and(ii) since a quasi-projective scheme over k which is universally closed over k must be projective over k .Part (iv) follows from [Est01, Cor. 15], which asserts that a simple torsion-free rank-1 sheaf is stablewith respect to a certain polarization, together with Remark 2.13, which asserts that it is enough toconsider general polarizations. (cid:3) If q is general, we set J X ( q ) := J ssX ( q ) = J sX ( q ) and we call it the fine compactified Jacobian withrespect to the polarization q . We denote by J X ( q ) the open subset of J X ( q ) parametrizing line bundleson X . Note that J X ( q ) is isomorphic to the disjoint union of a certain number of copies of the generalizedJacobian J ( X ) = Pic ( X ) of X .Since, for q general, J X ( q ) is an open subset of J X , the above Theorem 2.3 immediately yields thefollowing properties for fine compactified Jacobians of curves with locally planar singularities. Corollary 2.20.
Let X be a connected curve with locally planar singularities and q a general polarizationon X . Then(i) J X ( q ) is a reduced scheme with locally complete intersection singularities and embedded dimensionat most p a ( X ) at every point.(ii) J X ( q ) is dense in J X ( q ) . In particular, J X ( q ) has pure dimension equal to the arithmetic genus p a ( X ) of X .(iii) J X ( q ) is the smooth locus of J X ( q ) . Later, we will prove that J X ( q ) is connected (see Corollary 5.6) and we will give a formula for thenumber of its irreducible components in terms solely of the combinatorics of the curve X (see Corollary5.14). Remark . If q is not general, it may happen that J ssX ( q ) is not separated. However, it follows from[Ses82, Thm. 15, p. 155] that J ssX ( q ) admits a morphism φ : J ssX ( q ) → U X ( q ) onto a projective varietythat is universal with respect to maps into separated varieties; in other words, U X ( q ) is the biggestseparated quotient of J ssX ( q ). We call the projective variety U X ( q ) a coarse compactified Jacobian . Thefibers of φ are S-equivalence classes of sheaves, and in particular φ is an isomorphism on the open subset J sX ( q ) (see loc. cit. for details). Coarse compactified Jacobians can also be constructed as a special caseof moduli spaces of semistable pure sheaves, constructed by Simpson in [Sim94].Coarse compactified Jacobians behave quite differently from fine compactified Jacobians, even for anodal curve X ; for example(i) they can have (and typically they do have) fewer irreducible components than the number c ( X ) ofirreducible components of fine compactified Jacobians, see [MV12, Thm. 7.1];(ii) their smooth locus can be bigger than the locus of line bundles, see [CMKV15, Thm. B(ii)].(iii) their embedded dimension at some point can be bigger than 2 p a ( X ), see [CMKV15, Ex. 7.2].3. Varying the polarization
Fine compactified Jacobians of a connected curve X depend on the choice of a general polarization q .The goal of this section is to study the dependence of fine compactified Jacobians upon the choice of thepolarization. In particular, we will prove Theorem B, which says that there is always a finite number ofisomorphism classes (resp. homeomorphism classes if k = C ) of fine compactified Jacobians of a reducedcurve X even though this number can be arbitrarily large even for nodal curves.To this aim, consider the space of polarizations on X (3.1) P X := { q ∈ Q γ ( X ) : | q | ∈ Z } ⊂ R γ ( X ) , here γ ( X ) is the number of irreducible components of X . Define the arrangement of hyperplanes in R γ ( X ) (3.2) A X := X C i ⊆ Y x i = n Y ⊆ X,n ∈ Z where Y varies among all the subcurves of X such that Y and Y c are connected. By Remark 2.10, apolarization q ∈ P X is general if and only if q does not belong to A X . Moreover, the arrangement ofhyperplanes A X subdivides P X into chambers with the following property: if two general polarizations q and q ′ belong to the same chamber then ⌈ q Y ⌉ = ⌈ q ′ Y ⌉ for any subcurve Y ⊆ X such that Y and Y c are connected, hence J X ( q ) = J X ( q ′ ) by Remark 2.12. Therefore, fine compactified Jacobians of X correspond bijectively to the chambers of P X cut out by the hyperplane arrangement A X .Obviously, there are infinitely many chambers and therefore infinitely many different fine compactifiedJacobians. However, we are now going to show that there are finitely many isomorphism classes of finecompactified Jacobians. The simplest way to show that two fine compactified Jacobians are isomorphicis to show that there is a translation that sends one into the other. Definition 3.1.
Let X be a connected curve. We say that two compactified Jacobians J X ( q ) and J X ( q ′ )are equivalent by translation if there exists a line bundle L on X inducing an isomorphism J X ( q ) ∼ = −→ J X ( q ′ ) ,I I ⊗ L. Note however that, in general, there could be fine compactified Jacobians that are isomorphic withoutbeing equivalent by translation, see Section 7 for some explicit examples.
Proposition 3.2.
Let X be a connected curve. There is a finite number of fine compactified Jacobiansup to equivalence by translation. In particular, there is a finite number of isomorphism classes of finecompactified Jacobians of X .Proof. If two generic polarizations q and q ′ are such that q − q ′ ∈ Z γ ( X ) , then the multiplication by aline bundle of multidegree q − q ′ gives an isomorphism between J X ( q ′ ) and J X ( q ). Therefore, any finecompactified Jacobian of X is equivalent by translation to a fine compactified Jacobian J X ( q ) such that0 ≤ q C i < C i of X . We conclude by noticing that the arrangement ofhyperplanes A X of (3.2) subdivides the unitary cube [0 , γ ( X ) ⊂ R γ ( X ) into finitely many chambers. (cid:3) Nodal curves.
In this subsection, we study how fine compactified Jacobians vary for a nodal curve.Recall that the generalized Jacobian J ( X ) of a reduced curve X acts, via tensor product, on anyfine compactified Jacobian J X ( q ) and the orbits of this action form a stratification of J X ( q ) into locallyclosed subsets. This stratification was studied in the case of nodal curves by the first and third authorsin [MV12]. In order to recall these results, let us introduce some notation. Let X sing be the set of nodesof X and for every subset S ⊆ X sing denote by ν S : X S → X the partial normalization of X at the nodesbelonging to S . For any subcurve Y of X , set Y S := ν − S ( Y ). Note that Y S is the partial normalization of Y at the nodes S ∩ Y sing and that every subcurve of X S is of the form Y S for some uniquely determinedsubcurve Y ⊆ X . Given a polarization q on X , define a polarization q S on X S by setting q SY S := q Y forany subcurve Y of X . Clearly, if q is a general polarization on X then q S is a general polarization on X S . Moreover, consider the following subset of integral multidegrees on X S : B S ( q ) := { χ ∈ Z γ ( X S ) : | χ | = | q S | , χ Y S ≥ q Y S for any subcurve Y ⊆ X } , and for every χ ∈ B S ( q ) denote by J χX S the J ( X S )-torsor consisting of all the line bundles L on X S whose multi-Euler characteristic is equal to χ , i.e. χ ( L | Y S ) = χ Y S for every subcurve Y S ⊆ X S . Fact 3.3.
Let X be a connected nodal curve of arithmetic genus p a ( X ) = g and let J X ( q ) be a finecompactified Jacobian of X .(i) For every S ⊆ X sing , denote by J X,S ( q ) the locally closed subset (with reduced scheme structure) of J X ( q ) consisting of all the sheaves I ∈ J X ( q ) such that I is not locally free exactly at the nodes of S . Then(a) J X,S ( q ) = ∅ if and only if X S is connected;(b) J X,S ( q ) = ` S ⊆ S ′ J X,S ′ ( q ) . ii) The pushforward ν S ∗ along the normalization morphism ν S : X S → X gives isomorphisms J X,S ( q ) ∼ = J X S ( q S ) = a χ ∈ B S ( q ) J χX S ,J X,S ( q ) ∼ = J X S ( q S ) . (iii) The decomposition of J X ( q ) into orbits for the action of the generalized Jacobian J ( X ) is equal to J X ( q ) = a S ⊆ X sing χ ∈ B S ( q ) J χX S , where the disjoint union runs over the subsets S ⊆ X sing such that X S is connected.(iv) For every I ∈ J X,S ( q ) , the completion of the local ring O J X ( q ) ,I of the fine compactified Jacobian J X ( q ) at I is given by b O J X ( q ) ,I = k [[ Z , . . . , Z g −| S | ]] \ O ≤ i ≤| S | k [[ X i , Y i ]]( X i Y i ) . Proof.
Parts (i), (ii) and (iii) follow from [MV12, Thm. 5.1] keeping in mind the change of notation ofthis paper (where we use the Euler characteristic) with respect to the notation of [MV12] (where thedegree is used), see Remark 2.15.Part (iv) is a special case of [CMKV15, Thm. A] (see in particular [CMKV15, Example 7.1]), wherethe local structure of (possibly non fine) compactified Jacobians of nodal curves is described. (cid:3)
The set of J ( X )-orbits O ( J X ( q )) := { J χX S } on J X ( q ) forms naturally a poset (called the poset oforbits of J X ( q )) by declaring that J χX S ≥ J χ ′ X S ′ if and only if J χX S ⊇ J χ ′ X S ′ . Observe that the generalizedJacobian J ( X ) acts via tensor product on any coarse compactified Jacobian U X ( q ) (defined as in Remark2.21) and hence we can define the poset of orbits of U X ( q ). However, the explicit description of Fact 3.3fails for non fine compactified Jacobians.Clearly, the poset of orbits is an invariant of the fine compactified Jacobian endowed with the action ofthe generalized Jacobian. We will now give another description of the poset of orbits of J X ( q ) in termssolely of the singularities of the variety J X ( q ) without any reference to the action of the generalizedJacobian. With this in mind, consider a k -variety V , i.e. a reduced scheme of finite type over k . Defineinductively a finite chain of closed subsets ∅ = V r +1 ⊂ V r ⊂ . . . ⊂ V ⊂ V = V by setting V i equalto the singular locus of V i − endowed with the reduced scheme structure. The loci V i reg := V i \ V i +1 ,consisting of smooth points of V i , form a partition of V into locally closed subsets. We define the singularposet of V , denoted by Σ( V ), as the set of irreducible components of V i reg for 0 ≤ i ≤ r , endowed withthe poset structure defined by setting C ≥ C if and only if C ⊇ C . Proposition 3.4.
Let X be a connected nodal curve and let q be a general polarization on X . Then theposet of orbits O ( J X ( q )) is isomorphic to the singular poset Σ( J X ( q )) .In particular, if J X ( q ) ∼ = J X ( q ′ ) for two general polarizations q, q ′ on X then O ( J X ( q )) ∼ = O ( J X ( q ′ )) .Proof. According to Corollary 2.20(iii), the smooth locus of J X ( q ) is the locus J X ( q ) of line bundles;therefore, using Fact 3.3, the singular locus of J X ( q ) is equal to J X ( q ) = a ∅6 = S ⊆ X sing J X,S ( q ) = [ | S | =1 J X,S ( q ) ∼ = [ | S | =1 J X S ( q S ) . Applying again Corollary 2.20(iii) and Fact 3.3 and proceeding inductively, we get that J X ( q ) i = a | S |≥ i J X,S ( q ) = [ | S | = i J X,S ( q ) ∼ = [ | S | = i J X S ( q S ) . Therefore the smooth locus of J X ( q ) i is equal to J X ( q ) i reg = J X ( q ) i \ J X ( q ) i +1 = a | S | = i J X,S ( q ) = a | S | = i χ ∈ B S ( q ) J χX S . Since each subset J χX S is irreducible, being a J ( X S )-torsor, we deduce that the singular poset of J X ( q )is equal to its poset of orbits, q.e.d. (cid:3) oreover, as we will show in the next proposition, if our base field k is the field C of complex numbers,then the poset of orbits of a fine compactified Jacobian J X ( q ) is a topological invariant of the analyticspace J X ( q ) an associated to J X ( q ), endowed with the Euclidean topology. Proposition 3.5.
Let X be a connected nodal curve of arithmetic genus g = p a ( X ) and let q, q ′ be twogeneral polarizations on X . If J X ( q ) an and J X ( q ′ ) an are homeomorphic then O ( J X ( q )) ∼ = O ( J X ( q ′ )) .Proof. Let ψ : J X ( q ) an ∼ = −→ J X ( q ′ ) an be a homeomorphism. Consider a sheaf I ∈ J X,S ( q ) an for some S ⊆ X sing and denote by S ′ the unique subset of X sing such that ψ ( I ) ∈ J X,S ′ ( q ′ ) an . Fact 3.3(iv)implies that J X ( q ) an (resp. J X ( q ′ ) an ) is locally (analytically) isomorphic at I (resp. at ψ ( I )) to thecomplex analytic space given by the product of | S | (resp. | S ′ | ) nodes with a smooth variety of dimension g − | S | excision (resp. g − | S ′ | ). Using excision and Lemma 3.6, we get that ( dim Q H g ( J X ( q ) an , J X ( q ) an \ { I } , Q ) = 2 | S | , dim Q H g ( J X ( q ′ ) an , J X ( q ′ ) an \ { ψ ( I ) } , Q ) = 2 | S ′ | . Since ψ is a homeomorphism, we conclude that | S | = | S ′ | , or in other words that I ∈ a | S | = i J X,S ( q ) an for some i ≥ ⇒ ψ ( I ) ∈ a | S | = i J X,S ( q ′ ) an . Therefore, the map ψ induces a homeomorphism between a | S | = i J X,S ( q ) an ⊆ J X ( q ) and a | S | = i J X,S ( q ′ ) an ⊆ J X ( q ′ ) for any i ≥
0. Fact 3.3(ii) implies that we have the following decompositions into connectedcomponents a | S | = i J X,S ( q ) an = a | S | = i χ ∈ B S ( q ) ( J χX S ) an and a | S | = i J X,S ( q ′ ) an = a | S | = i χ ∈ B S ( q ′ ) ( J χX S ) an . Hence, ψ induces a bijection ψ ∗ : O ( J X ( q )) ∼ = −→ O ( J X ( q ′ )) between the strata of J X ( q ) and the strata of J X ( q ′ ) with the property that each stratum ( J χX S ) an of J X ( q ) an is sent homeomorphically by ψ onto thestratum ψ ∗ ( J χX S ) an of J X ( q ′ ) an . Therefore, the bijection ψ ∗ is also an isomorphism of posets, q.e.d. (cid:3) Lemma 3.6.
Let V be the complex subvariety of C k + n − k of equations x x = x x = . . . = x k − x k =0 , for some ≤ k ≤ n . Then dim Q H n ( V, V \ { } , Q ) = 2 k .Proof. Since V is contractible, by homotopical invariance of the cohomology of groups we have that(3.3) H n ( V, V \ { } , Q ) = H n − ( L, Q ) = H n − ( L, Q ) ∨ , where L is the link of the origin 0 in V , i.e. the intersection of V with a small sphere of C k + n − k centeredat 0. Observe that V is the union of 2 k vector subspaces of C k + n − k of dimension n : V ǫ • = h e ǫ , e ǫ , . . . , e k − ǫ k , e k +1 , . . . , e k + n − k i where ǫ • = ( ǫ , . . . , ǫ k ) ∈ { , } k , which intersect along vector subspaces of dimension less than or equal to n −
1. It follows that L is theunion of 2 k spheres { S , . . . , S k } of dimension 2 n − n −
3. Fix a triangulation of L that induces a triangulation of each sphere S i and oftheir pairwise intersections. Consider the natural map η : Q k ∼ = ⊕ k i =1 H n − ( S i , Q ) −→ H n − ( L, Q ) . Since there are no simplices in L of dimension greater than 2 n −
1, the map η is injective. Moreover,using the fact that the spheres S i only intersect along spheres of dimension less that or equal to 2 n − η is surjective. Indeed, let C ∈ Z n − ( L, Q ) be a cycle in L of dimension 2 n −
1, i.e.a simplicial (2 n − ∂ ( C ) vanishes. For every 1 ≤ i ≤ k , let C i be the chainobtained from C by erasing all the simplices that are not contained in the sphere S i . By construction,we have that C = P k i =1 C i . Therefore, using that C is a cycle, we get that (for every i ) ∂ ( C i ) = − X j = i ∂ ( C j ) . Observe now that ∂ ( C i ) is a (2 n − S i while P j = i ∂ ( C j ) is a (2 n − S j = i S j . Since S i intersects each S j with j = i in spheres of dimension less than or equal to (2 n − ), we conclude that ∂ ( C i ) = 0, or in other words that C i ∈ Z n − ( S i , Q ). Therefore, we get that η (cid:16)P k i =1 [ C i ] (cid:17) = [ C ], which shows that η is surjective.The assertion now follows from the equality (3.3) together with the fact that η is an isomorphism. (cid:3) Remark . We do not know of any example of two non isomorphic (or non homeomorphic if k = C )fine compactified Jacobians having isomorphic posets of orbits; therefore, we wonder if the converse ofthe second assertion of Proposition 3.4 or the converse of Proposition 3.5 might hold true.The poset of orbits of a fine compactified Jacobian J X ( q ) (or more generally of any coarse compactifiedJacobian U X ( q )) is isomorphic to the poset of regions of a certain toric arrangement of hyperplanes, as wenow explain. Let Γ X be the (connected) dual graph of the nodal curve X , i.e. the graph whose vertices V (Γ X ) correspond to irreducible components of X and whose edges E (Γ X ) correspond to nodes of X :an edge being incident to a vertex if the node corresponding to the former belongs to the irreduciblecomponent corresponding to the latter. We fix an orientation of Γ X , i.e. we specify the source and target s, t : E (Γ X ) → V (Γ X ) of each edge of Γ X . The first homology group H (Γ X , A ) of the graph Γ X withcoefficients in a commutative ring with unit A (e.g. A = Z , Q , R ) is the kernel of the boundary morphism(3.4) ∂ : C (Γ X , A ) = M e ∈ E (Γ X ) A · e −→ C (Γ X , A ) = M v ∈ V (Γ X ) A · v,e t ( e ) − s ( e ) . The map ∂ depends upon the choice of the orientation of Γ X ; however, H (Γ X , A ) does not dependupon the chosen orientation. Since the graph Γ X is connected, the image of the boundary map ∂ is thesubgroup C (Γ X , A ) := X v ∈ V (Γ X ) a v · v : X v ∈ V (Γ X ) a v = 0 ⊂ C (Γ X , A ) . When A = Q or R , we can endow the vector space C (Γ X , A ) with a non-degenerate bilinear form ( , )defined by requiring that ( e, f ) = ( e = f, e = f, for any e, f ∈ E (Γ X ). Denoting by H (Γ X , A ) ⊥ the subspace of C (Γ X , A ) perpendicular to H (Γ X , A ),we have that the boundary map induces an isomorphism of vector spaces(3.5) ∂ : H (Γ X , A ) ⊥ ∼ = −→ C (Γ X , A ) . Let now q be a polarization of total degree | q | = 1 − p a ( X ) = 1 − g and consider the element φ := X v ∈ V (Γ X ) φ v · v = X v ∈ V (Γ X ) (cid:18) q Y v + deg Y v ( ω X )2 (cid:19) · v ∈ C (Γ X , Q ) , where Y v is the irreducible component of X corresponding to the vertex v ∈ V (Γ X ). Using the isomor-phism (3.5), we can find a unique element ψ = P e ∈ E (Γ X ) ψ e · e ∈ H (Γ X , Q ) ⊥ such that ∂ ( ψ ) = φ .Consider now the arrangement of affine hyperplanes in H (Γ , R ) given by(3.6) V q := (cid:26) e ∗ = n + 12 − ψ e (cid:27) n ∈ Z ,e ∈ E (Γ X ) where e ∗ is the functional on C (Γ , R ) (hence on H (Γ X , R ) by restriction) given by e ∗ = ( e, − ). Thearrangement of hyperplanes V q is periodic with respect to the action of H (Γ X , Z ) on H (Γ , R ); hence,it induces an arrangement of hyperplanes in the real torus H (Γ X , R ) H (Γ X , Z ) , which we will still denote by V q and we will call the toric arrangement of hyperplanes associated to q . The toric arrangement V q of hyperplanes subdivides the real torus H (Γ X , R ) H (Γ X , Z ) into finitely many regions, which form naturally apartially ordered set (poset for short) under the natural containment relation. This poset is related tothe coarse compactified Jacobian U X ( q ) as follows. Fact 3.8 (Oda-Seshadri) . Let q be a polarization of total degree | q | = 1 − p a ( X ) = 1 − g on a connectednodal curve X . The poset of regions cut out by the toric arrangement of hyperplanes V q is isomorphicto the poset O ( J X ( q )) of J ( X ) -orbits on U X ( q ) . roof. See [OS79] or [Ale04, Thm. 2.9]. (cid:3)
The arrangement of hyperplanes V q determines wether the polarization q on X is generic or not, atleast if X does not have separating nodes, i.e. nodes whose removal disconnects the curve. Recall that atoric arrangement of hyperplanes is said to be simple if the intersection of r non-trivial hyperplanes inthe given arrangement has codimension at least r . Moreover, following [MV12, Def. 2.8], we say that apolarization q is non-degenerate if and only if q is not integral at any proper subcurve Y ⊂ X such that Y intersects Y c in at least one non-separating node. Note that a general polarization on a nodal curve X is non-degenerate and that the converse is true if X does not have separating nodes. Lemma 3.9.
Let q be a polarization of total degree | q | = 1 − p a ( X ) = 1 − g on a connected nodal curve X .(i) V q is simple if and only if q is non-degenerate. In particular, if q is general then V q is simple, andthe converse is true if X does not have separating nodes.(ii) If V q is simple then we can find a general polarization q ′ such that V q and V q ′ have isomorphicposet of regions.Proof. By [MV12, Thm. 7.1], q is non-degenerate if and only if the number of irreducible components ofthe compactified Jacobian U X ( q ) is the maximum possible which is indeed equal to the complexity c ( X )of the curve X , i.e. the number of spanning trees of the dual graph Γ X of X . By Fact 3.8, this happensif and only if the number of full-dimensional regions cut out by the toric arrangement V q is as big aspossible. This is equivalent, in turn, to the fact that V q is simple, which concludes the proof of (i).Now suppose that V q is simple. Then there exists a small Eucliden open neighborhood U of q in thespace P X of polarizations (see (3.1)) such that for every q ′ ∈ U the toric arrangement V q ′ of hyperplaneshas its poset of regions isomorphic to the poset of regions of V q . Clearly, such an open subset U willcontain a point q ′ not belonging to the arrangement of hyperplanes A X defined in (3.2); any such point q ′ will satisfy the conclusions of part (ii). (cid:3) Although Fact 3.8 and Lemma 3.9 are only stated for (fine) compactified Jacobians of total degree1 − p a ( X ), they can be easily extended to any compactified Jacobian since any (fine) compactified Jacobianof a curve X is equivalent by translation to a (fine) compactified Jacobian of total degree 1 − p a ( X )(although not to a unique one). In particular, combining Propositions 3.4 and 3.5, Fact 3.8 and Lemma3.9, we get the following lower bound for the number of non isomorphic (resp. non homeomorphic if k = C ) fine compactified Jacobians of a nodal curve X . Corollary 3.10.
Let X be a connected nodal curve. Then the number of non isomorphic (resp. nonhomeomorphic if k = C ) fine compactified Jacobians of X is bounded from below by the number of simpletoric arrangement of hyperplanes of the form V q whose posets of regions are pairwise non isomorphic. We end this section by giving a sequence of nodal curves { X n } n ∈ N of genus two such that the numberof simple toric arrangements of hyperplanes {V q } q ∈P X having pairwise non isomorphic posets of regionsbecomes arbitrarily large as n goes to infinity; this implies, by Corollary 3.10, that the number of nonisomorphic (resp. non homeomorphic if k = C ) fine compactified Jacobians can be arbitrarily large evenfor nodal curves, thus completing the proof of Theorem B from the introduction. Example . Consider a genus-2 curve X = X obtained from a dollar sign curve blowing up two ofits 3 nodes. Then the dual graph Γ X of X is as follows: ✸ s✛s ✸ sss s e e e e e Using the orientation depicted in the above Figure, a basis for H (Γ X , Z ) ∼ = Z is given by x := e + e + e and y := e + e + e . Therefore, the functionals on H (Γ X , R ) ∼ = R associated to the edgesof Γ X are given, in the above basis, by( e ) ∗ = ( e ) ∗ = x ∗ ( e ) ∗ = ( e ) ∗ = y ∗ e ∗ = x ∗ + y ∗ . hen each polarization q on X of total degree | q | = 1 − p a ( X ) = − V q of 5 lines in H (Γ X , R ) H (Γ X , Z ) ∼ = R Z of the form V η • = n x = η e , x = η e , y = η e , y = η e , x + y = η e o for some rational numbers η • which are determined by the polarization q , as explained in (3.6) above.Conversely, given such a toric arrangement V η • of 5 lines in R / Z , there is a polarization q on X of totaldegree | q | = 1 − p a ( X ) = − V q = V η • . Moreover, according to Lemma 3.9, the polarization q is general in X if and only if the arrangement V q = V η • is simple. Consider the following two simpletoric arrangements of 5 lines that are drawn on the unit square of R (two of the lines correspond to theedges of the unit square): ❅❅❅❅❅❅❅❅❅❅❅❅ ❅❅❅❅ ❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅❅❅ Then it is easy to check that the poset of regions of the two toric arrangements are not isomorphic:it suffices to note that on the one on the left there are 2 triangular two dimensional regions while onthe one on the right there are 4 triangular two dimensional regions. According to Corollary 3.10, thisimplies that there are at least two generic polarizations on X , q and q ′ , such that J X ( q ) and J X ( q ′ ) arenot isomorphic.More generally, blow up X further in order to obtain a genus-2 curve X n whose dual graph Γ X n is asfollows: sss s s s . . . e n +11 ss s e s s s s s e e n +12 e . . . In words, X n is obtained from the dollar sign curve by blowing up two of its nodes n times. Arguing asabove, the (simple) toric arrangements associated to the (general) polarizations q on X n of total degree | q | = 1 − p a ( X n ) = − n + 3 lines in R / Z of the form V η • = n x = η e i , y = η e j , x + y = η e o ≤ i,j ≤ n +1 for some rational numbers η • which depend on q . For every n < i ≤ n , consider two simple toricarrangements of hyperplanes of R / Z V + i := (cid:26) x = h n , y = k n , x + y = 2 i n + ǫ (cid:27) ≤ h,k ≤ n V − i := (cid:26) x = h n , y = k n , x + y = 2 i n − ǫ (cid:27) ≤ h,k ≤ n where ǫ is a sufficiently small rational number (the poset of regions of the above toric hyperplane arrange-ments do not actually depend on the chosen small value of ǫ ). In the next figure we have represented onthe unit square in R the toric arrangement V + i on the left and the toric arrangement V − i on the right. ❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅❅ i ni n
13 2 i n + ǫ ❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅❅❅ i ni n
13 2 i n − ǫ It is easy to see that the the number of triangular regions cut out on R / Z by V + i (resp. V − i ) is4( n − i )+2 (resp. 4( n − i )+4). This implies that the toric arrangements of hyperplanes {V + i , V − i } n/
The aim of this section is to study the deformation theory and the semiuniversal deformation spaceof a pair (
X, I ) where X is a (reduced) connected curve and I is rank-1 torsion-free simple sheaf on X .For basic facts on deformation theory, we refer to the book of Sernesi [Ser06].4.1. Deformation theory of X . The aim of this subsection is to recall some well-known facts aboutthe deformation theory of a (reduced) curve X .Let Def X (resp. Def ′ X ) be the local moduli functor of X (resp. the locally trivial moduli functor) of X in the sense of [Ser06, Sec. 2.4.1]. Moreover, for any p ∈ X sing , we denote by Def X,p the deformationfunctor of the complete local k -algebra b O X,p in the sense of [Ser06, Sec. 1.2.2]. The above deformationfunctors are related by the following natural morphisms:(4.1) Def ′ X → Def X → Def loc X := Y p ∈ X sing Def
X,p . Since X is reduced, the tangent spaces to Def ′ X , Def X and Def X,p where p ∈ X sing are isomorphic to(see [Ser06, Cor. 1.1.11, Thm. 2.4.1])(4.2) T Def ′ X := Def ′ X ( k [ ǫ ]) = H ( X, T X ) ,T Def X := Def X ( k [ ǫ ]) = Ext (Ω X , O X ) ,T Def ( X,p ) := Def ( X,p ) ( k [ ǫ ]) = ( T X ) p , where Ω X is the sheaf of K¨ahler differentials on X , T X := H om (Ω X , O X ) is the tangent sheaf of X and T X = E xt (Ω X , O X ) is the first cotangent sheaf of X , which is a sheaf supported on X sing by [Ser06,Prop. 1.1.9(ii)].The usual local-to-global spectral sequence gives a short exact sequence(4.3) 0 → H ( X, T X ) = T Def ′ X → Ext (Ω X , O X ) = T Def X →→ H ( X, E xt (Ω X , O X )) = M p ∈ X sing E xt (Ω X , O X ) p = T Def loc X → H ( X, T X ) = 0 , which coincides with the exact sequence on the tangent spaces induced by (4.1).By looking at the obstruction spaces of the above functors, one can give criteria under which theabove deformation functors are smooth (in the sense of [Ser06, Def. 2.2.4]). Fact 4.1. (i)
Def ′ X is smooth;(ii) If X has l.c.i. singularities at p ∈ X sing then Def
X,p is smooth;(iii) If X has l.c.i. singularities, then Def X is smooth and the morphism Def X → Def loc X is smooth. roof. Part (i): an obstruction space for Def ′ X is H ( X, T X ) by [Ser06, Prop. 2.4.6] and H ( X, T X ) = 0because dim X = 1. Therefore, Def ′ X is smooth.Part (ii) follows from [Ser06, Cor. 3.1.13(ii)].Part (iii): by [Ser06, Prop. 2.4.8] , an obstruction space for Def X is Ext (Ω X , O X ), which is zero by[Ser06, Example 2.4.9]. Therefore we get that Def X is smooth.Since Def loc X is smooth by part (ii) and the map of tangent spaces T Def X → T Def loc X is surjective by(4.3), the smoothness of the morphism Def X → Def loc X follows from the criterion [Ser06, Prop. 2.3.6]. (cid:3) Deformation theory of the pair ( X, I ) . The aim of this subsection is to review some fundamentalresults due to Fantechi-G¨ottsche-vanStraten [FGvS99] on the deformation theory of a pair (
X, I ), where X is a (reduced) curve and I is a rank-1 torsion-free sheaf on X (not necessarily simple).Let Def ( X,I ) be the deformation functor of the pair ( X, I ) and, for any p ∈ X sing , we denote byDef ( X,I ) ,p the deformation functor of the pair ( b O X,p , I p ). We have a natural commutative diagram(4.4) Def ( X,I ) / / (cid:15) (cid:15) Def loc(
X,I ) := Q p ∈ X sing Def ( X,I ) ,p (cid:15) (cid:15) Def X / / Def loc X := Q p ∈ X sing Def
X,p . Under suitable hypothesis, the deformation functors appearing in the above diagram (4.4) are smoothand the horizontal morphisms are smooth as well.
Fact 4.2 (Fantechi-G¨ottsche-vanStraten) . (i) The natural morphism Def ( X,I ) → Def loc(
X,I ) × Def loc X Def X is smooth. In particular, if X has l.c.i. singularities then the morphism Def ( X,I ) → Def loc(
X,I ) issmooth.(ii) If X has locally planar singularities at p ∈ X sing then Def ( X,I ) ,p is smooth. In particular, if X haslocally planar singularities then Def loc(
X,I ) and Def ( X,I ) are smooth.Proof. Part (i): the first assertion follows from [FGvS99, Prop. A.1] . The second assertion follows fromthe first one together with Fact 4.1(iii) which implies that the morphism Def loc( X,I ) × Def loc X Def X → Def loc(
X,I ) is smooth.Part (ii): the first assertion follows from [FGvS99, Prop. A.3] . The second assertion follows from thefirst together with part (i). (cid:3) Semiuniversal deformation space.
The aim of this subsection is to describe and study thesemiuniversal deformation spaces for the deformation functors Def X and Def ( X,I ) .According to [Ser06, Cor. 2.4.2], the functor Def X admits a semiuniversal formal couple ( R X , X ),where R X is a Noetherian complete local k -algebra with maximal ideal m X and residue field k and X ∈ \ Def X ( R X ) := lim ←− Def X (cid:18) R X m nX (cid:19) is a formal deformation of X over R X . Recall that this means that the morphism of functors(4.5) h R X := Hom( R X , − ) −→ Def X determined by X is smooth and induces an isomorphism of tangent spaces T R X := ( m X / m X ) ∨ ∼ = → T Def X (see [Ser06, Sec. 2.2]). The formal couple ( R X , X ) can be also viewed as a flat morphism of formal In loc. cit., it is assumed that the characteristic of the base field is 0. However, the statement is true in anycharacteristics, see [Vis, Thm. (4.4)]. In loc. cit., it is assumed that the base field is the field of complex numbers. However, a direct inspection reveals thatthe same argument works over any (algebraically closed) base field. As before, the argument of loc. cit. works over any (algebraically closed) base field. Some authors use the word miniversal instead of semiuniversal. We prefer to use the word semiuniversal in order tobe coherent with the terminology of the book of Sernesi [Ser06]. chemes(4.6) π : X →
Spf R X , where Spf denotes the formal spectrum, such that the reduced scheme X red underlying X (see [EGAI,Prop. 10.5.4]) is isomorphic to X (see [Ser06, p. 77]). Note that the semiuniversal formal couple ( R X , X )is unique by [Ser06, Prop. 2.2.7].Since X is projective and H ( X, O X ) = 0, Grothendieck’s existence theorem (see [Ser06, Thm. 2.5.13])gives that the formal deformation (4.6) is effective , i.e. there exists a deformation π : X →
Spec R X of X over Spec R X whose completion along X = π − ([ m X ]) is isomorphic to (4.6). In other words, wehave a Cartesian diagram(4.7) X (cid:15) (cid:15) (cid:31) (cid:127) / / (cid:3) X / / π (cid:15) (cid:15) (cid:3) X π (cid:15) (cid:15) Spec k ∼ = [ m X ] (cid:31) (cid:127) / / Spf R X / / Spec R X . Note also that the deformation π is unique by [Ser06, Thm. 2.5.11].Later on, we will need the following result on the effective semiuniversal deformation of a curve X with locally planar singularities. Lemma 4.3.
Assume that X has locally planar singularities. Let U be the open subset of Spec R X consisting of all the (schematic) points s ∈ Spec R X such that the geometric fiber X s of the universalfamily π : X →
Spec R X is smooth or has a unique singular point which is a node. Then the codimensionof the complement of U inside Spec R X is at least two.Proof. Since the natural morphism (see (4.1))Def X → Def loc X := Y p ∈ X sing Def
X,p is smooth by Fact 4.1(iii), it is enough to show that if Def
X,p has dimension at most one then p ∈ X sing iseither a smooth point or a node of X . This is stated in [Ser06, Prop. 3.1.5.] under the assumption thatchar( k ) = 0. However, a slight modification of the argument of loc. cit. works in arbitrary characteristic,as we are now going to show.First, since X has locally planar singularities at p , we can write b O X,p = k [[ x,y ]] f , for some power series f = f ( x, y ) ∈ k [[ x, y ]]. By [Ser06, p. 124], the tangent space to Def ( X,p ) is equal to T := T b O X,p = k [[ x, y ]]( f, ∂ x f, ∂ y f ) . Since Def
X,p is smooth by Fact 4.1(ii), then the dimension of Def
X,p is equal to dim k T .From the above description, it is clear that dim k T = 0 if and only if f contains some linear term,which happens if and only if p is a smooth point of X .Therefore, we are left with showing that p is a node of X (i.e. f can be taken to be equal to xy ) ifand only if dim k T = 1, which is equivalent to ( x, y ) = ( f, ∂ x f, ∂ y f ). Clearly, if f = xy then ∂ x f = y and ∂ y f = x so that ( x, y ) = ( f, ∂ x f, ∂ y f ) = ( xy, y, x ). Conversely, assume that ( x, y ) = ( f, ∂ x f, ∂ y f ).Then clearly f cannot have a linear term. Consider the degree two part f = Ax + Bxy + Cy of f . Bycomputing the partial derivatives and imposing that x, y ∈ ( f, ∂ x f, ∂ y f ), we get that the discriminant∆ = B − AC of f is different from 0. Then, acting with a linear change of coordinates, we can assumethat f = xy . Now, it is easily checked that via a change of coordinates of the form x x + g ( x, y ) and y y + h ( x ) with g ( x, y ) ∈ ( x, y ) and h ( x ) ∈ ( x ) , we can transform f into xy , and we are done. (cid:3) Consider now the functor J ∗X : { Spec R X − schemes } −→ { Sets } which sends a scheme T → Spec R X to the set of isomorphism classes of T -flat, coherent sheaves on X T := T × Spec R X X whose fibers over T are simple rank-1 torsion-free sheaves. The functor J ∗X containsthe open subfunctor J ∗X : { Spec R X − schemes } −→ { Sets } which sends a scheme T → Spec R X to the set of isomorphism classes of line bundles on X T .Analogously to Fact 2.2, we have the following act 4.4 (Altman-Kleiman, Esteves) . (i) The Zariski (equiv. ´etale, equiv. fppf ) sheafification of J ∗X is represented by a scheme J X endowedwith a morphism u : J X → Spec R X , which is locally of finite type and satisfies the existence part ofthe valuative criterion for properness. The scheme J X contains an open subset J X which representsthe Zariski (equiv. ´etale, equiv. fppf ) sheafification of J ∗X and the restriction u : J X → Spec R X issmooth.Moreover, the fiber of J X (resp. of J X ) over the closed point [ m X ] ∈ Spec R X is isomorphic to J X (resp. J X ).(ii) There exists a sheaf b I on X ×
Spec R X J X such for every F ∈ J ∗X ( T ) there exists a unique Spec R X -map α F : T → J X with the property that F = (id X × α F ) ∗ ( b I ) ⊗ π ∗ ( N ) for some N ∈ Pic( T ) , where π : X ×
Spec R X T → T is the projection onto the second factor. The sheaf b I is uniquely determinedup to tensor product with the pullback of an invertible sheaf on J X and it is called a universal sheaf on J X .Moreover, the restriction of b I to X × J X is equal to a universal sheaf as in Fact 2.2 (iii) .Proof. Part (i): the representability of the ´etale sheafification (and hence of the fppf sheafification) of J ∗X by an algebraic space J X locally of finite type over Spec R X follows from [AK80, Thm. 7.4], where itis proved for the moduli functor of simple sheaves, along with the fact that being torsion free and rank-1is an open condition. From [Est01, Cor. 52], it follows that J X becomes a scheme after an ´etale cover ofSpec R X . However, since R X is strictly henselian (being a complete local ring with algebraically closedresidue field), then Spec R X does not admit non trivial connected ´etale covers (see [BLR90, Sec. 2.3]);hence J X is a scheme. The scheme J X satisfies the existence part of the valuative criterion for propernessby [Est01, Thm. 32].The fact that J X represents also the Zariski sheafification of J ∗X follows from [AK79b, Thm. 3.4] oncewe prove that the morphism π : X →
Spec R X admits a section through its smooth locus. Indeed, let U be the smooth locus of the morphism π and denote by π ′ : U → Spec R X the restriction of π to U .Since X is assumed to be reduced, all the geometric fibers of π are reduced by [EGAIV3, Thm. 12.2.4];hence, we deduce that for every s ∈ Spec R X the open subset π ′− ( s ) is dense in π − ( s ). Now, since R X is a strictly henselian ring, given any point p ∈ π ′− ([ m X ]), we can find a section of π ′ : U → Spec R X passing through p (see [BLR90, Sec. 2.3, Prop. 5]), as required.Since J ∗X is an open subfunctor of J ∗X , it follows that J X contains an open subscheme J X whichrepresents the ´etale sheafification of J ∗X . The smoothness of J X over Spec R X follows from [BLR90, Sec.8.4, Prop. 2]. The last assertion of part (i) is obvious.Part (ii) is an immediate consequence of the fact that J X represents the Zariski sheafification of J ∗X (see also [AK79b, Thm. 3.4]). The last assertion of part (ii) is obvious. (cid:3) Let now I be a simple rank-1 torsion-free sheaf I on X , i.e. I ∈ J X ⊂ J X . If we denote by R ( X,I ) := b O J X ,I the completion of the local ring of J X at I and by m ( X,I ) its maximal ideal, then thereis a natural map j : Spec R ( X,I ) → J X which fits into the following Cartesian diagram(4.8) (id × j ) ∗ ( b I ) b IX ×
Spec R X Spec R ( X,I )id × j / / π × id (cid:15) (cid:15) (cid:3) X ×
Spec R X J X (cid:15) (cid:15) / / (cid:3) X π (cid:15) (cid:15) Spec R ( X,I ) j / / J X u / / Spec R X . Since I ∈ J X ⊂ J X , the map u ◦ j sends the closed point [ m X,I ] ∈ Spec b O J X ,I into the closed point[ m X ] ∈ Spec R X . In particular, we have that ( π × id) − ( m ( X,I ) ) = π − ( m X ) = X and the restrictionof (id × j ) ∗ ( b I ) to ( π × id) − ( m ( X,I ) ) = X is isomorphic to I by the universal property in Fact 4.4(ii). This result is stated in loc. cit. only for flat and proper morphisms with integral geometric fibers; however, the sameproof works assuming only reduced geometric fibers. he above diagram gives rise to a deformation of the pair ( X, I ) above Spec R ( X,I ) , which induces amorphism of deformation functors(4.9) h R ( X,I ) := Hom( R ( X,I ) , − ) −→ Def ( X,I ) . We can now prove the main result of this section.
Theorem 4.5.
Let X be a (reduced) curve and I a rank-1 torsion-free simple sheaf on X .(i) There exists a Cartesian diagram of deformation functors (4.10) h R ( X,I ) / / (cid:15) (cid:15) (cid:3) Def ( X,I ) (cid:15) (cid:15) h R X / / Def X , where the horizontal arrows realize R ( X,I ) and R X as the semiuniversal deformation rings for Def ( X,I ) and Def X , respectively.(ii) If X has l.c.i. singularities then R X is regular (i.e. it is a power series ring over k ).(iii) If X has locally planar singularities then R ( X,I ) is regular. In particular, the scheme J X is regular.Proof. Part (i): the fact that the diagram (4.10) is commutative follows from the definition of the map(4.9) and the commutativity of the diagram (4.8).Let us check that the above diagram (4.10) is Cartesian. Let A be an Artinian local k -algebra withmaximal ideal m A . Suppose that there exists a deformation ( e X, e I ) ∈ Def ( X,I ) ( A ) of ( X, I ) over A and a homomorphism φ ∈ Hom( R X , A ) = h R X ( A ) that have the same image in Def X ( A ). We haveto find a homomorphism η ∈ Hom( R ( X,I ) , A ) = h R ( X,I ) ( A ) that maps into φ ∈ h R X ( A ) and ( e X, e I ) ∈ Def ( X,I ) ( A ) via the maps of diagram (4.10). The assumption that the elements ( e X, e I ) ∈ Def ( X,I ) ( A )and φ ∈ h R X ( A ) have the same image in Def X ( A ) is equivalent to the fact that e X is isomorphicto X A := X ×
Spec R X Spec A with respect to the natural morphism Spec A → Spec R X induced by φ .Therefore the sheaf e I can be seen as an element of J ∗X (Spec A ). Fact 4.4(ii) gives a map α e I : Spec A → J X such that e I = (id X × α e I ) ∗ ( b I ), because Pic(Spec A ) = 0. Clearly the map α e I sends [ m A ] into I ∈ J X ⊂ J X and therefore it factors through a map β : Spec A → Spec R ( X,I ) followed by the map j of (4.8). Themorphism β determines the element η ∈ Hom( R ( X,I ) , A ) = h R ( X,I ) ( A ) we were looking for.Finally, the bottom horizontal morphism realizes the ring R X as the semiuniversal deformation ringfor Def X by the very definition of R X . Since the diagram (4.10) is Cartesian, the same is true for thetop horizontal arrow.Part (ii): R X is regular since the morphism h R X → Def X is smooth and Def X is smooth by Fact4.1(iii).Part (iii): R ( X,I ) is regular since the morphism h R ( X,I ) → Def ( X,I ) is smooth and Def ( X,I ) is smoothby Fact 4.2(ii). We deduce that the open subset U of regular points of J X contains the central fiber u − ([ m X ]) = J X , which implies that U = J X because u − ([ m X ]) contains all the closed points of J X ;hence J X is regular. (cid:3) Universal fine compactified Jacobians
The aim of this section is to introduce and study the universal fine compactified Jacobians relative tothe semiuniversal deformation π : X →
Spec R X introduced in § q on X as in Definition2.9. Indeed, we are going to show that the polarization q induces a polarization on each fiber of theeffective semiuniversal deformation family π : X →
Spec R X .With this aim, we will first show that the irreducible components of the fibers of the morphism π : X →
Spec R X are geometrically irreducible. For any (schematic) point s ∈ Spec R X , we denote by X s := π − ( s ) the fiber of π over s , by X s := X s × k ( s ) k ( s ) the geometric fiber over s and by ψ s : X s → X s the natural morphism. Lemma 5.1.
The irreducible components of X s are geometrically irreducible. Therefore we get a bijection ( ψ s ) ∗ : { Subcurves of X s } ∼ = −→ { Subcurves of X s } Z ⊆ X s ψ s ( Z ) ⊆ X s . roof. Let V ⊆ X be the biggest open subset where the restriction of the morphism π : X →
Spec R X is smooth. Since π is flat, the fiber V s of V over a point s ∈ Spec R X is the smooth locus of the curve X s = π − ( s ), which is geometrically reduced because the central curve X = π − ([ m X ]) is reduced. Inparticular, V s ⊆ X s and V s := V s × k ( s ) k ( s ) ⊆ X s are dense open subsets. Therefore, the irreduciblecomponents of X s (resp. of X s ) are equal to the irreducible components of V s (resp. of V s ). However, since V s is smooth over k ( s ) by construction, the irreducible components of V s coincide with the connectedcomponents of V s and similarly for V s . In conclusion, we have to show that the connected componentsof V s are geometrically connected for any point s ∈ Spec R X .We will need the following preliminary result.Claim: For any point s ∈ Spec R X , the irreducible components of V { s } := V ∩ π − ( { s } ) do notmeet on the central fiber V o := π − ([ m X ]) ∩ V and each of them is the closure of a unique irreduciblecomponent of V s .Indeed, observe that { s } is a closed integral subscheme of Spec R X , so that { s } = Spec T where T is a Noetherian complete local domain quotient of R X with residue field k = k ; hence, T is a strictlyHenselian local domain. This implies that Spec T is geometrically unibranch at its unique closed point o = [ m x ] (see [Stacks, Tag 06DM]). Since the morphism V { s } → { s } = Spec T is smooth, we infer that V { s } is geometrically unibranch along the central fiber V o (see [EGAIV2, Prop. 6.15.10]). This impliesthat two distinct irreducible components of V { s } do not meet along the central fiber V o , and the firstassertion of the Claim follows. The second assertion follows from the fact that, since V { s } → { s } is flat,each generic point of V { s } maps to the generic point s of { s } , q.e.d.Let now C be a connected component of V s , for some point s ∈ Spec R X . The closure C of C inside X will contain some irreducible component of the central fiber X o = X [ m X ] by the upper semicontinuity ofthe dimension of the fibers (see [EGAIV3, Lemma 13.1.1]) applied to the projective surjective morphism e C → { s } . Hence, C ∩ V will contain some (not necessarily unique) connected component C o of thecentral fiber V o = V [ m X ] . Since R X is a strictly henselian ring and V → Spec R X is smooth, given anypoint p ∈ C o ⊆ V o , we can find a section σ of V → Spec R X passing through p (see [BLR90, Sec. 2.3,Prop. 5]). By the Claim, C ∩ V is the unique irreducible component of V { s } containing the point p .Therefore, the restriction of σ at { s } must take values in C ∩ V . In particular, σ ( s ) is a k ( s )-rationalpoint of C . Now we conclude that C is geometrically connected by [EGAIV2, Cor. 4.5.14]. (cid:3) Consider now the set-theoretic map(5.1) Σ s : { Subcurves of X s } −→ { Subcurves of X }X s ⊇ Z ψ s ( Z ) ∩ X ⊆ X, where ψ s ( Z ) is the Zariski closure inside X of the subcurve ψ s ( Z ) ⊆ X s and the intersection ψ s ( Z ) ∩ X is endowed with the reduced scheme structure. Note that ψ s ( Z ) ∩ X has pure dimension one (in otherwords, it does not contain isolated points), hence it is a subcurve of X , by the upper semicontinuity ofthe local dimension of the fibers (see [EGAIV3, Thm. 13.1.3]) applied to the morphism ψ s ( Z ) → { s } and using the fact that ψ s ( Z ) has pure dimension one in X s .The map Σ s satisfies two important properties that we collect in the following Lemma 5.2. (i) If Z , Z ⊆ X s do not have common irreducible components then Σ s ( Z ) , Σ s ( Z ) ⊆ X do not havecommon irreducible components. In particular, Σ s ( Z c ) = Σ s ( Z ) c .(ii) If Z ⊆ X s is connected then Σ s ( Z ) ⊆ X is connected.Proof. Let us first prove (i). Since Z , Z are two subcurves of X s without common irreducible com-ponents then the subcurves ψ s ( Z ) and ψ s ( Z ) of X s do not have common irreducible components byLemma 5.1. As in the proof of Lemma 5.1, denote by V the biggest open subset of X on which therestriction of the morphism π is smooth. Then, since V s := V ∩ X s is the smooth locus of X s , wededuce that ψ s ( Z ) ∩ V and ψ s ( Z ) ∩ V are disjoint subsets of X s ∩ V each of which is a union ofconnected components of X s ∩ V . By the Claim in the proof of Lemma 5.1, the closures ψ s ( Z ) ∩ V and ψ s ( Z ) ∩ V do not intersect in the central fiber V o , or in other words Σ s ( Z ) ∩ V = ψ s ( Z ) ∩ V ∩ X andΣ s ( Z ) ∩ V = ψ s ( Z ) ∩ V ∩ X are disjoint. This implies that Σ s ( Z ) and Σ s ( Z ) intersect only in thesingular locus of X , and in particular they do not share any irreducible component of X . et us now prove (ii). Consider the closed subscheme (with reduced scheme structure) ψ s ( Z ) ⊆ X andthe projective and surjective morphism σ := π | ψ s ( Z ) : ψ s ( Z ) → { s } , where { s } ⊆ Spec R X is the closure(with reduced structure) of the schematic point s inside the scheme Spec R X . Note that Σ s ( Z ) is, bydefinition, the reduced scheme associated to the central fiber ψ s ( Z ) o := σ − ([ m X ]) of σ . By Lemma 5.1,the geometric generic fiber of σ is equal to ψ s ( Z ) × k ( s ) k ( s ) = Z , hence it is connected by assumption.Therefore, there is an open subset W ⊆ { s } such that σ − ( W ) → W has geometrically connected fibers(see [Stacks, Tag 055G]).Choose now a complete discrete valuation ring R , with residue field k , endowed with a morphism f : Spec R → { s } that maps the generic point η of Spec R to a certain point t ∈ W and the specialpoint 0 of Spec R to [ m X ] in such a way that the induced morphism Spec k (0) → Spec k ([ m x ]) is anisomorphism. Consider the pull-back τ : Y →
Spec R of the family σ : ψ s ( Z ) → { s } via the morphism f .By construction, the special fiber Y =: τ − (0) of τ is equal to ψ s ( Z ) o and the generic fiber Y η := τ − ( η )of τ is equal to the fiber product σ − ( t ) × Spec k ( t ) Spec k ( η ). In particular, the generic fiber Y η isgeometrically connected.Next, consider the closure Z := Y η of the generic fiber Y η inside Y , i.e. the unique closed subscheme Z of Y which is flat over Spec R and such that its generic fiber Z η is equal to Y η (see [EGAIV2, Prop. 2.8.5]).The special fiber Z of Z is a closed subscheme of Y = ψ s ( Z ) o which must contain the dense open subset X sm ∩ Σ s ( Z ) ⊆ Σ s ( Z ), where X sm is the smooth locus of X . Indeed, arguing as in the proof of Lemma5.1, through any point p of X sm ∩ Σ s ( Z ) there is a section of X ×
Spec R X Spec R → Spec R entirelycontained in Y , which shows that p must lie in the closure of Y η inside Y , i.e. in Z . Therefore, Σ s ( Z ) isalso the reduced scheme associated to the central fiber Z . Finally, since the morphism Z →
Spec R isflat and projective by construction and the generic fiber Z η = Y η is geometrically connected, we deducethat Z , and hence Σ s ( Z ), is (geometrically) connected by applying [EGAIV3, Prop. (15.5.9)] (whichsays that the number of geometrically connected components of the fibers of a flat and proper is lowersemicontinuous). (cid:3) We are now ready to show that a (general) polarization on X induces, in a canonical way, a (general)polarization on each geometric fiber of its semiuniversal deformation π : X →
Spec R X . Lemma-Definition 5.3.
Let s ∈ Spec R X and let q be a polarization on X . The polarization q s inducedby q on the geometric fiber X s is defined by q sZ := q Σ s ( Z ) ∈ Q for every subcurve Z ⊆ X s . If q is general then q s is general.Proof. Let us first check that q s is well-defined. i.e. that | q s | ∈ Z and that ( Z ⊆ X s ) q sZ is additive(see the discussion after Definition 2.8). Since Σ s ( X s ) = X , we have that | q s | = q s X s = q X = | q | ∈ Z . Moreover, the additivity of q s follows from the additivity of q using Lemma 5.2(i).The last assertion follows immediately from Remark 2.10 and Lemma 5.2. (cid:3) Given a general polarization q on X , we are going to construct an open subset of J X , proper overSpec R X , whose geometric fibers are fine compactified Jacobians with respect to the general polarizationsconstructed in the above Lemma-Definition 5.3. Theorem 5.4.
Let q be a general polarization on X . Then there exists an open subscheme J X ( q ) ⊆ J X which is projective over Spec R X and such that the geometric fiber of u : J X ( q ) → Spec R X over a point s ∈ Spec R X is isomorphic to J X s ( q s ) . In particular, the fiber of J X ( q ) → Spec R X over the closedpoint [ m X ] ∈ Spec R X is isomorphic to J X ( q ) . We call the scheme J X ( q ) the universal fine compactified Jacobian of X with respect to the polarization q . We denote by J X ( q ) the open subset of J X ( q ) parametrizing line bundles, i.e. J X ( q ) = J X ( q ) ∩ J X ⊆ J X . Proof.
This statement follows by applying to the effective semiuniversal family
X →
Spec R X a generalresult of Esteves ([Est01, Thm. A]). In order to connect our notations with the notations of loc. cit.,choose a vector bundle E on X such that q E = q (see Remark 2.16), so that our fine compactifiedJacobian J X ( q ) coincides with the variety J sE = J ssE in [Est01, Sec. 4]. ince an obstruction space for the functor of deformations of E is H ( X, E ⊗ E ∨ ) (see e.g. [FGA05,Thm. 8.5.3(b)]) and since this latter group is zero because X is a curve, we get that E can be extended to avector bundle E on the formal semiuniversal deformation X →
Spf R X of X . However, by Grothendieck’salgebraization theorem for coherent sheaves (see [FGA05, Thm. 8.4.2]), the vector bundle E is thecompletion of a vector bundle E on the effective semiuniversal deformation family π : X →
Spec R X of X . Note that the restriction of E to the central fiber of π is isomorphic to the vector bundle E on X .Denote by E s (resp. E s ) the restriction of E to the fiber X s (resp. the geometric fiber X s ).Claim: For any s ∈ Spec R X and any subcurve Z ⊆ X s , we have thatdeg Z ( E s ) = deg ψ s ( Z ) ( E s ) = deg Σ s ( Z ) ( E ) . Indeed, the first equality follows from the fact that Z is the pull-back of ψ s ( Z ) via the map Spec k ( s ) → Spec k ( s ) because of Lemma 5.1. In order to prove the second equality, consider the closed subscheme(with reduced scheme structure) ψ s ( Z ) ⊆ X and the projective and surjective morphism σ := π | ψ s ( Z ) : ψ s ( Z ) → { s } , where { s } ⊆ Spec R X is the closure of the schematic point s inside the scheme Spec R X .Note that the central fiber σ − ([ m x ]) := ψ s ( Z ) o of σ is a one-dimensional subscheme of X , which isgenerically reduced (because X is reduced) and whose underlying reduced curve is Σ s ( Z ) by definition.In particular, the 1-cycle associated to ψ s ( Z ) o coincides with the 1-cycle associated to Σ s ( Z ). Therefore,since the degree of a vector bundle on a subscheme depends only on the associated cycle, we have that(5.2) deg Σ s ( Z ) ( E ) = deg ψ s ( Z ) o ( E ) . Observe that there exists an open subset U ⊆ { s } such that σ | σ − ( U ) : σ − ( U ) → U is flat (by theTheorem of generic flatness, see [Mum66, Lecture 8]). Since the degree of a vector bundle is preservedalong the fibers of a flat morphism and clearly s ∈ U , we get that(5.3) deg ψ s ( Z ) ( E s ) = deg ψ s ( Z ) ( E ) = deg ψ s ( Z ) t ( E ) for any t ∈ U, where we set ψ s ( Z ) t := σ − ( t ).Choose now a complete discrete valuation ring R , with residue field k , endowed with a morphism f : Spec R → { s } that maps the generic point η of Spec R to a certain point t ∈ U and the specialpoint 0 of Spec R to [ m X ] in such a way that the induced morphism Spec k (0) → Spec k ([ m x ]) is anisomorphism. Consider the pull-back τ : Y →
Spec R of the family σ : ψ s ( Z ) → { s } via the morphism f and denote by F the pull-back to Y of the restriction of the vector bundle E to ψ s ( Z ). By construction,the special fiber Y =: τ − (0) of τ is equal to ψ s ( Z ) o and the generic fiber Y η := τ − ( η ) of τ is equal tothe fiber product ψ s ( Z ) t × Spec k ( t ) Spec k ( η ). Therefore, we have that(5.4) deg Y ( F ) = deg ψ s ( Z ) o ( E ) = deg ψ s ( Z ) o ( E ) and deg Y η ( F ) = deg ψ s ( Z ) t ( E ) . Next, consider the closure Z := Y η of the generic fiber Y η inside Y , i.e. the unique closed subscheme Z of Y which is flat over Spec R and such that its generic fiber Z η is equal to Y η (see [EGAIV2, Prop.2.8.5]). The special fiber Z of Z is a closed subscheme of Y = ψ s ( Z ) o which must contain the denseopen subset X sm ∩ Σ s ( Z ) ⊆ Σ s ( Z ), where X sm is the smooth locus of X . Indeed, arguing as in the proofof Lemma 5.1, through any point p of X sm ∩ Σ s ( Z ) there is a section of X ×
Spec R X Spec R → Spec R entirely contained in Y , which shows that p must lie in the closure of Y η inside Y , i.e. in Z . Therefore,the 1-cycle associated to Z coincides with the 1-cycle associated to Σ s ( Z ), from which we deduce that(5.5) deg Σ s ( Z ) ( E ) = deg Z ( F ) . Finally, since the morphism
Z →
Spec R is flat, we have that(5.6) deg Z ( F ) = deg Z η ( F ) = deg Y η ( F ) . By combining (5.2), (5.3), (5.4), (5.5) and (5.6), the Claim follows.The above Claim, together with Remark 2.16, implies that q E s = q s . Therefore, exactly as before,we get that a torsion-free rank-1 sheaf I on X , flat on Spec R X , is (semi)stable with respect to E inthe sense of [Est01, Sec. 1.4] if and only if for every s ∈ Spec R X the restriction I s of I to X s is(semi)stable with respect to q s in the sense of Definition 2.11. Since all the polarizations q s are generalby Lemma-Definition 5.3, we get that the open subscheme J X ( q ) := J s E = J ss E ⊂ J X parametrizingsheaves I ∈ J X whose restriction to X s is q s -semistable (or equivalently q s -stable) is a proper scheme We do not know if σ is flat, a property that would considerably simplify the proof of Claim. ver Spec R X by [Est01, Thm. A]. Moreover, J s E is quasi-projective over Spec R X by [Est01, Thm. C];hence it is projective over Spec R X . The description of the fibers of J X ( q ) → Spec R X is now clear fromthe definition of J X ( q ). (cid:3) If the curve X has locally planar singularities, then the universal fine compactified Jacobians of X have several nice properties that we collect in the following statement. Theorem 5.5.
Assume that X has locally planar singularities and let q be a general polarization on X .Then we have:(i) The scheme J X ( q ) is regular and irreducible.(ii) The surjective map u : J X ( q ) → Spec R X is projective and flat of relative dimension p a ( X ) .(iii) The smooth locus of u is J X ( q ) .Proof. The regularity of J X ( q ) follows from Theorem 4.5(iii). Therefore, in order to show that J X ( q ) isirreducible, it is enough to show that it is connected. Since the open subset J X ( q ) is dense by Corollary2.20, it is enough to prove that J X ( q ) is connected. However, this follows easily from the fact that J X ( q )is smooth over Spec R X and its generic fiber is the Jacobian of degree | q | of a smooth curve, hence it isconnected.Since also Spec R X is regular by Theorem 4.5(ii), the flatness of the map u : J X ( q ) → Spec R X willfollow if we show that all the geometric fibers are equi-dimensional of the same dimension (see [Mat89,Cor. of Thm 23.1, p. 179]). By Theorem 5.4, the geometric fiber of u over s ∈ Spec R X is isomorphicto J X s ( q s ) which has pure dimension equal to h ( X s , O X s ) = h ( X, O X ) = p a ( X ) by Corollary 2.20.The map u is projective by Theorem 5.4 and the fact that its smooth locus is equal to J X ( q ) followsfrom Corollary 2.20. (cid:3) The above result on the universal fine compactified Jacobians of X has also some very importantconsequences for the fine compactified Jacobians of X , that we collect in the following two corollaries. Corollary 5.6.
Assume that X has locally planar singularities and let q be a general polarization on X .Then J X ( q ) is connected.Proof. Consider the universal fine compactified Jacobian J X ( q ) and the natural surjective morphism u : J X ( q ) → Spec R X . According to Theorem 5.5(ii), u is flat and projective. Therefore, we can apply[EGAIV3, Prop. (15.5.9)] which says that the number of connected components of the geometric fibersof u is lower semicontinuous. Since the generic geometric fiber of u is the Jacobian of a smooth curve(by Theorem 5.4), hence connected, we deduce that also the fiber over the closed point [ m X ] ∈ Spec R X ,which is J X ( q ) by Theorem 5.4, is connected, q.e.d. (cid:3) Corollary 5.7.
Assume that X has locally planar singularities and let q be a general polarization on X .Then the universal fine compactified Jacobian u : J X ( q ) → Spec R X (with respect to the polarization q )has trivial relative dualizing sheaf. In particular, J X ( q ) has trivial dualizing sheaf.Proof. Observe that the relative dualizing sheaf, call it ω u , of the universal fine compactified Jacobian u : J X ( q ) → Spec R X is a line bundle because the fibers of u have l.c.i. singularities by Theorem 5.4and Corollary 2.20.Consider now the open subset U ⊆ Spec R X consisting of those points s ∈ Spec R X such that thegeometric fiber X s over s has at most a unique singular point which is a node (as in Lemma 4.3).CLAIM: ( ω u ) | u − ( U ) = O u − ( U ) .Indeed, Theorem 5.4 implies that the geometric fiber of J X ( q ) → Spec R X over a point s is isomorphicto J X s ( q s ). If X s is smooth or if it has a separating node, then J X s ( q s ) is an abelian variety, hence it hastrivial dualizing sheaf. If X s is irreducible with a node then J X s ( q s ) has trivial dualizing sheaf by [Ari11,Cor. 9]. Therefore, the fibers of the proper map u − ( U ) → U have trivial canonical sheaf. It follows that u ∗ ( ω u ) | U is a line bundle on U and that the natural evaluation morphism u ∗ u ∗ ( ω u ) | u − ( U ) → ( ω u ) | u − ( U ) is an isomorphism. Since Pic( U ) = 0, the line bundle u ∗ ( ω u ) | U is trivial, hence also ( ω u ) | u − ( U ) is trivial,q.e.d. he above Claim implies that ω u and O J X ( q ) agree on an open subset u − ( U ) ⊂ J X ( q ) whose com-plement has codimension at least two by Lemma 4.3. Since J X ( q ) is regular (hence S ) by Theorem 5.5,this implies that ω u = O J X ( q ) .The second assertion follows now by restricting the equality ω u = O J X ( q ) to the fiber J X ( q ) of u overthe closed point [ m X ] ∈ Spec R X . (cid:3) Note that a statement similar to Corollary 5.7 was proved by Arinkin in [Ari11, Cor. 9] for theuniversal compactified Jacobian over the moduli stack of integral curves with locally planar singularities.Finally, note that the universal fine compactified Jacobians are acted upon by the universal generalizedJacobian, whose properties are collected into the following
Fact 5.8 (Bosch-L¨utkebohmert-Raynaud) . There is an open subset of J X , called the universal generalizedJacobian of π : X →
Spec R X and denoted by v : J ( X ) → Spec R X , whose geometric fiber over anypoint s ∈ Spec R X is the generalized Jacobian J ( X s ) of the geometric fiber X s of π over s .The morphism v makes J ( X ) into a smooth and separated group scheme of finite type over Spec R X .Proof. The existence of a group scheme v : J ( X ) → Spec R X whose fibers are the generalized Jacobiansof the fibers of π : X →
Spec R X follows by [BLR90, Sec. 9.3, Thm. 7], which can be applied sinceSpec R X is a strictly henselian local scheme (because R X is a complete local ring) and the geometricfibers of π : X →
Spec R X are reduced and connected since X is assumed to be so. The result of loc.cit. gives also that the map v is smooth, separated and of finite type. (cid:3) X . The aim of this subsection is to study relative finecompactified Jacobians associated to a 1-parameter smoothing of a curve X and their relationship withthe N´eron models of the Jacobians of the generic fiber. As a corollary, we will get a combinatorial formulafor the number of irreducible components of a fine compactified Jacobian of a curve with locally planarsingularities.Let us start with the definition of 1-parameter regular smoothings of a curve X . Definition 5.9. A of X is a proper and flat morphism f : S → B =Spec R where R is a complete discrete valuation domain (DVR for short) with residue field k and quotientfield K and S is a regular scheme of dimension two, i.e. a regular surface, and such that the special fiber S k is isomorphic to X and the generic fiber S K is a K -smooth curve.The natural question one may ask is the following: which (reduced) curves X admit a 1-parameterregular smoothing? Of course, if X admits a 1-parameter regular smoothing f : S →
Spec R , then X is a divisor inside a regular surface S , which implies that X has locally planar singularities. Indeed, itis well known to the experts that this necessary condition turns out to be also sufficient. We include aproof here since we couldn’t find a suitable reference. Proposition 5.10.
A (reduced) curve X admits a 1-parameter regular smoothing if and only if X has locally planar singularities. More precisely, if X has locally planar singularities then there exists acomplete discrete valuation domain R (and indeed we can take R = k [[ t ]] ) and a morphism α : Spec R → Spec R X such that the pull-back (5.7) S / / f (cid:15) (cid:15) (cid:3) X π (cid:15) (cid:15) Spec R α / / Spec R X is a 1-parameter regular smoothing of X .Proof. We have already observed that the only if condition is trivially satisfied. Conversely, assume that X has locally planar singularities, and let us prove that X admits a 1-parameter regular smoothing.Consider the natural morphisms of deformation functors F : h R X → Def X → Def loc X = Y p ∈ X sing Def
X,p = Y p ∈ X Def
X,p , obtained by composing the morphism (4.1) with the morphism (4.5) and using the fact if p is a smoothpoint of X then Def X,p is the trivial deformation functor (see [Ser06, Thm. 1.2.4]). Observe that F is mooth because the first morphism is smooth since R X is a semiuniversal deformation ring for Def X andthe second morphism is smooth by Fact 4.1(iii).Given an element α ∈ h R X ( R ) = Hom(Spec R, Spec R X ) associated to a Cartesian diagram like in(5.7), the image of α into Def X,p ( R ) corresponds to the formal deformation of b O X,p given by the rightsquare of the following diagramSpec b O X,p / / (cid:15) (cid:15) (cid:3) Spf b O S ,p (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:3) Spec b O S ,p (cid:15) (cid:15) Spec k / / Spf R / / Spec R. Claim 1: The morphism f : S →
Spec R is a 1-parameter regular smoothing of X if and only if, forany p ∈ X , we have that(i) b O S ,p is regular;(ii) b O S ,p ⊗ R K is geometrically regular over K (i.e. b O S ,p ⊗ R K ′ is regular for any field extension K ⊆ K ′ ).Indeed, by definition, the surface S is regular if and only if the local ring O S ,q is regular for anyschematic point q ∈ S or, equivalently (see [Mat89, Thm. 19.3]), for any closed point q ∈ S . Clearly, theclosed points of S are exactly the closed points of its special fiber S k = X . Moreover the local ring O S,q is regular if and only if its completion b O S ,q is regular (see [Mat89, Thm. 21.1(i)]). Putting everythingtogether, we deduce that S is regular if and only if (i) is satisfied.Consider now the Spec R -morphism µ : ` p ∈ X Spec b O S ,p µ ′′ −→ ` p ∈ X Spec O S ,p µ ′ −→ S . The morphism µ is flat since any localization is flat (see [Mat89, Thm. 4.5]) and any completion is flat (see [Mat89, Thm.8.8]). Moreover, the image of µ is open because µ is flat and it contains the special fiber S k = X ⊂ S which contains all the closed points of S ; therefore, µ must be surjective, which implies that µ isfaithfully flat. Finally, µ has geometrically regular fibers (hence it is regular, i.e. flat with geometricallyregular fibers, see [Mat80, (33.A)]): this is obvious for µ ′ (because it is the disjoint union of localizationmorphisms), it is true for µ ′′ because each local ring O S ,p is a G-ring (in the sense of [Mat80, § µ to the generic point Spec K of Spec R , we get a morphism µ K : ` p ∈ X Spec ( b O S ,p ⊗ R K ) → S K which is also faithfully flat and regular, because both propertiesare stable under base change. Therefore, by applying [Mat80, (33.B), Lemma 1], we deduce that S K isgeometrically regular over K if and only if b O S ,p ⊗ R K is geometrically regular over K for any p ∈ X .Hence, S K is K -smooth (which is equivalent to the fact that S K is geometrically regular over K , because S K is of finite type over K by assumption) if and only if (ii) is satisfied, q.e.d.Suppose now that for any p ∈ X we can find an element of Def X,p ( R ) corresponding to a formaldeformation(5.8) Spec b O X,p / / (cid:15) (cid:15) (cid:3) Spf A (cid:15) (cid:15) / / (cid:3) Spec A (cid:15) (cid:15) Spec k / / Spf R / / Spec R such that A is a regular complete local ring, R → A is a local flat morphism and A ⊗ R K is K -formallysmooth. Then, using the smoothness of F , we can lift this element to an element α ∈ h R X ( R ) whoseassociated Cartesian diagram (5.7) gives rise to a 1-parameter regular smoothing of X by the aboveClaim.Let us now check this local statement. Since X has locally planar (isolated) singularities, we can write b O X,p = k [[ x, y ]]( f ) , for some reduced element 0 = f = f ( x, y ) ∈ ( x, y ) ⊂ k [[ x, y ]].Claim 2: Up to replacing f with f H for some invertible element H ∈ k [[ x, y ]], we can assume that(5.9) ∂ x f and ∂ y f do not have common irreducible factors , where ∂ x is the formal partial derivative with respect to x and similarly for ∂ y . ore precisely, we will show that there exists a, b ∈ k with the property that e f := (1 + ax + by ) f satisfies the conclusion of the Claim, i.e. ∂ x e f and ∂ y e f do not have common irreducible factors; it willthen follow that the same is true for a generic point ( a, b ) ∈ A ( k ). By contradiction, assume that(*) (1 + ax + by ) ∂ x f + af and (1 + ax + by ) ∂ y f + bf have a common irreducible factor for every a, b ∈ k. Observe that ( ∂ x f, ∂ y f ) = (0 , f would be a p -power in k [[ x, y ]] with p = char( k ) > f is reduced. So we can assume, without loss of generality that ∂ y f = 0.We now specialize condition (*) by putting b = 0 and using that 1 + ax is invertible in k [[ x, y ]], in orderto get that(**) (1 + ax ) ∂ x f + af and ∂ y f have a common irreducible factor for every a ∈ k. Since k is an infinite field (being algebraically closed) and 0 = ∂ y f has, of course, a finite number ofirreducible factors, we infer from (**) that there exists an irreducible factor q ∈ k [[ x, y ]] of ∂ y f such that q is also an irreducible factor of (1 + ax ) ∂ x f + af = ∂ x f + a ( x∂ x f + f ) for infinitely many a ∈ k . Thishowever can happen (if and) only if q divides f and ∂ x f . This implies that the hypersurface { f = 0 } issingular along the entire irreducible component { q = 0 } , which contradicts the hypothesis that { f = 0 } has an isolated singularities in (0 , f satisfies the conditions of (5.9). Let R := k [[ t ]] and consider thelocal complete k [[ t ]]-algebra A := k [[ x, y, t ]]( f − t ) . The k [[ x, y ]]-algebra homomorphism (well-defined since f ∈ ( x, y ))(5.10) A = k [[ x, y, t ]]( f − t ) −→ k [[ x, y ]] t f is clearly an isomorphism. Therefore, A is a regular local ring. Moreover, since f is not a zero-divisorin k [[ x, y ]], the algebra A is flat over k [[ t ]]. From now on, we will use the isomorphism (5.10) to freelyidentify A with k [[ x, y ]] seen as a k [[ t ]]-algebra via the map sending t into f .It remains to show that A ⊗ k [[ t ]] k (( t )) is geometrically regular over k (( t )). Since A ⊗ k [[ t ]] k (( t )) is thelocalization of A at the multiplicative system generated by ( t ), we have to check (by [Mat80, Def. in(33.A) and Prop. in (28.N)])) that, for any ideal m in the fiber of A over the generic point of k [[ t ]] (i.e.such that m ∩ k [[ t ]] = (0)), the local ring A m is formally smooth over k (( t )) for the m -adic topology on A m and the discrete topology on k (( t )) (see [Mat80, (28.C)] for the definition of formal smoothness). Sinceformal smoothness is preserved under localization (as it follows easily from [Mat80, (28.E) and (28.F)]),it is enough to prove that A m is formally smooth over k (( t )) for any closed point m of A ⊗ k [[ t ]] k (( t )).The closed points of A ⊗ k [[ t ]] k (( t )) correspond exactly to those prime ideals of A ∼ = k [[ x, y ]] of the form m = ( g ) for some irreducible element g ∈ k [[ x, y ]] that is not a factor of f . Indeed, any such ideal m of k [[ x, y ]] must be of height one and hence it must be principal (since k [[ x, y ]] is regular, see [Mat89, Thm.20.1, Thm. 20.3]), i.e. m = ( g ) for some g irreducible element of k [[ x, y ]]. Furthermore, the condition( g ) ∩ k [[ t ]] = (0) is satisfied if and only if g is not an irreducible factor of f . Therefore, we are left withproving the followingClaim 3: k [[ x, y ]] ( g ) is formally smooth over k (( t )) for any irreducible g ∈ k [[ x, y ]] that is not anirreducible factor of f .Observe first of all that k [[ x, y ]] ( g ) is formally smooth over k because k [[ x, y ]] is formally smooth over k (see [Mat80, (28.D), Example 3]) and formal smoothness is preserved by localization as observed before.Therefore, k [[ x, y ]] ( g ) is regular (see [Mat80, Thm. 61]).Consider now the residue field L = k [[ x, y ]] ( g ) / ( g ) of the local ring k [[ x, y ]] ( g ) , which is a field extensionof k (( t )). If L is a separable extension of k (( t )) (which is always the case if char( k ) = 0) then k [[ x, y ]] ( g ) is formally smooth over k (( t )) by [Mat80, (28.M)]. In the general case, using [Mat89, Thm. 66], theClaim is equivalent to the injectivity of the natural L -linear map(5.11) α : Ω k (( t )) /k ⊗ k (( t )) L → Ω k [[ x,y ]] ( g ) /k ⊗ k [[ x,y ]] ( g ) L = Ω k [[ x,y ]] /k ⊗ k [[ x,y ]] L, where in the second equality we have used that the K¨ahler differentials commute with the localizationtogether with the base change for the tensor product. Therefore, to conclude the proof of the Claim,it is enough to prove the injectivity of the map (5.11) if char( k ) = p >
0. Under this assumption (andrecalling that k is assumed to be algebraically closed), we have clearly that k (( t )) p = k (( t p )) ⊂ k (( t )), rom which it follows that t is a p-basis of k (( t )) /k in the sense of [Mat89, § k (( t )) /k is the k (( t ))-vector space of dimension one generated by dt . Hence, the injectivity of the above L -linear map α translates into α ( dt ⊗ = 0. Since the naturalmap k [[ t ]] → k [[ x, y ]] sends t into f , we can compute(5.12) α ( dt ⊗
1) = d ( f ) ⊗ ∂ x f dx + ∂ y f dy ) ⊗ ∈ Ω k [[ x,y ]] /k ⊗ k [[ x,y ]] L. Now observe that the map k [[ x, y ]] → k [[ x, y ]] ( g ) → L = k [[ x, y ]] ( g ) / ( g ) is also equal to the composition k [[ x, y ]] → k [[ x, y ]] / ( g ) → Quot( k [[ x, y ]] / ( g )) ∼ = L , where Quot denotes the quotient field. Moreover,since dx and dy generate a free rank-2 submodule of the k [[ x, y ]]-module Ω k [[ x,y ]] , the right hand side of(5.12) is zero if and only if g divides both ∂ x f and ∂ y f . Since this does not happen for our choice of f (see Claim 1), the proof is complete. (cid:3) From now till the end of this subsection, we fix a 1-parameter regular smoothing f : S → B = Spec R of X as in Proposition 5.10. Let P ic f denote the relative Picard functor of f (often denoted P ic S /B inthe literature, see [BLR90, Chap. 8] for the general theory) and let P ic df be the subfunctor parametrizingline bundles of relative degree d ∈ Z . The functor P ic f (resp. P ic df ) is represented by a scheme Pic f (resp. Pic df ) locally of finite presentation over B (see [BLR90, Sec. 8.2, Thm. 2]) and formally smoothover B (by [BLR90, Sec. 8.4, Prop. 2]). The generic fiber of Pic f (resp. Pic df ) is isomorphic to Pic( S K )(resp. Pic d ( S K )).Note that Pic df is not separated over B whenever X is reducible. The biggest separated quotient ofPic df coincides with the N´eron model N (Pic d S K ) of Pic d ( S K ), as proved by Raynaud in [Ray70, Sec.8] (see also [BLR90, Sec. 9.5, Thm. 4]). Recall that N (Pic d S K ) is smooth and separated over B ,the generic fiber N (Pic d S K ) K is isomorphic to Pic d S K and N (Pic d S K ) is uniquely characterized bythe following universal property (the N´eron mapping property, cf. [BLR90, Sec. 1.2, Def. 1]): every K -morphism q K : Z K → N (Pic d S K ) K = Pic d S K defined on the generic fiber of some scheme Z smoothover B admits a unique extension to a B -morphism q : Z → N (Pic d S K ). Moreover, N (Pic S K ) is a B -group scheme while, for every d ∈ Z , N (Pic d S K ) is a torsor under N (Pic S K ).Fix now a general polarization q on X and consider the Cartesian diagram(5.13) J f ( q ) / / (cid:15) (cid:15) (cid:3) J X ( q ) u (cid:15) (cid:15) B = Spec R α / / Spec R X We call the scheme J f ( q ) the f -relative fine compactified Jacobian with respect to the polarization q .Similarly, by replacing J X ( q ) with J X ( q ) in the above diagram, we define the open subset J f ( q ) ⊂ J f ( q ).Note that the generic fibers of J f ( q ) and J f ( q ) coincide and are equal to J f ( q ) K = J f ( q ) K = Pic d ( S K )with d := | q | + p a ( X ) −
1, while their special fibers are equal to J f ( q ) k = J X ( q ) and J f ( q ) k = J X ( q ),respectively. From Theorem 5.5, we get that the morphism J f ( q ) → B is flat and its smooth locus is J f ( q ). Therefore, the universal property of the N´eron model N (Pic d S K ) gives a unique B -morphism q f : J f ( q ) → N (Pic d S K ) which is the identity on the generic fiber. Indeed, J. L. Kass proved in [Kas09,Thm. A] that the above map is an isomorphism. Fact 5.11 (Kass) . For a 1-paramater regular smoothing f : X → B = Spec R as above and any generalpolarization q on X , the natural B -morphism q f : J f ( q ) → N (Pic | q | + p a ( X ) − S K ) is an isomorphism. From the above isomorphism, we can deduce a formula for the number of irreducible components of J X ( q ). We first need to recall the description due to Raynaud of the group of connected components ofthe N´eron model N (Pic ( S K )) (see [BLR90, Sec. 9.6] for a detailed exposition).Denote by C , . . . , C γ the irreducible components of X . A multidegree on X is an ordered γ -tuple ofintegers d = ( d C , . . . , d C γ ) ∈ Z γ . e denote by | d | = P γi =1 d C i the total degree of d . We now define an equivalence relation on theset of multidegrees on X . For every irreducible component C i of X , consider the multidegree C i =(( C i ) , . . . , ( C i ) γ ) of total degree 0 defined by( C i ) j = | C i ∩ C j | if i = j, − X k = i | C i ∩ C k | if i = j, where | C i ∩ C j | denotes the length of the scheme-theoretic intersection of C i and C j . Clearly, if we takea 1-parameter regular smoothing f : S → B of X as in Proposition 5.10, then | C i ∩ C j | is also equal tothe intersection product of the two divisors C i and C j inside the regular surface S .Denote by Λ X ⊆ Z γ the subgroup of Z γ generated by the multidegrees C i for i = 1 , . . . , γ . It is easyto see that P i C i = 0 and this is the only relation among the multidegrees C i . Therefore, Λ X is a freeabelian group of rank γ − Definition 5.12.
Two multidegrees d and d ′ are said to be equivalent , and we write d ≡ d ′ , if d − d ′ ∈ Λ X .In particular, if d ≡ d ′ then | d | = | d ′ | .For every d ∈ Z , we denote by ∆ dX the set of equivalence classes of multidegrees of total degree d = | d | .Clearly ∆ X is a finite group under component-wise addition of multidegrees (called the degree class group of X ) and each ∆ dX is a torsor under ∆ X . The cardinality of ∆ X is called the degree class number orthe complexity of X , and it is denoted by c ( X ).The name degree class group was first introduced by L. Caporaso in [Cap94, Sec. 4.1]. The namecomplexity comes from the fact that if X is a nodal curve then c ( X ) is the complexity of the dual graphΓ X of X , i.e. the number of spanning trees of Γ X (see e.g. [MV12, Sec. 2.2]). Fact 5.13 (Raynaud) . The group of connected component of the B -group scheme N (Pic S K ) is isomor-phic to ∆ X . In particular, the special fiber of N (Pic d S K ) for any d ∈ Z is isomorphic to the disjointunion of c ( X ) copies of the generalized Jacobian J ( X ) of X . For a proof, see the original paper of Raynaud [Ray70, Prop. 8.1.2] or [BLR90, Sec. 9.6].Finally, by combining Corollary 2.20, Fact 5.11 and Fact 5.13, we obtain a formula for the number ofirreducible components of a fine compactified Jacobian.
Corollary 5.14.
Assume that X has locally planar singularities and let q be any general polarizationon X . Then J X ( q ) has c ( X ) irreducible components. The above Corollary was obtained for nodal curves X by S. Busonero in his PhD thesis (unpublished)in a combinatorial way; a slight variation of his proof is given in [MV12, Sec. 3].Using the above Corollary, we can now prove the converse of Lemma 2.18(i) for curves with locallyplanar singularities, generalizing the result of [MV12, Prop. 7.3] for nodal curves. Lemma 5.15.
Assume that X has locally planar singularities. For a polarization q on X , the followingconditions are equivalent(i) q is general.(ii) Every rank-1 torsion-free sheaf which is q -semistable is also q -stable.(iii) Every simple rank-1 torsion-free sheaf which is q -semistable is also q -stable.(iv) Every line bundle which is q -semistable is also q -stable.Proof. (i) ⇒ (ii) follows from Lemma 2.18(i).(ii) ⇒ (iii) ⇒ (iv) are trivial.(iv) ⇒ (i): by contradiction, assume that q is not general. Then, by Remark 2.10, we can find aproper subcurve Y ⊆ X with Y and Y c connected and such that q Y , q Y c ∈ Z . This implies that we candefine a polarization q | Y on the connected curve Y by setting ( q | Y ) Z := q Z for any subcurve Z ⊆ Y .And similarly, we can define a polarization q | Y c on Y c .Claim 1: There exists a line bundle L such that L | Y is q | Y -semistable and L | Y c is q | Y c -semistable.Clearly, it is enough to show the existence of a line bundle L (resp. L ) on Y (resp. on Y c ) thatis q | Y -semistable (resp. q | Y c -semistable); the line bundle L that satisfies the conclusion of the Claim isthen any line bundle such that L | Y = L and L | Y c = L (clearly such a line bundle exists!). Let us provethe existence of L on Y ; the argument for L being the same. We can deform slightly the polarization | Y on Y in order to obtain a general polarization e q on Y . Corollary 5.14 implies that J Y ( e q ) is nonempty (since c ( Y ) ≥ J Y ( e q ) is non empty by Corollary 2.20(ii). In particular, we can finda line bundle L on Y that is e q -semistable. Remark 2.13 implies that L is also q | Y -semistable, and theClaim is proved.Observe that the line bundle L is not q -stable since χ ( L Y ) = χ ( L | Y ) = | q | Y | = q Y and similarly for Y c . Therefore, we find the desired contradiction by proving the followingClaim 2: The line bundle L is q -semistable.Let Z be a subcurve of X and set Z := Z ∩ Y , Z := Z ∩ Y c , W := Y \ Z and W := Y c \ Z .Tensoring with L the exact sequence0 → O Z → O Z ⊕ O Z → O Z ∩ Z → , and taking the Euler-Poincar´e characteristic we get χ ( L | Z ) = χ ( L | Z ) + χ ( L | Z ) − | Z ∩ Z | . Combining the above equality with the fact that L | Y (resp. L | Y c ) is q | Y -semistable (resp. q | Y c -semistable)by Claim 1 and using Remark 2.14, we compute(5.14) χ ( L | Z ) = χ ( L | Z ) + χ ( L | Z ) − | Z ∩ Z | ≤ q Z + | Z ∩ W | + q Z + | Z ∩ W | − | Z ∩ Z | ≤≤ q Z + | Z ∩ W | + | Z ∩ W | . Since X has locally planar singularity, we can embed X inside a smooth projective surface S (see 1.6).In this way, the number | Z i ∩ W i | is equal to the intersection number Z i · W i of the divisors Z i and W i inside the smooth projective surface S . Using that the intersection product of divisors in S is bilinear,we get that(5.15) | Z ∩ Z c | = Z · Z c = ( Z + Z ) · ( W + W ) ≥ Z · W + Z · W = | Z ∩ W | + | Z ∩ W | , where we used that Z · W ≥ Z and W do not have common components and similarly Z · W ≥
0. Substituting (5.15) into (5.14), we find that χ ( L | Z ) ≤ q Z + | Z ∩ Z c | , for every subcurve Z ⊆ X , which implies that L is q -semistable by Remark 2.14. (cid:3) It would be interesting to know if the above Lemma 5.15 holds true for every (reduced) curve X .6. Abel maps
The aim of this section is to define Abel maps for singular (reduced) curves. Although in the followingsections we will only use Abel maps for curves with locally planar singularities, the results of this sectionare valid for a broader class of singular curves, namely those for which every separating point is a node(see Condition (6.3)), which includes for example all Gorenstein curves.6.1.
Abel maps without separating points.
The aim of this subsection is to define the Abel mapsfor a reduced curve X without separating points (in the sense of 1.8).For every (geometric) point p on the curve X , its sheaf of ideals m p is a torsion-free rank-1 sheaf ofdegree − X . Also, if p is not a separating point of X , then m p is simple (see [Est01, Example 38]).Therefore, if X does not have separating points (which is clearly equivalent to the fact that δ Y ≥ Y of X ) then m p is torsion-free rank-1 and simple for any p ∈ X .Let I ∆ be the ideal of the diagonal ∆ of X × X . For any line bundle L ∈ Pic( X ), consider the sheaf I ∆ ⊗ p ∗ L, where p : X × X → X denotes the projection onto the first factor. The sheaf I ∆ ⊗ p ∗ L isflat over X (with respect to the second projection p : X × X → X ) and for any point p of X it holds I ∆ ⊗ p ∗ L | X ×{ p } = m p ⊗ L. Therefore, if X does not have separating points, then I ∆ ⊗ p ∗ L ∈ J ∗ X ( X ) where J ∗ X is the functor definedby (2.1). Using the universal property of J X (see Fact 2.2(iii)), the sheaf I ∆ ⊗ p ∗ L induces a morphism(6.1) A L : X → J X p m p ⊗ L. We call the map A L the (L-twisted) Abel map of X . rom the definition (2.4), it follows that a priori the Abel map A L takes values in the big compactifiedJacobian J χ ( L ) − X . Under the assumption that X is Gorenstein, we can prove that the Abel map A L takes always values in a fine compactified Jacobian contained in J χ ( L ) − X . Lemma 6.1.
Assume that X is Gorenstein. Then for every L ∈ Pic( X ) there exists a general polariza-tion q on X of total degree | q | = χ ( L ) − such that Im( A L ) ⊆ J X ( q ) .Proof. Consider the polarization q ′ on X defined by setting, for every irreducible component C i of X , q ′ C i = deg C i L − deg C i ( ω X )2 − γ ( X ) , where γ ( X ) denotes, as usual, the number of irreducible components of X . Note that for any subcurve Y ⊆ X we have that q ′ Y = deg Y ( L ) − deg Y ( ω X )2 − γ ( Y ) γ ( X ) and in particular | q ′ | = deg L − deg ω X − χ ( L ) − A L is q ′ -stable. Indeed, for any proper subcurve ∅ 6 = Y ( X and any point p ∈ X , we have that(6.2)deg Y ( m p ⊗ L ) − q ′ Y − deg Y ( ω X )2 = deg Y ( L ) + deg Y ( m p ) − deg Y ( L ) + γ ( Y ) γ ( X ) ≥ − γ ( Y ) γ ( X ) > − ≥ − δ Y , where we have used that γ ( Y ) > Y is not the empty subcurve and that δ Y ≥ X does nothave separating points by assumption. Therefore, A L ( p ) is q ′ -stable for every p ∈ X by Remark 2.15.Using Remark 2.13, we can deform slightly q ′ in order to obtain a general polarization q with | q | = | q ′ | and for which A L ( p ) is q -stable for every p ∈ X , which implies that Im A L ⊆ J X ( q ), q.e.d. (cid:3) Those fine compactified Jacobians for which there exists an Abel map as in the above Lemma 6.1 arequite special, hence they deserve a special name.
Definition 6.2.
Let X be a curve without separating points. A fine compactified Jacobian J X ( q ) issaid to admit an Abel map if there exists a line bundle L ∈ Pic( X ) (necessarily of degree | q | + p a ( X ))such that Im A L ⊆ J X ( q ).Observe that clearly the property of admitting an Abel map is invariant under equivalence by trans-lation (in the sense of Definition 3.1). Remark . It is possible to prove that a curve X without separating points and having at most twoirreducible components is such that any fine compactified Jacobian of X admits an Abel map.However, in Section 7 we are going to show examples of curves with more than two componentsand having a fine compactified Jacobian which does not admit an Abel map (see Proposition 7.4 andProposition 7.5). In particular, Proposition 7.4 shows that, as the number of irreducible components of X increases, fine compactified Jacobians of X that admit an Abel map became more and more rare.6.2. Abel maps with separating points.
The aim of this subsection is to define Abel maps for(reduced) curves X having separating points (in the sense of (1.8)) but satisfying the following(6.3) Condition ( † ) : Every separating point is a node.Indeed, there are plenty of curves that satisfy condition ( † ) due to the following result of Catanese (see[Cat82, Prop. 1.10]). Fact 6.4 (Catanese) . A (reduced) Gorenstein curve X satisfies condition ( † ) . Let us give an example of a curve that does not satisfy condition ( † ). Example . Consider a curve X made of three irreducible smooth components Y , Y and Y glued atone point p ∈ Y ∩ Y ∩ Y with linearly independent tangent directions, i.e. in a such a way that, locallyat p , the three components Y , Y and Y look like the three coordinate axes in A . A straightforwardlocal computation gives that δ Y = | Y ∩ ( Y ∪ Y ) | = 1, so that p is a separating point of X (in thesense of 1.8). However p is clearly not a node of X . Combined with Fact 6.4, this shows that X is notGorenstein at p ∈ X .Throughout this section, we fix a connected (reduced) curve X satisfying condition ( † ) and let { n , . . ., n r − } be its separating points. Since X satisfies condition ( † ), each n i is a node. Denote by e X thepartial normalization of X at the set { n , . . . , n r − } . Since each n i is a node, the curve e X is a disjoint nion of r connected reduced curves { Y , . . . , Y r } such that each Y i does not have separating points.Note also that X has locally planar singularities if and only if each Y i has locally planar singularities.We have a natural morphism(6.4) τ : e X = a i Y i → X. We can naturally identify each Y i with a subcurve of X in such a way that their union is X and thatthey do not have common irreducible components. In particular, the irreducible components of X arethe union of the irreducible components of the curves Y i . We call the subcurves Y i (or their image in X )the separating blocks of X .Let us first show how the study of fine compactified Jacobians of X can be reduced to the studyof fine compactified Jacobians of Y i . Observe that, given a polarization q i on each curve Y i , we get apolarization q on X such that for every irreducible component C of X we have(6.5) q C = ( q iC if C ⊆ Y i and C ∩ Y j = ∅ for all j = i,q iC − if C ⊆ Y i and C ∩ Y j = ∅ for some j = i. Note that | q | = P i | q i | + 1 − r so that indeed q is a polarization on X . We say that q is the polarization induced by the polarizations ( q , . . . , q r ) and we write q := ( q , . . . , q r ). Proposition 6.6.
Let X be a connected curve satisfying condition ( † ) .(i) The pull-back map τ ∗ : J X −→ r Y i =1 J Y i I ( I | Y , . . . , I | Y r ) , is an isomorphism. Moreover τ ∗ ( J X ) = Q i J Y i .(ii) Given a polarization q i on each curve Y i , consider the induced polarization q := ( q , . . . , q r ) on X as above. Then q is general if and only if each q i is general and in this case the morphism τ ∗ induces an isomorphism (6.6) τ ∗ : J X ( q ) ∼ = −→ Y i J Y i ( q i ) . (iii) If q is a general polarization on X then there exists a general polarization q ′ with | q ′ | = | q | on X which is induced by some polarizations q i on Y i and such that J X ( q ) = J X ( q ′ ) . Proof.
It is enough, by re-iterating the argument, to consider the case where there is only one separatingpoint n = n , i.e. r = 2. Therefore the normalization e X of X at n is the disjoint union of two connectedcurves Y and Y , which we also identify with two subcurves of X meeting at the node n . Denote by C (resp. C ) the irreducible component of Y (resp. Y ) that contains the separating point n . A warningabout the notation: given a subcurve Z ⊂ X , we will denote by Z c the complementary subcurve of Z inside X , i.e. X \ Z . In the case where Z ⊂ Y i ⊂ X for some i = 1 , Y i \ Z for thecomplementary subcurve of Z inside Y i .Part (i) is well-known, see [Est01, Example 37] and [Est09, Prop. 3.2]. The crucial fact is that if I is simple then I must be locally free at the separating point n ; hence τ ∗ ( I ) is still torsion-free, rank-1and its restrictions τ ∗ ( I ) | Y i = I | Y i are torsion-free, rank-1 and simple. Moreover, since n is a separatingpoint, the sheaf I is completely determined by its pull-back τ ∗ ( I ), i.e. there are no gluing conditions.Finally, I is a line bundle if and only if its pull-back τ ∗ ( I ) is a line bundle.Part (ii). Assume first that each q i is a general polarization on Y i for i = 1 ,
2. Consider a propersubcurve Z ⊂ X such that Z and Z c are connected. There are three possibilities:(6.7) Case I: C , C ⊂ Z c = ⇒ Z ⊂ Y i and Y i \ Z is connected (for some i = 1 , , Case II: C , C ⊂ Z = ⇒ Z c ⊂ Y i and Y i \ Z c is connected (for some i = 1 , , Case III: C i ⊂ Z and C − i ⊂ Z c = ⇒ Z = Y i and Z c = Y − i (for some i = 1 , . herefore, from the definition of q = ( q , q ), it follows that(6.8) q Z = q iZ in case I, | q | − q Z c = | q | − q iZ c in case II, | q i | −
12 in case III.In each of the cases I, II, III we conclude that q Z Z using that q i is general and that | q | , | q i | ∈ Z .Therefore q is general by Remark 2.10.Conversely, assume that q is general and let us show that q i is general for i = 1 ,
2. Consider a propersubcurve Z ⊂ Y i such that Z and Y i \ Z is connected. There are two possibilities:(6.9) ( Case A: C i Z = ⇒ Z c is connected,Case B: C i ⊂ Z = ⇒ ( Y i \ Z ) c is connected.Using the definition of q = ( q , q ), we compute(6.10) q iZ = ( q Z in case A, | q i | − q iY i \ Z = | q i | − q Y i \ Z in case B.In each of the cases A, B we conclude that q iZ Z using that q is general and | q i | ∈ Z . Therefore q i isgeneral by Remark 2.10.Finally, in order to prove (6.6), it is enough, using part (i), to show that a simple torsion-free rank-1sheaf I on X is q -semistable if and only if I | Y i is q i -semistable for i = 1 ,
2. Observe first that, since I islocally free at the node n (see the proof of part (i)), we have that for any subcurve Z ⊂ X it holds(6.11) χ ( I Z ) = ( χ ( I Z ∩ Y i ) = χ ( I Z ) if Z ⊆ Y i for some i,χ ( I Z ∩ Y ) + χ ( I Z ∩ Y ) − I | Y i is q i -semistable for i = 1 ,
2. Using (6.11), we get χ ( I ) = χ ( I Y ) + χ ( I Y ) − | q | + | q | − | q | . Consider a proper subcurve Z ⊂ X such that Z and Z c are connected. Using (6.7), (6.8) and (6.11), wecompute χ ( I Z ) − q Z = χ ( I Z ) − q iZ = χ ( I Z ∩ Y i ) − q iZ ∩ Y i in case I, χ ( I Z ∩ Y ) + χ ( I Z ∩ Y ) − − | q | + q iZ c == χ ( I Y i \ Z c ) + χ ( I Y − i ) − | q | − | q | + q iZ c = χ ( I Y i \ Z c ) − q iY i \ Z c in case II, χ ( I Y i ) − | q i | + 12 = 12 in case III.In each of the cases I, II, III we conclude that χ ( I Z ) − q Z ≥ I | Y i is q i -semistable. Therefore I is q -semistable by Remark 2.12.Conversely, assume that I is q -semistable. Using (6.11), we get that(6.12) | q | + | q | = | q | + 1 = χ ( I ) + 1 = χ ( I Y ) + χ ( I Y ) . Since I is q -semistable, inequalities (2.8) applied to Y i for i = 1 , χ ( I Y i ) ≥ q Y i = | q i | − . Since χ ( I Y i ) and | q i | are integral numbers, from (6.13) we get that χ ( I Y i ) ≥ | q i | which combined with(6.12) gives that χ ( I Y i ) = | q i | . Consider now a subcurve Z ⊂ Y i (for some i = 1 ,
2) such that Z and Y i \ Z are connected. Since I is locally free at n , we have that ( I | Y i ) Z = I Z . Using (6.9) and (6.10), wecompute χ (( I | Y i ) Z ) − q iZ = χ ( I Z ) − q Z in case A, χ ( I Y i \ Z c ) − χ ( I Y − i ) + 1 − | q i | + q Y i \ Z == χ ( I Y i \ Z c ) − q Y i \ Z c in case B. n each of the cases A, B we conclude that χ (( I | Y i ) Z ) − q iZ ≥ I is q -semistable. Therefore I | Y i is q i -semistable by Remark 2.12.Part (iii): note that a polarization q ′ on X is induced by some polarizations q i on Y i if and only if q ′ Y i + ∈ Z for i = 1 ,
2. For a general polarization q on Y , we have that ( | q | = q Y + q Y ∈ Z ,q Y i Z . Therefore, we can find unique integral numbers m , m ∈ Z and a unique rational number r ∈ Q with − < r < such that(6.14) q Y = m + 12 + r,q Y = m − − r. Define now the polarization q ′ on X in such a way that for an irreducible component C of X , we havethat q ′ C := q C if C = C , C ,q C − r if C = C ,q C + r if C = C . In particular for any subcurve Z ⊂ X , the polarization q ′ is such that(6.15) q ′ Z := q Z if either C , C ⊂ Z or C , C ⊂ Z c ,q Z − r if C ⊂ Z and C ⊂ Z c ,q Z + r if C ⊂ Z and C ⊂ Z c . Specializing to the case Z = Y , Y and using (6.14), we get that(6.16) q ′ Y = q Y − r = m + 12 ,q ′ Y = q Y + r = m − , | q ′ | = q ′ Y + q ′ Y = m + m = q Y + q Y = | q | . As observed before, this implies that q ′ is induced by two (uniquely determined) polarizations q and q on Y and Y , respectively, and moreover that | q ′ | = | q | .Let us check that q ′ is general. Consider a proper subcurve Z ⊂ X such that Z and Z c are connected.Using (6.7), (6.15) and (6.16), we compute that(6.17) q ′ Z = q Z in case I and II, q ′ Y i = q Y i + ( − i r = m i + ( − i +1
12 in case III.In each of the above cases I, II, III we get that q ′ Z Z using that q is general and that m i ∈ Z . Therefore q ′ is general by Remark 2.10.Finally, in order to check that J X ( q ) = J X ( q ′ ) we must show that a simple rank-1 torsion-free sheaf I on X with χ ( I ) = | q | = | q ′ | is q -semistable if and only if it is q ′ -semistable. Using Remark 2.12, itis sufficient (and necessary) to check that for, any proper subcurve Z ⊂ X such that Z and Z c areconnected, I satisfies (2.8) with respect to q Z if and only if it satisfies (2.8) with respect to q ′ Z . If Z belongs to case I or II (according to the classification (6.7)), this is clear by (6.17). If Z belongs to caseIII, i.e. if Z = Y i for some i = 1 ,
2, then, using (6.14) together with the fact that − < r < and m i , χ ( I Y i ) ∈ Z , we get that χ ( I Y i ) ≥ q Y i = m i + ( − i +1 (cid:18)
12 + r (cid:19) ⇐⇒ ( χ ( I Y i ) ≥ m i + 1 if i = 1 ,χ ( I Y i ) ≥ m i if i = 2 . Similarly using (6.16), we get that χ ( I Y i ) ≥ q ′ Y i = m i + ( − i +1 ⇐⇒ ( χ ( I Y i ) ≥ m i + 1 if i = 1 ,χ ( I Y i ) ≥ m i if i = 2 . his shows that I satisfies (2.8) with respect to q Y if and only if it satisfies (2.8) with respect to q ′ Y ,which concludes our proof. (cid:3) We can now define the Abel maps for X . Proposition 6.7.
Let X be a connected curve satisfying condition ( † ) as above.(i) For any line bundle L ∈ Pic( X ) , there exists a unique morphism A L : X → J X such that for any i = 1 , . . . , r it holds:(a) the following diagram is commutative Y i A Li / / _(cid:127) (cid:15) (cid:15) J Y i X A L / / J X τ ∗ ∼ = / / Q j J Y j π i b b b b ❉❉❉❉❉❉❉❉ where π i denotes the projection onto the i -th factor, τ ∗ is the isomorphism of Proposition 6.6 (i) and A L i is the L i -twisted map of (6.4) for L i := L | Y i .(b) The composition Y i ֒ → X A L −−→ J X τ ∗ −→ ∼ = Y j J Y j Q j = i π j −−−−−→ Y j = i J Y j is a constant map.Explicitly, the morphism A L is given for p ∈ Y i (with ≤ i ≤ r ) by τ ∗ ( A L ( p )) = ( L ( − n i ) , . . . , L i − ( − n ii − ) , m p ⊗ L i , L i +1 ( − n ii +1 ) , . . . , L r ( − n ir )) where for any h = k we denote by n hk the unique separating node of X that belongs to Y k and suchthat Y k and Y h belong to the distinct connected components of the partial normalization of X at n hk (note that such a point n hk exists and it is a smooth point of Y k ).(ii) Let q i be a general polarization on Y i for any ≤ i ≤ r and denote by q the induced (general)polarization on X . Then A L ( X ) ⊂ J X ( q ) ⇔ A L i ( Y i ) ⊂ J Y i ( q i ) for any ≤ i ≤ r. Proof.
Part (i): assume that such a map A L exists and let us prove its uniqueness. From (ia) and (ib)it follows that the composition f A L : e X = a i Y i τ −→ X A L −−→ J X τ ∗ −→ Y i J Y i is such that for every 1 ≤ i ≤ r and every p ∈ Y i it holds(6.18) ( f A L ) | Y i ( p ) = ( M i , . . . , M ii − , A L i ( p ) , M ii +1 , . . . , M ir )for some elements M ij ∈ J Y j for j = i . Moreover, if we set τ − ( n k ) = { n k , n k } then we must have that(6.19) f A L ( n k ) = f A L ( n k ) for any 1 ≤ k ≤ r − . Claim: The unique elements M ij ∈ J Y j (for any i = j ) such that the map f A L in (6.18) satisfies theconditions in (6.19) are given by M ij = L j ( − n ij ), where n ij are as above.The claim clearly implies the uniqueness of the map f A L , hence the uniqueness of the map A L . More-over, the same claim also shows the existence of the map A L with the desired properties: it is enough todefine f A L via the formula (6.18) and notice that, since the conditions (6.19) are satisfied, then the map f A L descends to a map A L : X → J X .It remains therefore to prove the Claim. Choose a separating node n k of X with inverse image τ − ( n k ) = { n k , n k } and suppose that n k ∈ Y i and n k ∈ Y j . Clearly, we have that n k = n ji and n k = n ij by construction and it is easily checked that(*) n ik = n jk for any k = i, j. rom condition (6.19) applied to n k , we deduce that(**) M ji = m n k ⊗ L i = L i ( − n ji ) ,M ij = m n k ⊗ L j = L j ( − n ij ) ,M ik = M jk for any k = i, j. By combining (*) and (**), it is easily checked that the unique elements M ij that satisfy condition (6.19)for every separating node are given by M ij = L j ( − n ij ), q.e.d.Part (ii) follows easily from the diagram in (ia) and the isomorphism (6.6). (cid:3) We call the map A L of Proposition 6.7(i) the ( L -twisted) Abel map of X . We can extend Definition6.2 to the case of curves satisfying condition ( † ) from (6.4). Definition 6.8.
Let X be a curve satisfying condition ( † ). We say that a fine compactified Jacobian J X ( q ) of X admits an Abel map if there exists L ∈ Pic( X ) (necessarily of degree | q | + p a ( X )) such thatIm A L ⊆ J X ( q ).By combining Propositions 6.6 and 6.7, we can easily reduce the problem of the existence of an Abelmap for a fine compactified Jacobian of X to the analogous question on the separating blocks of X . Corollary 6.9.
Let X be a curve satisfying condition ( † ) with separating blocks Y , . . . , Y r .(i) Let q be a general polarization of X and assume (without loss of generality by Proposition 6.6 (iii) )that q is induced by some general polarizations q i on Y i . Then J X ( q ) admits an Abel map if andonly if each J Y i ( q i ) admits an Abel map.(ii) If X is Gorenstein, then for any L ∈ Pic( X ) there exists a general polarization q on X of totaldegree | q | = χ ( L ) − such that Im A L ⊆ J X ( q ) .Proof. Part (i) follows from Proposition 6.7(ii). Part (ii) follows from Proposition 6.7(ii) together withLemma 6.1. (cid:3)
When is the Abel map A L an embedding? The answer is provided by the following result, whoseproof is identical to the proof of [CCE08, Thm. 6.3]. Fact 6.10 (Caporaso-Coelho-Esteves) . Let X be a curve satisfying condition ( † ) and L ∈ Pic( X ) . TheAbel map A L is an embedding away from the rational separating blocks (which are isomorphic to P )while it contracts each rational separating block Y i ∼ = P into a seminormal point of A L ( X ) , i.e. anordinary singularity with linearly independent tangent directions. Examples: Locally planar curves of arithmetic genus X satisfiesthe condition ( † ) and therefore, using Proposition 6.6 and Proposition 6.7, we can reduce the study offine compactified Jacobians and Abel maps to the case where X does not have separating points (orequivalently separating nodes). Under this additional assumption, a classification is possible. Fact 7.1.
Let X be a (reduced) connected singular curve without separating points, with locally planarsingularities and p a ( X ) = 1 . Then X is one of the curves depicted in Figure 1, which are called Kodairacurves.Proof. Since X has non separating points and p a ( X ) = 1 then X has trivial canonical sheaf by [Est01,Example 39]. These curves were classified by Catanese in [Cat82, Prop. 1.18]. An inspection of theclassification in loc. cit. reveals that the only such singular curves that have locally planar singularitiesare the ones depicted in Figure 1, i.e. the Kodaira curves. (cid:3) Note that the curves in Figure 1 are exactly the reduced fibers appearing in the well-known Kodairaclassification of singular fibers of minimal elliptic fibrations (see [BPV84, Chap. V, Sec. 7]). Thisexplains why they are called Kodaira curves.Abel maps for Kodaira curves behave particularly well, due to the following result proved in [Est01,Example 39]. ype I Type II Type
III (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)
Type IV Type I n , n ≥ Figure 1.
Kodaira curves.
Fact 7.2 (Esteves) . Let X be a connected curve without separating points and such that p a ( X ) = 1 .Then for any L ∈ Pic( X ) the image A L ( X ) ⊆ J X of X via the L -twisted Abel map is equal to a finecompactified Jacobian J X ( q ) of X and A L induces an isomorphism A L : X ∼ = −→ A L ( X ) = J X ( q ) . From the above Fact 7.2, we deduce that, up to equivalence by translation (in the sense of Definition3.1), there is exactly one fine compactified Jacobian that admits an Abel map and this fine compactifiedJacobian is isomorphic to the curve itself. This last property is indeed true for any fine compactifiedJacobian of a Kodaira curve, as shown in the following
Proposition 7.3.
Let X be a Kodaira curve. Then every fine compactified Jacobian of X is isomorphicto X .Proof. Let J X ( q ) be a fine compactified Jacobian of X . By Proposition 5.10, we can find a 1-parameterregular smoothing f : S → B = Spec R of X (in the sense of Definition 5.9), where R is a completediscrete valuation ring with quotient field K . Note that the generic fiber S K of f is an elliptic curve.Following the notation of § f -relative fine compactified Jacobian π : J f ( q ) → B with respect to the polarization q . Recall that π is a projective and flat morphism whose generic fiberis Pic | q | ( S K ) and whose special fiber is J X ( q ). Using Theorem 5.5, it is easy to show that if we choosea generic 1-parameter smoothing f : S → B of X , then the surface J f ( q ) is regular. Moreover, Fact5.11 implies that the smooth locus J f ( q ) → B of π is isomorphic to the N´eron model of the generic fiberPic | q | ( S K ). Therefore, using the well-known relation between the N´eron model and the regular minimalmodel of the elliptic curve Pic | q | ( S K ) ∼ = S K over K (see [BLR90, Chap. 1.5, Prop. 1]), we deduce that π : J f ( q ) → B is the regular minimal model of Pic | q | ( S K ). In particular, π is a minimal elliptic fibrationwith reduced fibers and therefore, according to Kodaira’s classification (see see [BPV84, Chap. V, Sec.7]), the special fiber J X ( q ) of π must be a smooth elliptic curve or a Kodaira curve.According to Corollary 5.14, the number of irreducible components of J X ( q ) is equal to the complexity c ( X ) of X . However, it is very easy to see that for a Kodaira curve X the complexity number c ( X ) isequal to the number of irreducible components of X . Therefore if c ( X ) ≥
4, i.e. if X is of Type I n with n ≥
4, then necessarily J X ( q ) is of type I n , hence it is isomorphic to X .In the case n ≤
3, the required isomorphism J X ( q ) ∼ = X follows from the fact that the smooth locusof J X ( q ) is isomorphic to a disjoint union of torsors under Pic ( X ) (see Corollary 2.20) and thatPic ( X ) = ( G m if X is of Type I or I n ( n ≥ , G a if X is of Type II, III or IV. (cid:3)
Let us now classify the fine compactified Jacobians for a Kodaira curve X , up to equivalence bytranslation, and indicate which of them admits an Abel map. X is of Type I or Type II ince the curve X is irreducible, we have that the fine compactified Jacobians of X are of the form J dX for some d ∈ Z . Hence they are all equivalent by translation and each of them admits an Abel map. X is of Type I n , with n ≥ X up to equivalence by translation and their behavior with respectto the Abel map are described in the following proposition. Proposition 7.4.
Let X be a Kodaira curve of type I n (with n ≥ ) and let { C , . . . , C n } be theirreducible components of X , ordered in such a way that, for any ≤ i ≤ n , C i intersects C i − and C i +1 ,with the cyclic convention that C n +1 := C .(i) Any fine compactified Jacobian is equivalent by translation to a unique fine compactified Jacobianof the form J X ( q ) for a polarization q that satisfies (*) q = q , . . . , q n − , − n − X i =1 q i ! with ≤ q i < , (**) s X i = r q i Z for any ≤ r ≤ s ≤ n − , (***) q i = k i n with k i = 1 , . . . , n − , for any ≤ i ≤ n − . In particular, there are exactly ( n − fine compactified Jacobians of X up to equivalence bytranslation.(ii) The unique fine compactified Jacobian, up to equivalence by translation, that admits an Abel mapis J X (cid:18) n − n , . . . , n − n , − ( n − n (cid:19) . Proof.
Part (i): given any polarization q ′ , there exists a unique polarization q that satisfies conditions(*) and such that q − q ′ ∈ Z n . Since any connected subcurve Y ⊂ X is such that Y or Y c is equal to C r ∪ . . . ∪ C s (for some 1 ≤ r ≤ s ≤ n − q that satisfies (*) is general ifand only if it satisfies (**). Hence any fine compactified Jacobian is equivalent by translation to a unique J X ( q ), for a polarization q that satisfies (*) and (**). Consider now the arrangement of hyperplanes in R n − given by ( s X i = r q i = n ) for all 1 ≤ r ≤ s ≤ n − n ∈ Z . This arrangement of hyperplanes cuts the hypercube [0 , n − into finitely many chambers. Notice that a polarization q satisfies (*) and (**) if and only if it belongs tothe interior of one of these chambers. Arguing as in the proof of Proposition 3.2, it is easy to see that twopolarizations q and q ′ satisfying (*) and (**) belong to the same chamber if and only if J X ( q ) = J X ( q ′ ).Now it is an entertaining exercise (that we leave to the reader) to check that any chamber containsexactly one polarization q that satisfies (***). This proves the first claim of part (i). The second claimin part (i) is an easy counting argument that we again leave to the reader.Part (ii): if we take a line bundle L of multidegree deg L = (1 , . . . , , − ( n − A L ⊆ J := J X ( n − n , . . . , n − n , − ( n − n ). Therefore, from Fact 7.2, it followsthat J is the unique fine compactified Jacobian, up to equivalence by translation, that admits an Abelmap. (cid:3) X is of Type III
Since X has two irreducible components, then every fine compactified Jacobian of X admits an Abelmap by Remark 6.3. By Fact 7.2, all fine compactified Jacobians of X are therefore equivalent bytranslation. X is of Type IV The fine compactified Jacobians of X up to equivalence by translation and their behavior with respectto the Abel map are described in the following proposition. roposition 7.5. Let X be a Kodaira curve of type IV .(i) Any fine compactified Jacobian of X is equivalent by translation to either J := J X (cid:0) , , − (cid:1) or J := J X (cid:0) , , − (cid:1) .(ii) J admits an Abel map while J does not admit an Abel map.Proof. Part (i) is proved exactly as in the case of the Kodaira curve of type I (see Proposition 7.4(i)).Part (ii): if we take a line bundle L of multidegree deg L = (1 , , −
1) then Im A L ⊆ J as follows fromthe proof of Lemma 6.1. Therefore J admits an Abel map.Let us now show that J does not admit an Abel map. Suppose by contradiction that there exists a linebundle L of multidegree deg L = ( d , d , d ) such that A L ( p ) = m p ⊗ L ∈ J where p denotes the uniquesingular point of X . The stability of m p ⊗ L with respect to the polarization ( q , q , q ) = ( , , − )gives for any irreducible component C i of X : d i = d i − C i ( m p ⊗ L ) + 1 = χ (( m p ⊗ L ) C i ) > q i . We deduce that d ≥ d ≥ d ≥
0. However if Im A L ⊂ J then the total degree of L must beone, which contradicts the previous conditions. (cid:3) Remark . Realize a Kodaira curve X of Type IV as the plane cubic with equation y ( x + y )( x − y ) = 0.One can show that the singular points in J and in J correspond to two sheaves that are not locallyisomorphic: the singular point of J is the sheaf I := m p ⊗ L where m p is the ideal sheaf of the point p defined by ( x, y ) and L is any line bundle on X of multidegree (1 , , − J is thesheaf I := I Z ⊗ M where I Z is the ideal sheaf of the length 2 subscheme defined by ( x, y ) and M isany line bundle on X of multidegree (1 , , e X the seminormalization of X (explicitly e X can be realized as the union of threelines in projective space meeting in one point with linearly independent directions) and π : e X → X is thenatural map. Using Table 2 of [Kas12] (where the unique singularity of e X is called e D and the uniquesingular point of X is classically called D ), it can be shown that, up to the tensorization with a suitableline bundle on X , I is the pushforward of the trivial line bundle on e X while I is the pushforward ofthe canonical sheaf on e X , which is not a line bundle since e X is not Gorenstein (see Example 6.5). Remark . Simpson (possibly coarse) compactified Jacobians of Kodaira curves have been studied byA. C. L´opez Mart´ın in [LM05, Sec. 5], see also [LM06], [LRST09].
Acknowledgments.
We are extremely grateful to E. Esteves for several useful conversations on finecompactified Jacobians and for generously answering several mathematical questions. We thank E.Sernesi for useful conversations on deformation theory of curves with locally planar singularities and E.Markman for asking about the embedded dimension of compactified Jacobians. We are very grateful tothe referees for their very careful reading of the paper and for the many suggestions and questions thathelped in improving a lot the presentation.This project started while the first author was visiting the Mathematics Department of the Universityof Roma “Tor Vergata” funded by the “Michele Cuozzo” 2009 award. She wishes to express her gratitudeto Michele Cuozzo’s family and to the Department for this great opportunity.The three authors were partially supported by the FCT (Portugal) project
Geometria de espa¸cos demoduli de curvas e variedades abelianas (EXPL/MAT-GEO/1168/2013). M. Melo and F. Viviani werepartially supported by the FCT projects
Espa¸cos de Moduli em Geometria Alg´ebrica (PTDC/MAT/111332/2009) and
Comunidade Portuguesa de Geometria Alg´ebrica (PTDC/MAT-GEO/0675/2012). A.Rapagnetta and F. Viviani were partially supported by the MIUR project
Spazi di moduli e applicazioni (FIRB 2012). F. Viviani was partially supported by CMUC - Centro de Matem´atica da Universidade deCoimbra.
References [Ale04] V. Alexeev:
Compactified Jacobians and Torelli map . Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, 1241-1265.12, 18[AIK76] A. B. Altman, A. Iarrobino, S. L. Kleiman:
Irreducibility of the compactified Jacobian.
Real and complexsingularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 1–12. Sijthoff andNoordhoff, Alphen aan den Rijn, 1977. 8, 10[AK79a] A. B. Altman, S. L. Kleiman:
Bertini theorems for hypersurface sections containing a subscheme.
Comm.Algebra 7 (1979), no. 8, 775–790. 6 AK80] A. B. Altman, S. L. Kleiman:
Compactifying the Picard scheme . Adv. Math. 35 (1980), 50–112. 2, 4, 7, 8, 9,23[AK79b] A. B. Altman, S. L. Kleiman:
Compactifying the Picard scheme II . American Journal of Mathematics 101(1979), 10–41. 7, 23[Ari11] D. Arinkin:
Cohomology of line bundles on compactified Jacobians . Math. Res. Lett. 18 (2011), no. 06, 1215–1226. 2, 28, 29[Ari13] D. Arinkin:
Autoduality of compactified Jacobians for curves with plane singularities . J. Algebraic Geom. 22(2013), 363–388. 2[BPV84] W. Barth, C. Peters, A. Van de Ven:
Compact complex surfaces.
Ergebnisse der Mathematik und ihrer Gren-zgebiete (3) [Results in Mathematics and Related Areas (3)], 4. Springer-Verlag, Berlin, 1984. 40, 41[Bat99] V. V. Batyrev:
Birational Calabi-Yau n-folds have equal Betti numbers.
New trends in algebraic geometry(Warwick, 1996), 1–11, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999. 3[BNR89] A. Beauville, M. S. Narasimhan, S. Ramanan:
Spectral curves and the generalised theta divisor . J. Reine Angew.Math. 398 (1989), 169–179. 1[BLR90] S. Bosch, W. L¨utkebohmert, M. Raynaud:
N´eron models . Ergebnisse der Mathematik und ihrer Grenzgebiete(3) [Results in Mathematics and Related Areas (3)], 21. Springer-Verlag, Berlin, 1990. 7, 23, 25, 29, 32, 33, 41[BGS81] J. Brian¸con, M. Granger, J.P. Speder:
Sur le sch´ema de Hilbert d’une courbe plane.
Ann. Sci. ´Ecole Norm.Sup. (4) 14 (1981), no. 1, 1–25. 8[Cap94] L. Caporaso:
A compactification of the universal Picard variety over the moduli space of stable curves . J. Amer.Math. Soc. 7 (1994), 589–660. 33[CCE08] L. Caporaso, J. Coelho, E. Esteves:
Abel maps for Gorenstein curves . Rend. Circ. Mat. Palermo (2) 57 (2008),33-59. 4, 5, 40[CE07] L. Caporaso, E. Esteves:
On Abel maps of stable curves.
Michigan Math. J. 55 (2007), no. 3, 575–607. 4, 5[CMK12] S. Casalaina-Martin, J. L. Kass:
A Riemann singularity theorem for integral curves.
Amer. J. Math. 134 (2012),no. 5, 1143–1165. 10[CMKV15] S. Casalaina-Martin, J. L. Kass, F. Viviani:
The Local Structure of Compactified Jacobians . Proc. LondonMath. Soc. 110 (2015), 510–542. 12, 13, 15[Cat82] F. Catanese:
Pluricanonical-Gorenstein-curves.
Enumerative geometry and classical algebraic geometry (Nice,1981), pp. 51–95, Progr. Math., 24, Birkh¨auser Boston, Boston, MA, 1982. 4, 11, 35, 40[CL10] P. H. Chaudouard, G. Laumon:
Le lemme fondamental pond´er´e. I. Constructions g´eom´etriques.
Compos.Math. 146 (2010), no. 6, 1416–1506. 2[CL12] P. H. Chaudouard, G. Laumon:
Le lemme fondamental pond´er´e. II. ´Enonc´es cohomologiques.
Ann. Math. (2)176 (2012), No. 3, 1647–1781. 2[CP10] J. Coelho, M. Pacini:
Abel maps for curves of compact type.
J. Pure Appl. Algebra 214 (2010), no. 8, 1319–1333.4, 5[DP12] R. Donagi, T. Pantev:
Langlands duality for Hitchin systems.
Invent. Math. 189 (2012), 653–735. 2[EGAI] A. Grothendieck, J. Dieudonn´e:
El´ements de G´eom´etrie Alg´ebrique I . Publ. Math. Inst. Hautes ´Etudes Sci. 4(1960). 22[EGAIV2] A. Grothendieck, J. Dieudon´e:
El´ements de G´eom´etrie Alg´ebrique IV-Part 2 . Publ. Math. Inst. Hautes ´EtudesSci. 24 (1965). 25, 26, 27[EGAIV3] A. Grothendieck, J. Dieudon´e:
El´ements de G´eom´etrie Alg´ebrique IV-Part 3 . Publ. Math. Inst. Hautes ´EtudesSci. 28 (1966). 23, 25, 26, 28[Est01] E. Esteves:
Compactifying the relative Jacobian over families of reduced curves.
Trans. Amer. Math. Soc. 353(2001), 3045–3095. 1, 2, 4, 7, 12, 13, 23, 26, 27, 28, 34, 36, 40[Est09] E. Esteves:
Compactified Jacobians of curves with spine decompositions.
Geom. Dedicata 139 (2009), 167–181.5, 36[EGK00] E. Esteves, M. Gagn´e, S. Kleiman:
Abel maps and presentation schemes.
Special issue in honor of RobinHartshorne. Comm. Algebra 28 (2000), no. 12, 5961–5992. 4[EGK02] E. Esteves, M. Gagn´e, S. Kleiman:
Autoduality of the compactified Jacobian . J. Lond. Math. Soc. 65 (2002),No. 3, 591–610. 4[EK05] E. Esteves, S. Kleiman:
The compactified Picard scheme of the compactified Jacobian.
Adv. Math. 198 (2005),no. 2, 484–503. 4[FGvS99] B. Fantechi, L. G¨ottsche, D. van Straten:
Euler number of the compactified Jacobian and multiplicity of rationalcurves.
J. Algebraic Geom. 8 (1999), no. 1, 115–133. 4, 21[FGA05] B. Fantechi, L. G¨ottsche, L. Illusie, S. L. Kleiman, N. Nitsure, A. Vistoli:
Fundamental algebraic geometry.Grothendieck’s FGA explained.
Mathematical Surveys and Monographs, 123. American Mathematical Society,Providence, RI, 2005. 7, 27[Har94] R. Hartshorne:
Generalized divisors on Gorenstein schemes.
Proceedings of Conference on Algebraic Geometryand Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992). K -Theory 8 (1994), no. 3, 287–339. 9[Hit86] N. Hitchin: Stable bundles and integrable systems.
Duke Math. J. 54 (1987), no. 1, 91–114. 1[Kas09] J. L. Kass:
Good Completions of N´eron Models . PhD thesis, Harvard, 2009. 10, 32[Kas12] J. L. Kass:
An Explicit Non-smoothable Component of the Compactified Jacobian.
Journal of Algebra 370(2012), 326–343. 10, 43[Kas13] J. L. Kass:
Degenerating the Jacobian: the N´eron Model versus Stable Sheaves.
Algebra and Number Theory7 (2013), No. 2, 379–404. 3[Kas15] J. L. Kass:
The compactified Jacobian can be nonreduced.
Bull. Lond. Math. Soc. 47 (2015), no. 4, 686–692. 10[Kaw02] Y. Kawamata: D -equivalence and K -equivalence. J. Differential Geom. 61 (2002), 147-171. 3 KK81] H. Kleppe, S.L. Kleiman:
Reducibility of the compactified Jacobian.
Compositio Math. 43 (1981), no. 2, 277–280.10[Kle81] H. Kleppe:
The Picard Scheme of a Curve and its Compactification.
PhD thesis, Massachusetts Institute ofTechnology, Cambridge, Massachusetts, 1981. 10[LM05] A. C. L´opez-Mart´ın:
Simpson Jacobians of reducible curves.
J. Reine Angew. Math. 582 (2005), 1–39. 43[LM06] A. C. L´opez-Mart´ın:
Relative Jacobians of elliptic fibrations with reducible fibers.
J. Geom. Phys. 56 (2006),no. 3, 375–385. 43[LRST09] A. C. L´opez-Mart´ın, D. H. Ruip´erez, D. S´anchez G´omez, C. Tejero Prieto:
Moduli spaces of semistable sheaveson singular genus curves . Int. Math. Res. Not. IMRN 2009, no. 23, 4428–4462 43[Mat80] H. Matsumura: Commutative algebra.
Second edition. Mathematics Lecture Note Series, 56. Ben-jamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. 30, 31[Mat89] H. Matsumura:
Commutative ring theory.
Translated from the Japanese by M. Reid. Second edition. CambridgeStudies in Advanced Mathematics, Vol. 8. Cambridge University Press, Cambridge, 1989. 28, 30, 31, 32[MRV1] M. Melo, A. Rapagnetta, F. Viviani:
Fourier-Mukai and autoduality for compactified Jacobians. I . PreprintarXiv:1207.7233v2. 1, 2[MRV2] M. Melo, A. Rapagnetta, F. Viviani:
Fourier-Mukai and autoduality for compactified Jacobians. II . PreprintarXiv:1308.0564v1. 1, 2, 3[MV12] M. Melo, F. Viviani:
Fine compactified Jacobians . Math. Nach. 285 (2012), no. 8-9, 997–1031. 3, 11, 12, 13, 14,15, 18, 33[MSV] L. Migliorini, V. Schende, F. Viviani:
A support theorem for Hilbert schemes of planar curves II.
PreprintarXiv:1508.07602v1. 3[Muk81] S. Mukai:
Duality between D ( X ) and D ( ˆ X ) with its application to Picard sheaves. Nagoya Math. J. 81 (1981),153–175. 2[Mum66] D. Mumford:
Lectures on curves on an algebraic surface.
Princeton University Press, 1966. 27[Nit91] N. Nitsure:
Moduli space of semistable pairs on a curve . Proc. London Math. Soc. (3) 62 (1991), no. 2, 275–300.1[Ngo06] B. C. Ngˆo:
Fibration de Hitchin et endoscopie . Invent. Math. 164 (2006), no. 2, 399–453. 2[Ngo10] B. C. Ngˆo:
Le lemme fondamental pour les alg`ebres de Lie . Publ. Math. Inst. Hautes ´Etudes Sci. No. 111(2010), 1–169. 2[OS79] T. Oda, C.S. Seshadri:
Compactifications of the generalized Jacobian variety . Trans. Amer. Math. Soc. 253(1979), 1–90. 3, 18[Ray70] M. Raynaud:
Sp´ecialisation du foncteur de Picard.
Inst. Hautes ´Etudes Sci. Publ. Math. 38 (1970), 27–76. 3,32, 33[Reg80] C. J. Rego:
The compactified Jacobian.
Ann. Sci. `Ecole Norm. Sup. (4) 13 (1980), no. 2, 211–223. 10[Sch98] D. Schaub:
Courbes spectrales et compactifications de jacobiennes . Math. Z. 227 (1998), no. 2, 295–312. 1[Ser06] E. Sernesi:
Deformations of algebraic schemes . Grundlehren der Mathematischen Wissenschaften 334. Springer-Verlag, Berlin, 2006. 20, 21, 22, 29[Ses82] C. S. Seshadri:
Fibr´es Vectoriels sur les Courbes Alg´ebriques.
Ast´erisque Vol. 96 (Soc. Math. France, Montrouge,1982). 13[Sim94] C. T. Simpson:
Moduli of representations of the fundamental group of a smooth projective variety. I.
Inst.Hautes ´Etudes Sci. Publ. Math. No. 79 (1994), 47–129. 13[Stacks]
Stacks Project. http://stacks.math.columbia.edu 25, 26[Vis] A. Vistoli:
The deformation theory of local complete intersections.