The Local Structure of Compactified Jacobians
aa r X i v : . [ m a t h . AG ] O c t THE LOCAL STRUCTURE OF COMPACTIFIED JACOBIANS
SEBASTIAN CASALAINA-MARTIN, JESSE LEO KASS, AND FILIPPO VIVIANI
Abstract.
This paper studies the local geometry of compactified Jacobians. The main result is apresentation of the completed local ring of the compactified Jacobian of a nodal curve as an explicitring of invariants described in terms of the dual graph of the curve. The authors have investigatedthe geometric and combinatorial properties of these rings in previous work, and consequences forcompactified Jacobians are presented in this paper. Similar results are given for the local structureof the universal compactified Jacobian over the moduli space of stable curves. Introduction
This paper studies the local geometry of compactified Jacobians associated to nodal curves.These are projective varieties that play a role similar to that of the Jacobian variety for a non-singular curve. Recall that a Jacobian can be viewed as the moduli space of line bundles (offixed degree) on a non-singular curve. A compactified Jacobian is an analogous parameter spaceassociated to a nodal curve. A major barrier to constructing these spaces is that, while the modulispace of fixed degree line bundles on a nodal curve exists, it typically does not have nice properties:often it has infinitely many connected components (i.e. is not of finite type), and these componentsfail to be proper. To construct a well-behaved compactified Jacobian, one must modify the moduliproblem. There are a number of different ways to do this, and the literature on the compactificationproblem is vast (e.g. [21], [15], [36], [4], [40], [7], [41], [37], [22], [16], [29], [18]).Geometric Invariant Theory (GIT) provides a general framework for these types of compactifi-cation problems, and in this approach, a compactified Jacobian ¯ J ( X ) of a nodal curve X is con-structed as a coarse moduli space of certain line bundles together with their degenerations: rank1, torsion-free sheaves. The sheaves parameterized by ¯ J ( X ) = ¯ J φ ( X ) are those rank 1, torsion-freesheaves that are semistable with respect to a numerical polarization φ (see Definitions 2.1 and 2.2).As explained in [30, Sec. 2], this semistability condition generalizes the other know semistabilityconditions that appear in the work of Oda-Seshadri [36], Seshadri [40], Esteves [16] as well as themore familiar slope semistability condition with respect to an ample line bundle that appears inthe work of Simpson [41]. In general, compactified Jacobians are non-fine moduli spaces becausetypically there are non isomorphic semistable sheaves I and I ′ that correspond to the same point[ I ] = [ I ′ ] ∈ ¯ J ( X ). Indeed, this happens precisely when two Jordan-H¨older filtrations of I and I ′ have the same stable factors. If a Jordan-H¨older filtration of I splits, i.e. if I is the direct sum ofstable sheaves supported on subcurves of X , we say that I is polystable. Therefore, given a point x ∈ ¯ J ( X ), there exists a unique polystable sheaf I such that [ I ] = x ∈ ¯ J ( X ); see § g does not admit a universal curve, this does not imply that the compactified Jacobians fit into Mathematics Subject Classification.
Primary 14D20, 14H40, Secondary 14D15, 14H20.The first author was supported by NSF grant DMS-1101333. The second author was supported by NSF grantDMS-0502170. The third author is supported by the MIUR project
Spazi di moduli e applicazioni (FIRB 2012), byCMUC and by the FCT-grants PTDC/MAT-GEO/0675/2012 and EXPL/MAT-GEO/1168/2013. family over M g . However, Caporaso [7] (and later Pandharipande [37]) has constructed a familyΦ : ¯ J d,g → M g (which we call the universal compactified Jacobian) of projective schemes that ex-tends the Jacobian of the generic genus g curve. The fiber of Φ over a stable curve X is isomorphicto a certain compactified Jacobian of X , modulo its automorphism group (see Fact 2.12).The main result of this paper describes the local geometry of both a compactified Jacobian ¯ J ( X )of a nodal curve X , and of the universal compactified Jacobian ¯ J d,g at a point corresponding to anautomorphism-free stable curve. Theorem A.
Let X be a nodal curve of arithmetic genus g ( X ) , let I be a rank , torsion-free sheafon X , and let Σ be the set of nodes where I fails to be locally free. Set Γ = Γ X (Σ) to be the dualgraph of any curve obtained from X by smoothing the nodes not in Σ . Fix an arbitrary orientationon Γ , and denote by V (Γ) , E (Γ) , and s, t : E (Γ) → V (Γ) the vertices, edges and source and targetmaps respectively. Set b (Γ) = E (Γ) − V (Γ) + 1 . Let T Γ := Y v ∈ V (Γ) G m , [ A (Γ) := k [[ X e , Y e : e ∈ E (Γ)]]( X e Y e : e ∈ E (Γ)) and [ B (Γ) := k [[ X e , Y e , T e : e ∈ E (Γ)]]( X e Y e − T e : e ∈ E (Γ)) . Define an action of the torus T Γ on [ A (Γ) and [ B (Γ) by the rule that λ = ( λ v ) v ∈ V (Γ) ∈ T Γ acts as λ · X e = λ s ( e ) X e λ − t ( e ) , λ · Y e = λ t ( e ) Y e λ − s ( e ) and λ · T e = T e . Define complete local rings R I := [ A (Γ)[[ W , . . . , W g ( X ) − b (Γ) ]] and R ( X,I ) = [ B (Γ)[[ W , . . . , W g − − b (Γ) − E (Γ) ]] , with actions of T Γ induced by the actions on [ A (Γ) and [ B (Γ) , and the trivial action on the remaininggenerators.(i) Suppose ¯ J ( X ) is a compactified Jacobian of X . If [ I ] ∈ ¯ J ( X ) with I polystable, then there isan isomorphism b O ¯ J ( X ) , [ I ] ∼ = R T Γ I between the completed local ring of ¯ J ( X ) at [ I ] and the T Γ -invariant subring of R I .(ii) If X is a stable curve with trivial automorphism group and [( X, I )] ∈ ¯ J d,g with I polystable,then there is an isomorphism b O ¯ J d,g , [( X,I )] ∼ = R T Γ ( X,I ) between the completed local ring of ¯ J d,g at [( X, I )] and the T Γ -invariant subring of R ( X,I ) . Theorem A is a consequence of Theorems 5.10 and 6.1 (see also Remarks 5.9, 6.2). We discussthe proof in more detail below. The rings [ A (Γ) appearing above are further studied in [13]. In thenotation of that paper, [ A (Γ) is the completion of the ring A (Γ) defined in [13, Theorem A], and theaction of T Γ in both papers is the same. It is shown in [13, Theorem A] that the invariant subring A (Γ) T Γ is isomorphic to the cographic ring R (Γ) defined in [13, Definition 1.4]. In particular,the completed local ring of the compactified Jacobian is isomorphic to a power series ring over acompletion of the cographic toric face ring R (Γ). A number of geometric properties of cographicrings are established in [13], and some consequences for compactified Jacobians are discussed inTheorem B below. The geometric and combinatorial properties of the rings [ B (Γ) T Γ will be describedin more detail by the authors in [12]. Theorem B.
Let ¯ J ( X ) be a compactified Jacobian of a nodal curve X .(i) ¯ J ( X ) has Gorenstein, semi log-canonical (slc) singularities. In particular, ¯ J ( X ) is seminor-mal. ii) Let [ I ] ∈ ¯ J ( X ) with I polystable. Then [ I ] lies in the smooth locus of ¯ J ( X ) if and only if I fails to be locally free only at separating nodes of the dual graph of X . The proof is given at the end of §
6. In [13] it is shown that a number of further propertiesof cographic rings can be determined from elementary combinatorics of the graph Γ = Γ X (Σ)introduced in Theorem A. For instance, that paper provides combinatorial formulas giving theembedding dimension and the multiplicity of b O ¯ J ( X ) ,x , as well as a description of the irreduciblecomponents and the normalization of this ring. The reader is directed to § J ( X ) at a stable sheaf is isomorphic to a completed product of nodes and smooth factors. Using Theorem A and theresults of [13] one can construct examples of compactified Jacobians whose structure at a strictlysemi-stable point is more complicated (see esp. § J ( X ) of a nodal curve X (the case of the universal compactifiedJacobian ¯ J d,g is similar). To begin with, there is a well-known explicit description of the miniversaldeformation ring R I of a rank 1, torsion-free sheaf I on a nodal curve X (see Corollary 3.17), andwe use that description to define an explicit action of Aut( I ) on R I (see Theorems 5.10). We provethe main result by showing that, when [ I ] ∈ ¯ J ( X ) with I polystable, the ring of invariants R Aut( I ) I is isomorphic to the completed local ring of ¯ J ( X ) at [ I ].In order to establish this last point, we use the GIT construction of ¯ J ( X ) together with theLuna Slice Theorem and a theorem of Rim. Recall that the compactified Jacobian is constructedas a GIT quotient of a suitable Quot scheme Quot( O ⊕ rX ) by the action of SL r (see Corollary 2.10).We check that the complete local ring R x of a slice (which exists by Luna Slice Theorem) at apolystable point x = [ O ⊕ rX ։ I ] ∈ Quot( O ⊕ rX ) is a miniversal deformation ring for I (Lemma 6.4).Thus R x is (non-canonically) isomorphic to R I . By definition, the stabilizer G x of x (which isdescribed in Lemma 6.6) acts on the ring R x , and it follows from the definition of a slice that theinvariant ring R G x x is isomorphic to the complete local ring of the GIT quotient at the image of thepoint x . We complete the proof by using a theorem of Rim (Fact 5.4) to identify the action of G x on R x to our explicit action of Aut( I ) on R I , completing the proof (see Theorems. 5.10 and 6.1).Theorem B is one consequence of Theorem A. Other consequences will be found in the upcomingarticle [12]. There the authors will use Theorem A to describe the singularities of ¯ J d,g . Moreprecisely, they will prove that ¯ J d,g has canonical singularities provided char( k ) = 0. When X does not admit a non-trivial automorphism, the authors will prove this result by using the explicitdescription of the completed local ring in Theorem A, and in general, they will reduce the proofto a similar argument using a generalization of the Reid–Tai–Shepherd-Barron criterion for toricsingularities. The results in [12] will extend the work of Bini, Fontanari and the third author [6],where it is shown that ¯ J d,g has canonical singularities when gcd( d + 1 − g, g −
2) = 1, a conditionequivalent to the condition that ¯ J d,g has finite quotient singularities. Under the same assumptionon d and g , the same authors computed the Kodaira dimension and the Itaka fibration of ¯ J d,g ([6,Thm. 1.2]), and in [12], the present authors will extend that computation to all d, g .The authors also hope to use the results of this paper to study the singularities of the thetadivisor of a nodal curve. The theta divisor is an ample effective Cartier divisor on the canonicalcompactified Jacobian of degree g −
1, parametrizing sheaves with a non-trivial section (see [2], [9],[10]). The case of integral nodal curves has been studied by the first two authors in [11], where ananalogue of the Riemann singularity theorem is proved. The authors are currently investigating howto extend the Riemann singularity theorem to non-integral nodal curves, based upon the explicitlocal description of the compactified Jacobian obtained in this paper. his paper suggests two technical questions for future study. In Theorem 6.1(ii) the curve X isassumed to be automorphism-free. It is particularly difficult to described the local structure of ¯ J d,g when X admits an automorphism of order equal to p , the characteristic of k . When X admits suchan automorphism, Aut( X, I ) is reductive but not linearly reductive. Linear reductivity is crucialin two places: in the proof of Theorem 6.1, which uses the Luna Slice Theorem, and Theorem 5.10,which uses a result of Rim. It would be interesting to know if suitable generalizations of Rim’sTheorem and the Luna Slice Theorem hold for reductive groups such as Aut(
X, I ). We discussthese issues after the proofs of the two theorems.Positive characteristic issues also appear in Fact 2.12, which relates the fibers of ¯ J d,g → M g tocompactified Jacobians. That result is only stated in characteristic 0, and it would be interestingto know if the result remains valid in positive characteristic. This is discussed in greater detailimmediately after the proof of the fact.There are approaches to describing the local structure of a compactified Jacobian different fromthe approach taken here. Alexeev has proven in [2, Thm. 5.1] that compactified Jacobians arestable semi-abelic varieties in the sense of [1], and consequently can be described using Mumford’sconstruction [32] of degenerations of abelian varieties. In Mumford’s approach (that has beenfurther developed in [33], [34], [3], [1]), one compactifies a semi-abelian variety by first formingthe projectivization of a (non-finitely generated) graded algebra and then quotienting out by alattice. This procedure provides direct access to the local structure of the compactification, andthus Alexeev’s work provides another approach to studying the local structure of compactifiedJacobians. It would be interesting to compare the descriptions arising from this approach to thedescriptions given in this paper, but we do not pursue this topic here.The results of Theorem B are related to some results in the literature. Specifically, it was knownthat ¯ J ( X ) is seminormal [2, Thm. 5.1] and Gorenstein [3, Lemma 4.1]. In personal correspondence,Alexeev has explained to the authors that the techniques of those papers can also be used establishthe fact that ¯ J ( X ) is semi-log canonical. The description of the smooth locus of ¯ J ( X ) is certainlywell-known to the experts (see e.g. [7, Thm. 6.1(3)], [8, Thm. 7.9(iii)], [9, Fact 4.1.5(iv)], [30,Fact 1.19(ii)]); however, it seems that a proof has not appeared in print.This paper is organized as follows. We review the definition and the GIT construction of (univer-sal) compactified Jacobians in § §
3, where we develop the deformation theoryneeded to compute deformation rings parameterizing deformations of a rank 1, torsion-free sheaf.These rings admit natural actions of automorphism groups, which are described in the next twosections. The structure of the automorphism groups is studied in §
4, and then those results areused in § § § Acknowledgements.
We would like to thank Robert Lazarfeld for helpful expository suggestions.This work began when the authors were visiting the MSRI, in Berkeley, for the special semester inAlgebraic Geometry in the spring of 2009; we would like to thank the organizers of the program aswell as the institute for the excellent working conditions and the stimulating atmosphere.
Conventions.1.1. k will denote an algebraically closed field (of arbitrary characteristic). All schemes are k -schemes, and all morphisms are implicitly assumed to respect the k -structure. A curve is a connected, complete, reduced scheme (over k ) of pure dimension 1. We denoteby ω X the dualizing sheaf of X . The genus g ( X ) of a curve X is g ( X ) := h ( X, O X ). .3. A subcurve Y of a curve X is a closed k -scheme Y ֒ → X that is reduced and of pure dimension1 (but possibly disconnected). A subcurve Y ⊆ X is said to be proper if it non empty and differentfrom X . Given a subcurve Y , the complementary subcurve Y c is defined to be X \ Y , or, in otherwords, Y c is the subcurve which is the union of all the irreducible components of X that are notcontained in Y . A family of curves is a proper, flat morphism X → T whose geometric fibers are curves. A family of coherent sheaves on a family of curves X → T is a O T -flat, finitely presented O X -module I . A coherent sheaf I on a curve X is said to be:(i) of rank I has rank 1 at every generic point;(ii) pure if for every non-zero subsheaf J ⊆ I the dimension of the support of J is equal to thedimension of the support of I ;(iii) torsion-free if it is pure and the support of I is X .The degree of a torsion-free, rank 1 sheaf I on a curve X is defined to be deg I := χ ( I ) − χ ( O X ),where χ denote the Euler characteristic.2. Preliminaries on GIT and compactified Jacobians
Here we review the definition and the construction of compactified Jacobians of a fixed nodalcurve as well as of the universal compactified Jacobian, with the goal of collecting the results neededto prove Theorem 6.1.2.1.
Geometric Invariant Theory.
The (universal) compactified Jacobians are coarse modulispaces of sheaves constructed using Geometric Invariant Theory (GIT) and, in the proof of ourresults, we will need to make use of their construction, and not just the fact of their existence.Therefore, we will quickly review some background from GIT.Recall that GIT is a tool for constructing a quotient of a quasi-projective variety Q by the actionof a reductive group G . Given an auxiliary ample line bundle O (1) together with a lift of the actionof G on Q to an action on O (1) (i.e. a linearization), there is distinguished open subscheme Q ss of Q that consists of points that are semi-stable with respect to the linearized action. The significance of Q ss is that it admits a categorical quotient that we define to be the GIT quotient of Q , written Q//G . That is, there exists a pair ( Q ss /G, π ) consisting of a quasi-projective variety Q ss /G and a G -invariant map π : Q ss → Q ss /G with the property that π is universal among all G -invariant maps out of Q ss . When the characteristicof k is 0, the pair ( Q ss , π ) is actually a universal categorical quotient , i.e. for any morphism T → Q ss /G the base change morphism π T : Q ss × Q ss /G T → T is again a categorical quotient.The local structure of Q//G is described by the Luna Slice Theorem, which compares
Q//G tothe quotient of a certain model G -space. For the remainder of § Q is affine, so Q = Q ss and Q//G is the categorical quotient ([31, Thm. 1, p. 27]). The model scheme is G × H V ,whose definition we now review. Suppose H ⊂ G is a reductive subgroup and V a scheme with aleft H -action. The product G × V carries an H -action defined by h · ( g, x ) := ( gh − , h · x ) , and we write G × H V for the categorical quotient. This quotient admits a left action of G definedby the translation action on the first factor. The two projections out of G × V induce morphisms p : G × H V → V /H and q : G × H V → G/H. he first map is G -invariant and realizes V /H as the quotient of G × H V by G . The map q isequivariant and can often be described as a contraction onto an orbit. To be precise, suppose weare given an element v ∈ V fixed by H . One may verify that the image of ( e, v ) ∈ G × V in G × H V has stabilizer H , and the associated orbit map defines a section of q .The Luna Slice Theorem provides sufficient conditions for Q//G to be ´etale locally isomorphicto
V /H for a suitable H and V . More precisely, let x be a point of Q with stabilizer H . Given anyaffine, locally closed subscheme V ⊂ Q that contains x and is stabilized by H (i.e. H · V ⊂ V ), theaction map induces a G -equivariant morphism G × H V → Q . We say that V is a slice at x if thefollowing conditions are satisfied:(1) the morphism G × H V → Q is ´etale;(2) the image of G × H V → Q is an open affine U ⊂ Q that is π -saturated (i.e. for each u ∈ U , π − ( π ( u )) ⊆ U );(3) the induced morphism ( G × H V ) /G → U/G is ´etale;(4) the induced morphism G × H V → U × U/G
V /H is an isomorphism.Note in particular that condition (3) together with the observation above on the map p impliesthat there is an ´etale morphism(2.1) V /H ´et −→ Q//G.
The original Luna Slice Theorem [24, p. 97] states that in characteristic zero a slice exists providedthat x is (GIT-) polystable , i.e. the orbit of x is closed. When x has a closed orbit, Matsushima’scriterion implies that the stabilizer H is reductive ([27] for k = C ; [38] for k arbitrary). Bardsley andRichardson have extended the Luna Slice Theorem to arbitrary characteristic. With no assumptionson char( k ), they prove that a slice exists provided the orbit of x is closed and the stabilizer H isreduced and linearly reductive ([5, Prop. 7.6]; the condition in loc. cit. that the orbit is separableis equivalent to our condition that H is reduced).2.2. Compactified Jacobians of nodal curves.
In this subsection, we review the definition ofcompactified Jacobians of a nodal curve X .Any (known) compactified Jacobian of X parametrizes torsion-free, rank-1 sheaves on X thatare semistable with respect to some polarization. There are several ways to define a polarizationand the associated semistability condition on X . The most general definition is stated in terms ofnumerical polarizations (following [30, Sec. 2.4]): all the other know semistability conditions arespecial case of this one, see [30, Sec. 2]. Definition 2.1.
Let X be a nodal curve with irreducible components { X , . . . , X γ } . A numericalpolarization on X is a γ -tuple of rational numbers φ = { φ i = φ X i } γi =1 ∈ Q γ , one for eachirreducible component of X , such that | φ | := P i φ i ∈ Z . For any subcurve Y of X , we set φ Y = P X i ⊆ Y φ X i ∈ Q . For any subcurve Y ⊂ X such that φ Y − Y ∩ Y c )2 ∈ Z , we define anumerical polarization φ Y on Y by setting( φ Y ) Y i := φ Y i − Y i ∩ Y c )2 for any irreducible component Y i of Y. The semistability of a torsion-free, rank 1 sheaf on X with respect to a numerical polarization φ is defined as it follows. Definition 2.2.
Let X be a nodal curve and let φ = ( φ i ) be a numerical polarization on X .(i) A torsion-free, rank 1 sheaf I on X is said to be φ -semistable if deg I = | φ | and(2.2) deg( I Y ) ≥ φ Y − Y ∩ Y c )2 , or any subcurve Y ⊆ X , where I Y denotes the biggest torsion-free quotient of the restriction I | Y of I to Y .(ii) A torsion-free, rank 1 sheaf I on X is said to be φ -stable if it is φ -semistable and the inequality(2.2) is strict for every proper subcurve ∅ 6 = Y ( X .(iii) A torsion-free, rank 1 sheaf I on X is said to be φ -polystable if it is φ -semistable and for allsubcurves Y for which inequality (2.2) is an equality then it holds that I = I Y ⊕ I Y c .The φ -semistability condition is stated as a lower bound on the multidegree of I . However, thisimplies also an upper bound on the multidegree of I , as we observe in the following Remark. Remark 2.3.
Let φ be a numerical polarization on a nodal curve X and let I be a torsion-freerank 1 sheaf on X of degree deg I = | φ | . Then I is φ -semistable if and only if(2.3) deg( I Y ) ≤ φ Y + Y ∩ Y c )2 − { p ∈ Y ∩ Y c : I fails to be locally free at p } . holds for every subcurve Y of X . Indeed, this follows from the two easily checked formulas ( deg( I Y ) + deg( I Y c ) = deg I − { p ∈ Y ∩ Y c : I fails to be locally free at p } ,φ Y + φ Y c = | φ | . The condition of being polystable is better understood in terms of the
Jordan-H¨older filtration.Recall that, given a φ -semistable sheaf I , a Jordan-H¨older filtration of I is a filtration0 = I q +1 ( I q ( . . . ( I ( I = I, with the following properties:(1) The sheaf I k is a rank 1, torsion-free sheaf supported on a subcurve Z k ⊂ X and φ Z k -semistable, for every 0 ≤ k ≤ q .(2) The quotient sheaf I k /I k +1 is a rank 1, torsion-free sheaf supported on the subcurve Y k = Z k \ Z k +1 and φ Y k -stable, for every 0 ≤ k ≤ q .Jordan-H¨older filtrations exist for every φ -semistable sheaf I but they are not unique; however, thegraded sheaf Gr( I ) := I /I ⊕ . . . ⊕ I q /I q +1 depends only on I (see e.g. [16, § § I is polystable if andonly if I ∼ = Gr( I ). Moreover, we say that two φ -semistable sheaves I and I ′ are S-equivalent (orJordan-H¨older equivalent) if Gr( I ) ∼ = Gr( I ′ ). Therefore, every φ -semistable sheaf is S-equivalent toa unique φ -polystable sheaf, namely Gr( I ).With the above definitions, we can now introduce the φ -compactified Jacobian functor ¯ J ♯φ ( X ) : k -Sch → Setswhich associates to a k -scheme T the set of families of coherent sheaves on X × k T → T that arefiberwise rank 1, torsion-free and φ -semi-stable. Fact 2.4 (Oda-Seshadri, Seshadri) . There exists a projective variety ¯ J φ ( X ) , called the φ -compa-ctified Jacobian or simply compactified Jacobian , that co-represents the functor ¯ J ♯φ ( X ) . More-over, two sheaves I, I ′ ∈ ¯ J ♯φ ( X )( k ) define the same k -point [ I ] = [ I ′ ] ∈ ¯ J φ ( X ) if and only if I and I ′ are S-equivalent. In particular, every k -point of ¯ J φ ( X ) is equal to [ I ] , for a unique φ -polystablesheaf I . Recall that the fact that ¯ J φ ( X ) co-represents ¯ J ♯φ ( X ) means that there exist a natural trans-formation of functors π : ¯ J ♯φ ( X ) → Hom( − , ¯ J φ ( X )) which is universal with respect to natural ransformations from ¯ J ♯φ ( X ) to the functor of points of k -schemes. Given a point I ∈ ¯ J ♯φ ( X )( k ), weset [ I ] := π ( I ) ∈ Hom(Spec k, ¯ J φ ( X )) = ¯ J φ ( X )( k ). Proof.
This is proved by Oda-Seshadri in [36, Thms 11.4 and 12.14] and Seshadri in [40, Thm.15]. Note that in loc. cit. the authors use two different definitions of φ -(semi)stability, which arehowever equivalent to our definition as discussed in [2, § (cid:4) Remark 2.5.
If the numerical polarization φ is such that φ Y − Y ∩ Y c )2 Z for every propersubcurve ∅ 6 = Y ( X (in which case we say that φ is general ), then it follows from Definition 2.2that every φ -semistable sheaf is also φ -stable. Hence, ¯ J φ ( X ) is a fine moduli space parametrizing φ -stable sheaves and we say that ¯ J φ ( X ) is a fine compactified Jacobian . Such compactified Jacobiansare studied in [30] and in [28].We now compare the φ -semistability condition introduced above with the (more familiar) notionof slope semistability. Recall that, given a nodal curve X and a polarization L on X , i.e. an ampleline bundle on X , the slope µ L ( I ) of a coherent sheaf I with respect to L is defined to be a/r ,where a and r are coefficients of the Hilbert polynomial P L ( I, t ) := r · t + a of I with respect to L . Definition 2.6.
Let X be a nodal curve and L be a polarization on X .(i) The sheaf I is said to be slope semistable (resp. slope stable ) with respect to the polar-ization L if it is pure and satisfies µ L ( I ) ≤ µ L ( J ) (resp. < ) for all pure non-trivial quotients I ։ J with 1-dimensional support Supp( J ).(ii) The sheaf I is said to be slope polystable if it is slope semistable and isomorphic to a directsum of slope stable sheaves.With the above definitions, we can now introduce the Simpson Jacobian functor of degree d to be the functor ¯ J ♯L,d ( X ) : S -Sch. → Setswhich sends a k -scheme T into the set of families of coherent sheaves on X × k T → T that arefiberwise rank 1, torsion-free of degree d and slope semistable with respect to the polarization L . Fact 2.7 (Simpson) . There exists a projective scheme ¯ J L,d ( X ) , called the Simpson compactifiedJacobian , that co-represents the functor ¯ J ♯L,d ( X ) .Proof. This follows easily from the work of Simpson [41]. However, for later use, we need to reviewthe explicit GIT construction. Consider the polynomial P d ( t ) := deg( L ) · t + d + 1 − g ( X ) , which is the Hilbert polynomial, with respect to the polarization O X (1) := L , of any rank 1,torsion-free sheaf of degree d . Let M ♯ ( O X , P d ) : k -Sch. → Sets to be the functor which associatesto a k -scheme T the set of isomorphism classes of coherent sheaves on X × k T , flat over T , thatare fiberwise slope semistable with Hilbert polynomial P d ( t ) with respect to L . Note that ¯ J ♯d,L ( X )is the subfunctor of M ♯ ( O X , P d ) parametrizing families of torsion-free, rank 1 sheaves.Following Simpson’s construction [41] (note that Simpson [41] works over k = C but his con-struction has been extended over an arbitrary base field k by Maruyama [26] and Langer [23]),choose b sufficiently large and set r := P d ( b ) = d · b + 1 − g ( X ). Consider the Quot schemeQuot( O X ( − b ) ⊕ r , P d ( t )) parametrizing quotients O X ( − b ) ⊕ r ։ I , where I is a coherent sheaf on X of Hilbert polynomial P d ( t ) with respect to O (1) (see [20] and [35] for details on Quot schemes).There is a closed and open subscheme ([41, p. 66]) Q ◦ ⊆ Quot( O X ( − b ) ⊕ r , P d ) that parameterizesquotient maps q : O X ( − b ) ⊕ r ։ I satisfying the following additional conditions: H ( X, I ( b )) = 0; • q ⊗ H ( X, O ⊕ rX ) → H ( X, I ( b )) is an isomorphism; • I ( b ) is generated by its global section.The natural linearized action of SL r on the Quot scheme Quot( O X ( − b ) ⊕ r , P d ( t )) restricts to alinearized action on Q ◦ and the GIT stability for this action is naturally related to slope stability.Specifically, a point of the Quot scheme corresponding to q : O ( − b ) ⊕ r ։ I is GIT (resp. semi,poly)stable if and only if I is (resp. semi, poly)stable with respect to the polarization L (see [41,Cor. 1.20, Thm. 1.19, Pf. of Thm. 1.21]). Therefore, the projective GIT quotient M( O X , P d ) := Q ◦ // SL r = ( Q ◦ ) ss /SL r naturally co-represents the functor M ♯ ( O X , P d ).Consider now the locus Q ◦◦ ⊆ Q ◦ parametrizing quotients q : O X ( − b ) ⊕ r ։ I such that • I is a rank 1, torsion-free sheaf on X .This is a SL r -invariant subset that is closed and open in Q ◦ by [37, Lemma 8.1.1]. Therefore,the image of Q ◦◦ in the GIT quotient Q ◦ // SL r , which we set to be equal to ¯ J L,d ( X/S ), must beclosed-and-open in M( O X , P d ) by [31, p. 8, Remark 6]. By construction, the projective scheme¯ J L,d ( X/S ) co-represents the functor ¯ J ♯L,d ( X/S ). (cid:4) Simpson compactified Jacobians are a special case of Oda-Seshadri φ -compactified Jacobians. Fact 2.8 (Alexeev) . Let X be a nodal curve endowed with a polarization L and fix d ∈ Z . Considerthe numerical polarization φ such that (2.4) φ X i := deg( L | X i )deg( L ) (cid:18) d − deg( ω X )2 (cid:19) + deg( ω X | X i )2 , for each irreducible component X i of X . Then a rank , torsion-free sheaf I of degree d on X isslope semistable (resp. stable, resp. polystable) with respect to L if and only if it is φ -semistable(resp. φ -stable, φ -polystable).In particular, we have that ¯ J ♯L,d ( X ) = ¯ J ♯φ ( X ) , which implies ¯ J L,d ( X ) ∼ = ¯ J φ ( X ) .Proof. This is proved by Alexeev in [2], where it shown that a torsion-free, rank 1 sheaf I of degree d is slope (semi)stable (with respect to the polarization L ) if and only if, for any subcurve i : Y ֒ → X ,we have that µ L ( I ) ≤ µ L ( i ∗ ( I Y )), where I Y is the biggest torsion-free quotient the restriction i ∗ ( I ) = I | Y . By the definition of the slope µ L , we get(2.5) deg( I ) − / ω X )deg( L ) = µ L ( I ) ≤ µ L ( i ∗ ( I Y )) = deg( I Y ) − / ω X | Y ) + 1 / Y ∩ Y c )deg( L | Y ) , where we used the formula ω X | Y = ω Y ( Y ∩ Y c ). Equation (2.5) can be rewritten as(2.6) deg( I Y ) ≥ deg( L | Y )deg( L ) (cid:18) deg( I ) − deg( ω X )2 (cid:19) + deg( ω X | Y )2 − Y ∩ Y c )2 = φ Y − Y ∩ Y c )2 , which says that I is φ -(semi)stable with respect to the numerical polarization defined by (2.4). Thefact that slope polystability correspond to φ -polystability follows easily from the above. (cid:4) Remark 2.9.
Let X be a nodal curve of genus g and consider the compactified Jacobians of X .(i) There are φ -compactified Jacobians of degree d that are not Simpson Jacobians of degree d . The most extreme case is d = g −
1. As it follows from (2.4), every Simpson compactifiedJacobian of degree g − φ -compactified Jacobian ¯ J φ ( X ) such that φ X i =deg( ω X | X i )2 for every irreducible component X i of X (a very special compactified Jacobian,called the canonical compactified Jacobian of degree g −
1, that was studied in detail in [2,Sec. 3], [9], [10]). However, there are many φ -compactified Jacobians of degree d = g − lso in degree d = g −
1, there are, in general, φ -compactified Jacobians that are notSimpson compactified Jacobians. For example, let X be the genus 2 nodal curve that consistsof two rational components meeting in three nodes. For a numerical polarization φ = ( φ , φ )such that | φ | = φ + φ = 0, then one can compute that ¯ J φ ( X ) has two irreducible componentsif φ , φ ∈ / Z and three irreducible components otherwise. On the other hand, given anample line bundle L with bidegree ( a, b ), the associated φ -parameter (see (2.4)) is φ = (1 / − b/ ( a + b ) , / − a/ ( a + b )) , and φ , φ cannot belong to 1 / Z because a, b >
0. Therefore, every Simpson compactifiedJacobian of degree 0 has three irreducible components.(ii)
Every φ -compactified Jacobian of degree d is isomorphic to a Simpson Jacobian of degree d ′ ,for some d ′ ≫ d . Indeed given a numerical polarization φ , pick a line bundle M of sufficiently small degreeon each irreducible component X i of X in such a way that a i := φ X i − deg( M | X i ) − deg( ω X | X i )2 > . Moreover, pick a sufficiently divisible natural number e ∈ N such that b i := e a i d + g − ∈ Z for every i. Finally, choose a line bundle L of total degree e such that deg( L | X i ) = b i and observe that L is ample since deg( L | X i ) = b i >
0. With the above choices, we get that(2.7) ψ X i := φ X i − deg( M | X i ) = (cid:18) d − deg( ω X )2 (cid:19) deg( L | X i )deg( L ) + deg( ω X | X i )2 . Therefore, using Fact 2.8, we get the isomorphism¯ J φ ( X ) ∼ = −→ ¯ J ψ ( X ) ∼ = ¯ J L,d ′ ( X ) I I ⊗ M − , where d ′ = | ψ | = | φ | − deg M .We record in the following corollary a presentation of any compactified Jacobian of a nodal curveas a GIT quotient of an open subset of a suitable Quot scheme. Such a GIT description will becrucial in proving Theorem A(i). Corollary 2.10 (GIT presentation of compactified Jacobians) . Let X be a nodal curve of genus g and let ¯ J ( X ) be any compactified Jacobian of X . There exists a Quot scheme Quot( O ⊕ rX , P d ( t )) ,parametrizing quotients q : O ⊕ rX ։ I with Hilbert polynomial P d ( t ) = d · t +1 − g with respect to someample line bundle O X (1) , with an open and closed SL r -invariant subscheme U ⊆ Quot( O ⊕ rX , P d ( t )) parameterizing the quotients q : O ⊕ rX ։ I with the property that(1) H ( X, I ) = 0 ,(2) q : H ( X, O ⊕ rX ) → H ( X, I ) is an isomorphism,(3) I is generated by the global sections,(4) I is a torsion-free, rank sheaf,in such a way that ¯ J ( X ) ∼ = U// SL r = U ss / SL r , where the GIT quotient on the right hand side is taken with respect to the natural linearized actionof SL r . roof. By Remark 2.9(ii), it is enough to prove the Corollary for a Simpson compactified Jaco-bian ¯ J L,e ( X ) (with e ≫ J L,e ( X ) admits a GITdescription as Q ◦◦ // SL r , where Q ◦◦ is the open and closed subscheme of a suitable Quot schemeQuot( O X ( − b ) ⊕ r , P e ( t )), parametrizing the quotients q : O X ( − b ) ⊕ r ։ I such that • H ( X, I ( b )) = 0; • q ⊗ H ( X, O ⊕ rX ) → H ( X, I ( b )) is an isomorphism; • I is a rank 1, torsion-free sheaf on X .The isomorphism Φ : Quot( O X ( − b ) ⊕ r , P e ( t )) ∼ = −→ Quot( O ⊕ rX , P e + b deg O X (1) ( t )) , [ q : O X ( − b ) ⊕ r ։ I ] [ q : O ⊕ rX ։ I ( b )] , sends Q ◦◦ isomorphically onto the open subset U ⊆ Quot( O ⊕ rX , P e + b deg O X (1) ( t )) parametrizingquotients q : O ⊕ rX ։ I satisfying the three conditions (1), (2), (4) and moreover¯ J L,e ( X ) ∼ = Q ◦◦ // SL r ∼ = −→ U// SL r . (cid:4) The universal compactified Jacobian.
In this subsection, we review the definition andthe construction of the universal degree d compactified Jacobian ¯ J d,g → M g over the moduli spaceof stable curves M g of genus g ≥ J d,g is originally due to Caporaso [7] in terms of balanced line bundles onquasi-stable curves. Later, Pandharipande [37] re-interpreted ¯ J d,g in terms of rank 1, torsion-freesemistable sheaves on stable curves. We will focus on Pandharipande’s later construction becausethis description most naturally relates to the other compactified Jacobians we discuss here. For adescription of Caporaso’s approach, we direct the interested reader to [7] and [37, § d and g ≥
2, the universal compactified Jacobian functor ¯ J ♯d,g : k -Sch. → Setsis defined to be the functor sending a k -scheme T to the set of isomorphism classes of families X → T of stable curves of genus g together with a family of coherent sheaves which is fiberwisetorsion-free, rank 1 of degree d and slope semistable with respect to the relative dualizing linebundle. Fact 2.11 (Pandharipande [37]) . The functor ¯ J ♯d,g is co-representable by a projective scheme ¯ J d,g ,called the universal compactified Jacobian , which is endowed with a forgetful projective mor-phism Φ : ¯ J d,g → M g .Proof. This follows from the work of Pandharipande [37], where the projective scheme ¯ J d,g isconstructed via GIT. Since we will need this GIT description in the proof of Theorem A(ii), we willnow review the relevant GIT set-up.To begin, we may assume d is sufficiently large because tensoring with the dualizing sheaf definesa canonical isomorphism between ¯ J ♯d,g and J ♯d +2 g − ,g . Thus, let d be large and fixed. Set N :=10(2 g − − g and e := 10(2 g − P ( t ) := e · t + d + 1 − g and set r := P (0).Inside of the Hilbert scheme of degree e curves in P N , we can consider the locally closed subscheme H g parameterizing non-degenerate, 10-canonically embedded stable curves. The product H g × P N contains the universal 10-canonically embedded curve X g , and associated to this family is therelative Quot scheme Quot( O ⊕ rX g , P ( t )), parametrizing quotients q : O ⊕ rX g ։ E such that E is acoherent sheaf on X g , flat over H g , with the property that on each fiber of X g → H g the Hilbert olynomial (with respect to the polarization given the embedding X g ֒ → H g × P N ) of E is equalto P ( t ).The product group SL r × SL N +1 acts on this Quot scheme by making SL r act on O ⊕ rX g by changingbases, SL N +1 act on P N by changing projective coordinates, and then making SL r × SL N +1 act onthe Quot scheme by the product action. The action of SL r × SL N +1 admits a natural linearizationcoming from the construction of the relative Quot scheme (see [35]).Inside Quot( O ⊕ rX g , P ( t )), there is an invariant closed-and-open subset Q ◦ parameterizing torsion-free, rank 1 quotients ([37, Lemma 8.1.1]). It is shown in [37, Thm. 8.2.1, Thm. 9.1.1] that a point[ q : O ⊕ rX g ։ E ] ∈ Q ◦ is GIT semistable if and only if E is relatively semistable with respect to therelative dualizing sheaf. Therefore, the GIT (projective) quotient(2.8) ¯ J d,g := Q ◦ // SL r × SL N +1 . co-represents ¯ J ♯d,g and, by construction, it is endowed with a forgetful projective morphism Φ :¯ J d,g → M g . (cid:4) The fibers of the forgetful morphism Φ : ¯ J d,g → M g are related to compactified Jacobians ofstable curves with respect to their canonical polarization. Fact 2.12.
Assume char( k ) = 0 . Then the fiber of Φ : ¯ J d,g → M g over a stable curve X is (2.9) Φ − ( X ) ∼ = ¯ J ω X ,d ( X ) / Aut( X ) . Proof.
This fact is surely well-known (see e.g. [2, § k ) = 0, the GIT quotient (2.8) is a universal categorical quotient (see § J d,g co-represents the functor ¯ J ♯d,g universally , i.e. for any scheme T → ¯ J d,g the base change functor ¯ J ♯d,g × Hom( − , ¯ J d,g ) Hom( − , T ) is co-represented by T . Applying this propertyto the inclusion Φ − ( X ) ֒ → ¯ J d,g , we deduce that Φ − ( X ) co-represents the functor( ¯ J ♯d,g ) | X := ¯ J ♯d,g × Hom( − , ¯ J d,g ) Hom( − , Φ − ( X )) : k -Sch. → Setswhich associates to a k -scheme S the set of isomorphism classes of iso-trivial families p : X → S withfiber X together with a coherent sheaf I , flat over S , which is fiberwise torsion-free, rank 1 and ω X /S -semistable of degree d . Therefore, there is a natural transformation of functors ¯ J ♯ω X ,d → ( ¯ J ♯d,g ) | X which factors through the natural action of Aut( X ) on ¯ J ♯ω X ,d (2.10) η : ¯ J ♯ω X ,d / Aut( X ) → ( ¯ J ♯d,g ) | X . It is easily since that η is a local isomorphism in the ´etale topology (using that every iso-trivial familybecomes trivial after an ´etale base change); therefore, passing to the varieties that co-represent theabove functors, we get an isomorphism(2.11) η : ¯ J ω X ,d / Aut( X ) ∼ = −→ Φ − ( X ) . (cid:4) Remark 2.13.
If char( k ) ≫ g , then the stabilizers of the GIT quotient (2.8) are linearly reductive(by Lemma 6.6(ii) and Corollary 4.3), which implies that the above GIT quotient is a universalcategorical quotient, and so the proof of Fact 2.12 still goes through. We ignore the question ofwhether, in small characteristic, the GIT quotient (2.8) remains universal and Fact 2.12 still holdstrue. . Deformation Theory
In the previous section, we studied the representability properties of global moduli functorsparameterizing rank 1, torsion-free sheaves on a (fixed or varying) nodal curve. This sectionfocuses on the analogous local topic: the pro-representability properties of deformation functorsparameterizing infinitesimal deformations of a rank 1, torsion-free sheaf on a (fixed or varying)nodal curve. The main result is Corollary 3.17, which explicitly describes miniversal deformationsrings parameterizing such deformations. The corollary is used in § The deformation functors.
We begin by reviewing the deformation functors of interest.
Definition 3.1.
Suppose we are given a k -scheme S , a finitely presented O S -module F , and alocal k -algebra A with residue field k . A deformation of the pair ( S, F ) over A is a quadruple( S A , F A , i, j ) that consists of(1) a flat A -scheme S A ;(2) a A -flat, finitely presented O S A -module F A ;(3) an isomorphism i : S A ⊗ A k ∽ −→ S ;(4) an isomorphism j : i ∗ ( F A ⊗ A k ) ∽ −→ F of O S -modules.The trivial deformation of a pair ( S, F ) over A is defined to be the quadruple ( S ⊗ k A, F ⊗ k A, i can , j can ). Here i can and j can are defined to be the canonical maps. If ( S ′ A , F ′ A , i ′ , j ′ ) is a seconddeformation of the pair ( S, F ), then an isomorphism from ( S A , F A , i, j ) to ( S ′ A , F ′ A , i ′ , j ′ ) is definedto be a pair ( φ, ψ ) that consists of(1) an isomorphism φ : S A ∽ −→ S ′ A over A such that i ′ ◦ ( φ ⊗
1) = i ;(2) an isomorphism ψ : φ ∗ ( F A ) ∽ −→ F ′ A of O S A ′ -modules such that j ′ ◦ i ′∗ ( ψ ⊗
1) = j .A deformation of the scheme S over A is defined by omitting the data of F A and j from thedefinition of a deformation of a pair. Similarly, an isomorphism from one deformation ( S A , i ) of S to another ( S ′ A , i ′ ) is defined by omitting ψ from Definition 3.1. The scheme S always admitsthe trivial deformation over A given by the pair ( S ⊗ k A, i can ).A deformation of a sheaf F over A is defined to be a pair ( F A , j ) such that the quadruple( S ⊗ k A, F A , i can , j ) is a deformation of the pair ( S, F ). An isomorphism from one deformation of F to another is defined to be a deformation of the associated deformations of the pair ( S, F ). Thetrivial deformation of the pair (
S, F ) may be considered as a trivial deformation of F .Let Art k be the category of artin local k -algebras with residue field k . Recall that a deformationfunctor is a functor F : Art k → Sets of artin rings with the property that F ( k ) is a singleton set.We study the following deformation functors. Definition 3.2.
Define functors Def S , Def F , Def ( S,F ) : Art k → Sets byDef ( S,F ) ( A ) := { iso. classes of deformations of ( S, F ) over A } , (3.1) Def S ( A ) := { iso. classes of deformations of S over A } , Def F ( A ) := { iso. classes of deformations of F over A } . The automorphism groups Aut(
S, F ), Aut( S ), and Aut( F ) act on appropriate deformationsfunctors, and this action will be studied in §
4. The reader should be familiar with the definitionsof Aut( S ) and Aut( F ), but perhaps not of Aut( S, F ). Definition 3.3. An automorphism of ( S, F ) is a pair ( σ, τ ) that consists of:(1) an automorphism σ : S ∽ −→ S ;(2) an isomorphism of sheaves τ : σ ∗ F ∽ −→ F . he group of automorphisms of ( S, F ), denoted by Aut(
S, F ), fits into the exact sequence(3.2) 0 → Aut( F ) → Aut(
S, F ) → Aut( S )( σ, τ ) σ. These automorphism groups act naturally on their respective functors.
Definition 3.4.
Let (
S, F ) be a given pair. Then we define the natural action of • Aut(
S, F ) on Def ( S,F ) by making an element ( σ, τ ) ∈ Aut(
S, F ) acts as( S A , F A , i, , j ) ( S A , F A , σ ◦ i, τ ◦ σ ∗ ( j )) . Here τ ◦ σ ∗ ( j ) is the composition σ ∗ i ∗ ( F A ⊗ A k ) σ ∗ ( j ) −→ σ ∗ ( F ) τ −→ F ; • Aut( S ) on Def S by making an element σ ∈ Aut( S ) acts as ( S A , i ) ( S A , σ ◦ i ); • Aut( F ) on Def F by making an element τ ∈ Aut( F ) acts as ( F A , j ) ( F A , τ ◦ j ) . Later we will relate the above deformation functors to the Quot scheme, so it is convenient tointroduce the deformation functors arising from the Quot scheme. To avoid irrelevant foundationalissues, we only define the deformation functors associated to nodal curves.
Definition 3.5.
Let X be a nodal curve; F a coherent sheaf on X ; and q : O ⊕ rX ։ F a surjection. A deformation of the pair ( X, q ) over A ∈ Art k is a quadruple ( X A , i, q A , j ) where q A : O rX A ։ F A is a surjection such that ( X A , F A , i, j ) is a deformation of ( X, F ) in the sense of Definition 3.1.Furthermore, we require that the isomorphism j : i ∗ ( F A ⊗ A k ) ∽ −→ F respects quotient maps, in thesense that q = j ◦ i ∗ ( q A ⊗ X ′ A , i ′ , q ′ A , j ′ ) of ( X, q ) over A , an isomorphism from ( X A , i, q A , j )to ( X ′ A , i ′ , q ′ A , j ′ ) is defined to be a pair ( φ, ψ ) consisting of(1) an isomorphism φ : X A ∽ −→ X ′ A over A ;(2) an isomorphism ψ : φ ∗ ( F A ) ∽ −→ F ′ A of O A ′ -modules such that ψ ◦ φ ∗ ( q A ) = q ′ A .A deformation of q over A ∈ Art k is defined to be a deformation of ( X, q ) of the form( X ⊗ k A, i can , q A , j ), where ( X ⊗ k A, i can ) is the trivial deformation. An isomorphism from onedeformation of q to another is defined to be an isomorphism of the associated deformations of( X, q ).The deformation functors Def q and Def ( X,q ) are defined in the expected manner. Definition 3.6.
We define functors Def q , Def ( X,q ) : Art k → Sets byDef ( X,q ) ( A ) := { iso. classes of deformations of ( X, q ) over A } , (3.3) Def q ( A ) := { iso. classes of deformations of q over A } . To study ¯ J d,g , we also need a slight generalization of Def ( X,q ) . Definition 3.7.
Suppose that X is a stable curve; F a coherent sheaf; q : O ⊕ rX ։ F a quotientmap; and p : X ֒ → P N is a 10-canonical embedding. A deformation of the pair ( p, q ) over A ∈ Art k is a quadruple ( p A , i, q A , j ), where p A : X A ֒ → P NA is closed embedding and ( X A , i, q A , j )is a deformation of the pair ( X, q ). We further require • the line bundles O X A (1) and ω ⊗ X A /A are isomorphic; • p A ⊗ p ◦ i .Given a second deformation ( p ′ A , i ′ , q ′ A , j ′ ) of ( p, q ), we define an isomorphism from the firstdeformation to the second to be an isomorphism ( φ, ψ ) of the associated deformations of ( X, q )with the property that p A = p ′ A ◦ φ. efinition 3.8. Define the functor Def ( p,q ) : Art k → Sets byDef ( p,q ) ( A ) := { iso. classes of deformations of ( p, q ) over A } . Note that there are forgetful transformations Def q → Def F and Def ( p,q ) → Def ( X,q ) → Def ( X,F ) that are formally smooth once F is sufficiently positive (see Lemma 6.3).The deformation functors we study are parameterized by complete local k -algebras. There areseveral different ways in which a complete local k -algebra can parameterize a deformation functor.We say that a functor Def : Art k → Sets is pro-representable if it is isomorphic to the formalspectrum functor(3.4) Spf( R ) : Art k → Sets A Hom loc ( R, A ) , for some complete local k -algebra R with residue field k . A pair ( R, π ) consisting of such an algebra R and an isomorphism π : Spf( R ) ∽ −→ Def is said to be a universal deformation ring for Def. Aneasy application of Yoneda’s lemma shows that if (
R, π ) and ( R ′ , π ′ ) are both universal deformationrings for Def, then there is a canonical isomorphism R ∼ = R ′ . An exercise in unraveling definitionsshows that the completed local ring of an appropriate Quot scheme is a deformation ring for Def q ,and similarly for Def ( X,q ) .The functors Def F and Def ( X,F ) are not always pro-representable, but do satisfy the weakercondition of admitting a miniversal deformation ring. Suppose that we are given a pair ( R, π )consisting of a complete local k -algebra R and a natural transformation π : Spf( R ) → Def. Wesay that (
R, π ) is a versal deformation ring for Def if π is formally smooth. If π has theadditional property that it induces an isomorphism on tangent spaces, then we say that ( R, π )is a miniversal (or semiuniversal) deformation ring . One can show that if (
R, π ) and ( R ′ , π ′ )are both miniversal deformation rings for Def, then R is isomorphic to R ′ , but in contrast to thesituation for deformation rings, there is no distinguished isomorphism R ∼ = R ′ . We now proceed toconstruct miniversal deformation rings for Def I and Def ( X,I ) .3.2. The miniversal deformation rings.
The existence of miniversal deformation rings for Def I and Def ( X,I ) can be deduced from theorems of Schlessinger, but for later computations, we willwant an explicit description of these rings. We derive such a description by relating Def I andDef ( X,I ) to the analogous deformation functors associated to the node O . We begin by fixing somenotation for the node. Definition 3.9.
The standard node O is the complete local k -algebra k [[ x, y ]] / ( xy ). The nor-malization of the standard node is denoted ˜ O .As a subring of the total ring of fractions Frac( O ), the normalization of O is equal to ˜ O = O [ x/ ( x + y )]. It follows that the quotient ˜ O / O is a 1-dimensional k -vector space spanned by theimage of x/ ( x + y ). Recall that ˜ O is also isomorphic to the ring k [[ x ]] ⊕ k [[ y ]], and the inclusion O → ˜ O factors as k [[ x, y ]]( xy ) → k [[ x ]] ⊕ k [[ y ]] ∽ −→ k [[ x, y ]]( xy ) (cid:20) xx + y (cid:21) where the first map is given by h ( x, y ) ( h ( x, , h (0 , y )) and the second map is given by ( f, g ) ( f x + gy ) / ( x + y ).Over O , there are exactly two rank 1, torsion-free modules up to isomorphism: the free moduleand a unique module that fails to be locally free. A proof of this statement can be found in [15],where it is deduced from [43, Thm. 3.1]. There are several ways to describe the module that failsto be locally free. efinition 3.10. The unique rank 1, torsion-free module I over O that fails to be locally freecan be described as any one of the following modules:(1) the ideal ( x, y ) ⊂ O , considered as an O -module,(2) the extension ˜ O ⊃ O , considered as an O -module,(3) the O -module with presentation h e, f : y · e = x · f = 0 i .An isomorphism from the 3rd module to the 1st module is given by e x , f y , while anisomorphism from the 3rd to the 2nd is given by e x/ ( x + y ), f y/ ( x + y ). In passingfrom one model of I to another, we will always implicitly identify the modules via these specificisomorphisms.3.2.1. Formal smoothness and reduction to the case of nodes. If I is a rank 1, torsion-free sheaf ona nodal curve X , then the study of Def I and Def ( X,I ) reduces to the study of Def I and Def ( O ,I ) .Indeed, say that Σ is the set of nodes where I fails to be locally free. For a given e ∈ Σ, let X e denote the spectrum of the completed local ring b O X,e and I e the pullback of I to X e . There areforgetful transformations relating global deformations to local deformations:Def ( X,I ) → Y e ∈ Σ Def ( X e ,I e ) , (3.5) Def I → Y e ∈ Σ Def I e , Def X → Y e ∈ Σ Def X e . All of these transformations are formally smooth. Indeed, for the last transformation, this is [14,Prop. 1.5]. That result together with [17, A.1-4] shows that the first transformation is formallysmooth. Essentially the same argument also shows that the middle transformation is formallysmooth, and this is a special case of [17, B.1].We now construct deformation rings for Def I and Def ( O ,I ) . We begin by parameterizingdeformations of ( O , I ). Definition 3.11.
Define S = S ( O , I ) := k [[ t, u, v ]] / ( uv − t ) . The deformation ( O S , I S , i, j ) of( O , I ) over S is defined by setting • O S := S [[ x, y ]] / ( xy − t ); • I S equal to the O S -module with presentation(3.6) I S := h ˜ e, ˜ f : y · ˜ e = − u · ˜ f , x · ˜ f = − v · ˜ e i ; • i : O S ⊗ S k ∽ −→ O equal to the isomorphism that is the identity on the variables x and y ; • j : i ∗ ( I S ⊗ S k ) ∽ −→ I equal to the isomorphism given by rules ˜ e ⊗ e and ˜ f ⊗ f .Deformations of I alone are parameterized similarly. Definition 3.12.
Define S = S ( I ) := k [[ u, v ]] / ( uv ). The algebraic deformation ( I S , j ) of I over S is defined by setting • O S = S [[ x, y ]] / ( xy ); • I S equal to the O S - module with presentation(3.7) I := h ˜ e, ˜ f : y · ˜ e = − u · ˜ f , x · ˜ f = − v · ˜ e i ; • j : i ∗ ( I S ⊗ S k ) ∽ −→ I equal to isomorphism given by rules ˜ e ⊗ e and ˜ f ⊗ f . emark 3.13. It may be more intuitive to describe the deformations in geometric terms. Thereis a versal deformation (resp. trivial deformation) X → B of the node, with base B = Spec k [ u, v, t ] / ( uv − t ) (resp. B = Spec k [ u, v ] / ( uv ))and total space X = B × Spec k [ x, y ] / ( xy − t ) (resp. X = B × Spec k [ x, y ] / ( xy )) . The module I S (resp. I S ) is essentially the “universal” ideal I = ( x − u, y − v ) ⊆ Γ( X , O X )considered as a module as in Definition 3.10 (3). Lemma 3.14. S is a miniversal deformation ring for Def ( O ,I ) . More precisely, the algebraicdeformation ( O S , i, I S , j ) defines a transformation Spf( S ) → Def ( O ,I ) that realizes S as theminiversal deformation ring for Def I . Similarly, S is a miniversal deformation ring for Def I .Proof. The claim concerning the ring S was established in the course of proving Proposition 2.6of [11]. The same argument holds for S provided that one replaces the standard irreducible, nodalplane cubic used in that proof with a general pencil containing such a curve. (cid:4) Given a rank 1, torsion-free sheaf I that fails to be locally free at a set of nodes Σ, there is asimple relation between Def I and Q e ∈ Σ Def I e . Definition 3.15.
Let Def l . t .I ⊂ Def I be the subfunctor parameterizing deformations that map tothe trivial deformation under Def I → Q e ∈ Σ Def I e . Define Def l . t . ( X,I ) similarly. Elements of thesedeformation functors (valued in a given ring) are called locally trivial deformations (over thatring). Lemma 3.16.
Let X be a nodal curve; Σ a set of nodes; g : X Σ → X the map that normalizes thenodes Σ ; and I := g ∗ ( L ) the direct image of a line bundle L on X Σ . Then the rule (3.8) Def L ( A ) −→ Def l . t .I ( A )( L A , j ) (( g × id) ∗ ( L ) , ( g × id) ∗ ( i )) for any A ∈ Art k , defines an isomorphism Def L ∼ = −→ Def l . t .I .Proof. The map Def L → Def I defined by Eqn. (3.8) has the property that the composition Def L → Def I → Q e ∈ Σ Def I e is the trivial map, so there is an induced map Def L → Def l . t .I . Studying themap Def I → Q Def I e and the associated map on tangent-obstruction theories, one can show usingthe local-to-global spectral sequence for Ext that Def l . t .I is formally smooth with tangent space T (Def l . t .I ) = H (End( I )). This vector space is just H ( X Σ , O X Σ ) (see e.g. the proof of Lemma 4.2),which can be identified with the tangent space to Def L in such a way that T (Def L ) → T (Def l . t .I ) isthe identity. By formal smoothness, it follows that Def L → Def l . t .I is an isomorphism. (cid:4) Let us denote by R the miniversal deformation ring of Def I and by R the miniversal deformationring of Def ( X,I ) (which exists by, say, [17, § A]). Lemma 3.14 together with the discussion followingEqn. (3.5) allows us to describe the miniversal deformation rings R and R as follows. Corollary 3.17.
Let X be a nodal curve; I a rank , torsion-free sheaf on X ; and Σ the setof nodes where I fails to be locally free. For every e ∈ Σ , fix an identification of ( ˆ O X,e , I e ) with ( O , I ) . Then the forgetful transformations in Eqn. (3.5) induce inclusions [ O e ∈ Σ k [[ U ← e , U → e ]] / ( U ← e U → e ) ∼ = dO e ∈ Σ S ֒ → R , [ O e ∈ Σ k [[ U ← e , U → e , T e ]] / ( U ← e U → e − T e ) ∼ = dO e ∈ Σ S ֒ → R , nd each inclusion realizes the larger ring as a power series ring over the smaller ring. Automorphism Groups and Their Actions
Automorphism groups appeared in the previous section, where we defined group actions on defor-mation functors (Def. 3.4). Here we study the structure of these groups with the aim of collectingresults to use in §
5. There we will study the problem of lifting the action of an automorphismgroup on a deformation functor to an action on a miniversal deformation ring. The existence of alift follows from a theorem of Rim if the automorphism group is known to be linearly reductive.Thus, the focus of this section is on showing that the automorphism groups of interest are linearlyreductive.We begin by studying automorphisms of the node O (Def. 3.9) and its unique rank 1, torsion-free module I that fails to be locally free (Def. 3.10). The automorphism group Aut( X , I ) fitsinto the exact sequence(4.1) 0 −−−−→ Aut( I ) −−−−→ Aut( X , I ) −−−−→ Aut( X ) −−−−→ , and the group Aut( I ) admits the following explicit description. Lemma 4.1.
Consider I as the normalization ˜ O . Then the natural action of ˜ O ∗ on I inducesan isomorphism ˜ O ∗ ∽ −→ Aut( I ) .Proof. We claim that every O -linear map φ : I → I is ˜ O -linear. It is enough to show that φ commutes with multiplication by x/ ( x + y ), and this is clear: for all s ∈ I , we have( x + y ) · φ ( x/ ( x + y ) · s ) = φ ( x · s ) = x · φ ( s ) . Dividing by x + y , we obtain the desired equality. Thus, Aut( I ) coincides with the group of˜ O -linear automorphisms, which equals ˜ O ∗ . (cid:4) The action of ˜ O ∗ can also be described in terms of the presentation from Definition 3.10. Atypical element f ∈ ˜ O ∗ can be uniquely written as f = α xx + y + β yx + y + g ( x, y ), with α, β ∈ k ∗ and g ( x, y ) ∈ ( x, y ) ⊂ O , and this element acts by e ( α + g ( x, e, f ( β + g (0 , y )) f. We now turn to the global picture. Let I be a rank 1, torsion-free sheaf on a nodal curve X . SetΣ equal to the set of nodes where I fails to be locally free. In analogy with Eqn. (4.1), Aut( X, I )fits into the following exact sequence:(4.2) 0 −−−−→
Aut( I ) −−−−→ Aut(
X, I ) −−−−→ Aut( X ) . We describe Aut(
X, I ) by describing the outermost groups.Consider first Aut( X ). Without more information, we can only describe the rough features ofthis group. For X stable (the main case of interest), Aut( X ) is a finite, reduced group scheme ([14,Thm. 1.11]), and if we additionally assume that X is general and of genus g ≥
3, then this group istrivial. However, Aut( X ) can be highly non- trivial for special curves: see [42] for a sharp boundon the cardinality of Aut( X ) in terms of the genus g , and for a description of the curves attainingthe bounds.The group Aut( I ) admits the following explicit description. In the notation from § I ) → Aut( I e ) for every e ∈ Σ, and we use this map to describe Aut( I ). Lemma 4.2.
Let X be a nodal curve; I a rank , torsion-free sheaf; Σ the set of points where I fails to be locally free; and g : X Σ → X the map that normalizes the nodes Σ . Then there is a nique isomorphism H ( X Σ , O ∗ X Σ ) ∼ = Aut( I ) that extends the inclusion of H ( X, O ∗ X ) in Aut( I ) and makes the diagram H ( X Σ , O ∗ X Σ ) ∼ = / / (cid:15) (cid:15) Aut( I ) (cid:15) (cid:15) ˜ O ∗ X,e ∼ = / / Aut( I e ) commute for all e ∈ Σ . Here ˜ O X,e is the normalization of the completed local ring at e , thehorizontal maps are isomorphisms, and the vertical maps are restrictions.Proof. Given I , we prove the stronger statement that End( I ) is canonically isomorphic to g ∗ ( O X Σ ).Because I is torsion-free, End( I ) injects into End( I ⊗ Frac( O X )), which equals Frac( O X ) as I isrank 1. Thus, End( I ) is a finitely generated, commutative O X - algebra satisfying O X ⊂ End( I ) ⊂ Frac( O X ). Furthermore, an application of the Cayley-Hamilton Theorem shows that a local sectionof End( I ) satisfies a monic equation whose coefficients are local sections of O X . We may concludethat End( I ) ⊂ ν ∗ ( O ˜ X ), where ν : ˜ X → X is the (full) normalization. To complete the proof, it isenough to show that the support of ν ∗ ( O ˜ X ) / End( I ) is precisely Σ. However, this can be checkedon the level of completed stalks, and so we may deduce the claim from the Lemma 4.1. The resultnow follows by taking global sections of End( I ) and passing to units. (cid:4) One consequence of the previous two lemmas is that many of the groups appearing in thispaper are linearly reductive. Recall that the ground field k may have positive characteristic, andin positive characteristic linear reductivity is a strong condition to impose. Indeed, while manyalgebraic groups (e.g. GL r , SL r , . . . ) are linearly reductive in characteristic 0, Nagata has shownthat the only linearly reductive groups in characteristic p > G whose identitycomponent G is a multiplicative torus and whose ´etale quotient G/G has prime-to- p order. Wenow list the groups we have shown satisfy this condition. Corollary 4.3.
Let X be a nodal curve and I a rank , torsion-free sheaf. Then the followinggroups are reduced and linearly reductive: • the automorphism group Aut( I ) ; • the quotient group Aut( I ) / (1 + ( x, y ) O ) ; • the automorphism group Aut(
X, I ) when X is stable and does not admit an order p =char( k ) automorphism.Proof. Lemma 4.1 shows Aut( I ) / (1 + ( x, y ) O ) is a multiplicative torus, and Lemma 4.2 shows thesame is true for Aut( I ). Given this, an inspection of Eqn. (4.2) proves that Aut( X, I ) is linearlyreductive. (cid:4) Group Actions on Rings
In this section we show that, in the cases of interest, the actions on deformation functors fromDefinition 3.4 lift to unique actions on miniversal deformation rings (Fact 5.4), which we thencompute (Thm. 5.10). These results are used in §
6, where we show that the action on the miniver-sal deformation ring can be described using the GIT construction of the compactified Jacobian(Lemma 6.4, Lemma 6.6). We then use this observation to deduce the main theorem of the paper(Thm. 6.1). Key to this section are the linear reductivity results from the previous section.We begin by showing that certain actions are trivial.
Lemma 5.1.
The action of x, y ) O ⊂ Aut( I ) on Def I is trivial. roof. Suppose we are given A ∈ Art k and a deformation ( I A , j ) of I over A . Given τ ∈ x, y ) O ,we must show that ( I A , j ) and ( I A , τ − ◦ j ) are isomorphic deformations. But this is clear: τ liesin O , and multiplication by τ ⊗ ∈ O ⊗ k A defines an isomorphism ( I A , j ) ∽ −→ ( I A , τ − ◦ j ). (cid:4) Essentially the same argument proves the following two lemmas.
Lemma 5.2.
Let X be a nodal curve and I a rank , torsion-free sheaf on X . Then the subgroup G m ⊂ Aut( I ) of scalar automorphisms acts trivially on Def I . Under the inclusion (4.2) , G m alsoacts trivially on Def ( X,I ) .Proof. We give a proof for Def I ; the case of Def ( X,I ) is similar, and left to the reader. If ( I A , j ) isa deformation of I and τ ∈ G m ⊂ Aut( I ) a scalar automorphism, then τ trivially extends to anautomorphism ˜ τ of I A that defines an isomorphism of ( I A , j ) with ( I A , τ − ◦ j ). (cid:4) Lemma 5.3.
Let X be a nodal curve and I a rank , torsion-free sheaf. Then Aut( I ) acts triviallyon the subfunctor Def l . t .I ⊂ Def I . Under the inclusion (4.2) , Aut( I ) also acts trivially on thesubfunctor Def l . t . ( X,I ) ⊂ Def ( X,I ) .Proof. The lemma is a consequence of Lemmas 4.1 and 3.16. (cid:4)
We may now invoke a theorem of Rim to show that the actions uniquely lift to actions onminiversal deformation rings.
Fact 5.4 (Rim [39]) . Let X be a nodal curve and I a rank , torsion-free sheaf. Then:(i) there is a unique action of Aut( I ) on the miniversal deformation ring S (resp. S ) that makesthe map Spf( S ) → Def I (resp. Spf( S ) → Def ( O ,I ) ) equivariant and has the property thatthe subgroup x, y ) O ⊂ ˜ O ∗ = Aut( I ) acts trivially;(ii) there is a unique action of Aut( I ) on the miniversal deformation ring R of Def I that makes Spf( R ) → Def I equivariant;(iii) there is a unique action of Aut(
X, I ) on the miniversal deformation ring R of Def ( X,I ) thatmakes Spf( R ) → Def ( X,I ) equivariant, provided that X is stable and it does not admit anorder p = char( k ) automorphism.Proof. This is a special case of [39, p. 225]. Indeed, the functors Def I , Def I , Def ( O ,I ) and Def ( X,I ) are all examples of a deformation functor F associated to a “homogeneous fibered category ingroupoid” satisfying a finiteness condition. Given an action of a linearly reductive group on such acategory, there is an induced action on F , and Rim’s Theorem asserts that there exists a miniversaldeformation ring R that admits an action of G making Spf( R ) → F equivariant. Furthermore, asan algebra with G -action, R is unique up to a (non-unique) isomorphism.One may verify that the actions on Def I , Def I , Def ( O ,I ) and Def ( X,I ) are defined on the levelof groupoids. The claims concerning R and R follows immediately because we have shown thatAut( I ) and Aut( X, I ) are linearly reductive. The group Aut( I ) is certainly not linearly reductive,but Lemma 5.1 asserts that this group acts through its linearly reductive quotient Aut( I ) / (1 +( x, y ) O ). Case (i) then follows as well. (cid:4) The actions described by the Fact 5.4 are, of course, unique only up to a non-unique isomorphism.Because of the non-uniqueness, it is not immediate that the group action is functorial. This issueis addressed in the lemma below.
Lemma 5.5.
Let X be a nodal curve and I a rank , torsion-free sheaf. For every point e ∈ X where I fails to be locally free, fix an isomorphism between ( b O X,e , I ⊗ b O X,e ) and ( O , I ) . Then therestriction transformations (5.1) Def I → Y e ∈ Σ Def I resp. Def ( X,I ) → Y e ∈ Σ Def ( O ,I )20 ift to transformations of miniversal deformation rings Spf( R ) → Y e ∈ Σ Spf( S ) resp. Spf( R ) → Y e ∈ Σ Spf( S ) that are equivariant with respect to the homomorphism (5.2) Aut( I ) → Y e ∈ Σ Aut( I ) and the actions of Aut( I ) and Aut( I ) described in Fact 5.4.Proof. The only condition that is not immediate is that the natural transformations can be chosento be equivariant. We give the proof for Spf( R ) and leave the task of extending the argument toSpf( R ) to the interested reader.As Spf( S ) → Def I is formally smooth, there exists a lift Spf( R ) → Q Spf( S ) of the forgetfultransformation Def I → Q Def I , and such a lift is automatically formally smooth. Writing R asa power series ring over b ⊗ S , it is easy to see that there exists an action of Aut( I ) on Spf( R )that makes Spf( R ) → Q Spf( S ) equivariant and has the property that the induced action on thetangent space T (Spf( R )) coincides with the natural action on T (Def I ). To complete the proof, wemust show that this action makes Spf( R ) → Def I equivariant, and hence satisfies the conditionsof Fact 5.4.Consider the composition Spf( R ) → Q Spf( S ) → Q Def I . This transformation is formallysmooth and hence realizes R as a (non-minimal) versal deformation ring for Def I . Furthermore,the constructed action of Aut( I ) on R makes Spf( R ) → Q Def I equivariant and induces thestandard action on T( R ) = T(Def I ). A second action on R with this property is the uniqueaction that makes Spf( R ) → Def I equivariant. An inspection of Rim’s proof shows that theuniqueness statement in Fact 5.4 still holds if the miniversality hypothesis is weakened to versality,provided the action on the tangent space is specified. In particular, there is an automorphism of R transforming the first action into the second. We can conclude that the map in (5.2) and the actionin Fact 5.4 can be chosen so that Spf( S ) → Def I is equivariant. This completes the proof. (cid:4) We now compute the actions described by Fact 5.4. Let us start with the action of Aut( I ) on S . Lemma 5.6.
In terms of the presentation from Definitions 3.12, 3.11, define an action of
Aut( I ) on S and S by making τ = a xx + y + b yx + y + g ∈ Aut( I ) act as u ab − · u, v a − b · v, t t. Here a, b ∈ k ∗ and g ∈ ( x, y ) ˜ O . Then this action is the unique action described by Fact 5.4 (i).Proof. We give a proof for the case of S ; the case of S is similar, and left to the reader. The ruleabove is easily seen to define an action of Aut( I ) on S with the property that 1 + ( x, u ) O actstrivially, so we need only show that this action makes Spf( S ) → Def I into an equivariant map.In fact, it is enough to verify this for the subgroup of Aut( I ) that consists of elements of the form τ := a xx + y + b yx + y because this subgroup maps isomorphically onto Aut( I ) / (1 + ( x, y ) O ).Given such a τ , what is the pullback of the miniversal deformation ( I S , i ) under τ ? It is themodule with presentation(5.3) h ˜ e ′ , ˜ f ′ : y · ˜ e ′ = − a − bu · ˜ f ′ , x · ˜ f ′ = − ab − v · ˜ e ′ i , together with the identification j sending ˜ e ′ e , ˜ f ′ f . One isomorphism between this deforma-tion and the deformation ( I S , τ − ◦ j ) is˜ e ′ b − ˜ e, ˜ f ′ a − ˜ f . his completes the proof. (cid:4) We now turn our attention to the action of Aut( I ) on R . It is convenient to introduce somecombinatorial language. Definition 5.7.
Let e ∈ Σ be a node that lies on the intersection of the irreducible components v and w . Write ← e for the pair ( v, w ) and → e for the pair ( w, v ). Define s, t : { → e , ← e } → { v, w } to beprojection onto the first component and onto the second component respectively.This notation is intended to be suggestive of graph theory. We may consider v and w as beingvertices of the dual graph Γ X that are connected by an edge corresponding to e . The pairs ← e and → e should be thought of as orientations of this edge, and the maps s and t are the “source” and“target” maps sending an oriented edge to its source vertex and its target vertex respectively. Therelation with graph theory is developed more systematically by the authors in [13].The group Aut( I ) can also be described using similar notation. Definition 5.8.
Let X be a nodal curve, I a rank 1, torsion-free sheaf, Σ the set of nodes where I fails to be locally free, and V the set of irreducible components of X . Define T Σ to be the subgroup T Σ ⊂ Y v ∈ V G m that consists of sequences ( λ v ) with the property that λ v = λ v for every two components v and v whose intersection contains some node not in Σ. Remark 5.9.
The torus T Σ is isomorphic to Aut( I ) = H ( X Σ , O ∗ X Σ ) (Lemma 4.2). Indeed, theelement λ = ( λ v ) ∈ T Σ corresponds to the regular function f ∈ H ( X Σ , O ∗ X Σ ) that is equal tothe constant λ v on the component v . It is convenient to have the following explicit isomorphismof Aut( I ) with a split torus. Let Γ X be the dual graph of X and let Γ = Γ X (Σ) be the dualgraph of a curve obtained from X by smoothing the nodes not in Σ. There is a map of vertices c : V (Γ X ) → V (Γ) ([13, § φ : T Γ := Y v ∈ V (Γ) G m ∽ −→ T Σ = Aut( I ) ⊆ Y w ∈ V (Γ X ) G m defined as follows. Given ( g v ) ∈ Q v ∈ V (Γ) G m , set φ (( g v )) w = g c ( w ) for each w ∈ V (Γ X ).We use the description of Aut( I ) as T Σ to describe the action of Aut( I ) on R and on R . Theorem 5.10.
Let X be a nodal curve; I a rank , torsion-free sheaf; Σ the set of nodes where I fails to be locally free; and g Σ := h ( X Σ , O X Σ ) the arithmetic genus of X Σ . Then:(i) Define an action of T Σ = Aut( I ) on R (Σ) := k [[ { U ← e , U → e : e ∈ Σ } ; W , . . . , W g Σ ]] / ( U ← e U → e : e ∈ Σ) by making λ ∈ T Σ act as (5.4) U → e λ s ( → e ) · U → e · λ − t ( → e ) , U ← e λ s ( ← e ) · U ← e · λ − t ( ← e ) , W i W i . Then there exists an isomorphism R ∼ = R (Σ) that identifies the above action of T Σ on R (Σ) with the action of Aut( I ) on R from Fact 5.4.(ii) Suppose Aut( X ) is trivial, and define an action of T Σ = Aut( X, I ) on R (Σ) := k [[ { U ← e , U → e , T e : e ∈ Σ } ; W , . . . , W m ]] / ( U ← e U → e − T e : e ∈ Σ) for some m ∈ Z ≥ by making λ ∈ T Σ act as in (5.4) and as T e T e . Then there exists anisomorphism R ∼ = R (Σ) that identifies the above action of T (Σ) on R (Σ) with the actionof Aut(
X, I ) on R from Fact 5.4. emark 5.11. Let Γ = Γ X (Σ) be the dual graph of any curve obtained from X by smoothing thenodes not in Σ. Then one can check that in the notation of the theorem above, g Σ = g ( X ) − b (Γ).It is also easy to see that the action of T Γ on R I and R ( X,I ) defined in Theorem A agrees with theaction of T Σ defined above. Proof.
This is a consequence of results already proven in his section. We only prove the statementabout R and leave the task of extending the proof to R to the interested reader.Suggestively set(5.5) S (Σ) := k [[ U ← e , U → e : e ∈ Σ]] / ( U ← e U → e : e ∈ Σ) . This is a miniversal deformation ring for Q Def I , where the product runs over the elements ofΣ. If we fix an isomorphism between ( b O X,e , I ⊗ b O X,e ) and ( O , I ) for every node e ∈ Σ, then byCorollary 3.17 and Lemma 5.5, there exists an equivariant map S (Σ) ֒ → R realizing R as a powerseries ring over S (Σ). To complete the proof, we need to show that there exists an expression of R as a power series ring generated by variables invariant under the group action.Thus, consider the map from the cotangent space of Spf( R ) to the cotangent space of Def l . t .I .This is an equivariant map, and the action of Aut( I ) on the target space is trivial (Lemma 5.3).Because Aut( I ) is linearly reductive, we can find invariant elements ¯ W , . . . , ¯ W g Σ ∈ R whoseimages in the cotangent space m / m map isomorphically onto the cotangent space of Def l . t .I .Letting W , . . . , W g Σ denote indeterminates, define a map φ : S (Σ)[[ W , . . . , W g Σ ]] → R by sending W i to ¯ W i . The target and source of φ are isomorphic, and the induced map on tangentspaces is an isomorphism, hence φ itself must be an isomorphism.Furthermore, if we make T Σ act on S (Σ)[[ W , . . . , W g Σ ]] by making the group act trivially onthe indeterminates, then φ is equivariant. The ring S (Σ)[[ W , . . . , W g Σ ]], together with this groupaction, is nothing other than R (Σ), so the proof is complete. (cid:4) Observe that the theorem computes the action of Aut(
X, I ) on R when X is automorphism-free. It would interesting to compute the action when X is stable, but possibly admits non-trivialautomorphisms. Indeed, such a result (combined with a suitable extension of Theorem 6.1) wouldallow us to remove the hypothesis that X does not have an automorphism from Theorem A. When X does not admit an automorphism of order p = char( k ), Fact 5.4 states that there is a uniqueaction of Aut( X, I ), so the problem is to modify the action described in Theorem 5.10 to incorporateAut( X ). The case where X admits an order p = char( k ) automorphism is more challenging forthen we can no longer cite Rim’s work to assert that Aut( X, I ) acts on R or to assert that suchan action, if it exists, is unique. Simply knowing if R still admits an unique action of Aut( X, I )would be interesting. More generally, it would be interesting to know if Rim’s Theorem remainstrue if the assumption that the group G acting is linearly reductive is weakened.6. Luna Slice Argument
We now prove that the invariant subrings in Theorem 5.10 are isomorphic to the completed localrings of the compactified Jacobians. The main result is the following.
Theorem 6.1.
Let X be a nodal curve and I a rank , torsion-free sheaf.(i) Let ¯ J ( X ) be a compactified Jacobian of X and assume that I is polystable with respect to theassociated stability condition. Then the action from Fact 5.4 of Aut( I ) on the deformationring R parameterizing deformations of I satisfies b O ¯ J ( X ) , [ I ] ∼ = R Aut( I )1 . ii) Assume X is stable and does not admit an order p automorphism, and I is slope polystablewith respect to the dualizing sheaf ω X . Then the action of Aut(
X, I ) on the deformation ring R satisfies b O ¯ J d,g , [( X,I )] ∼ = R Aut(
X,I )2 . In the theorem, the isomorphisms between the complete local rings are non-canonical, but this isnecessarily so as the rings R and R are themselves only defined up to non-canonical isomorphism. Remark 6.2.
Observe that Theorem 6.1, together with Theorem 5.10, establishes Theorem A(see also Remarks 5.9, 5.11). An elementary argument in GIT shows that the ring [ B (Γ) T Γ definedin Theorem A has dimension b (Γ) + E (Γ). Since ¯ J d,g has dimension 4 g −
3, it follows that m = 4 g − − b (Γ) − E (Γ) in Theorem 5.10.The proof of Theorem 6.1 is given at the end of the section, where it is deduced from the followingsequence of lemmas. Lemma 6.3.
Let X be a nodal curve; I a rank , torsion-free sheaf; and q : O ⊕ rX ։ I a surjection.If H ( X, I ) = 0 , then the forgetful morphism
Def q → Def I is formally smooth. Assume furtherthat X is stable and p : X ֒ → P N is a -canonical embedding. Then Def ( p,q ) → Def ( X,I ) is formallysmooth.Proof. We prove the statement about Def q → Def I and leave the proof for Def ( p,q ) → Def ( X,I ) to the interested reader. Given a surjection B ։ A of artin local k -algebras, a deformation( I B , j ) of I over B , and a deformation ( q A , j ) of q such that the associated deformation of I isisomorphic to ( I B ⊗ B A, j ⊗ q B , j ) extending( q A , j ) and inducing ( I B , j ). A filtering argument shows that the vanishing H ( X, I ) = 0 impliesthat H ( X B , I B ) → H ( X A , I A ) is surjective. Now suppose s , . . . , s r ∈ H ( I A ) is the image of thestandard basis for H ( X A , O ⊕ rX A ). If we lift these elements to ˜ s , . . . , ˜ s r ∈ H ( X B , I B ) and define q B : O ⊕ rX B ։ I B to be the map the sends the i -th standard basis element to ˜ s i , then ( q B , j ) has thedesired properties. (cid:4) We now relate R and R to the appropriate Quot schemes. Lemma 6.4.
Let X be a nodal curve X ; I a rank , torsion-free sheaf I ; q : O ⊕ rX ։ I a quotientmap corresponding to a point ˜ x ∈ Quot( O ⊕ rX ) . Assume: • H ( X, I ) = 0 ; • q : H ( X, O ⊕ rX ) → H ( X, I ) is an isomorphism.(i) If Z is a slice through ˜ x in some invariant affine open neighborhood ˜ x ∈ U ⊆ Quot( O ⊕ rX ) ,then the completed local ring b O Z, ˜ x of Z at ˜ x is a miniversal deformation ring for Def I .(ii) Assume additionally that X is stable. Let p : X ֒ → P N be a -canonical embedding with ( p, q ) corresponding to the point ˜ y of the relative Quot scheme Quot( O ⊕ rX g ) (as in the proofof Fact 2.11). If Z is a slice through ˜ y in some invariant affine open neighborhood ˜ y ∈ V ⊆ Quot( O ⊕ rX g ) , then the completed local ring of b O Z, ˜ y of Z at ˜ y is a miniversal deformation ringfor Def ( X,I ) .Proof. We prove the statement relating Quot( O ⊕ rX ) to Def I and leave the task of extending theargument to Def ( X,I ) to the interested reader. The necessary changes are primarily notational(e.g. the action of SL r must be replaced with that of SL r × SL N +1 ).Temporarily set F equal to the functor pro-represented by b O Z,x . There is a natural forgetfulmap Def q → Def I , and our goal is to show that the restriction of this map to F is formally smoothand an isomorphism on tangent spaces. We do this by proving that F ( A ) → Def I ( A ) is injective or A = k [ ǫ ] and has the same image as Def q ( A ) → Def I ( A ) for all A ∈ Art k . Because Def q → Def I is formally smooth (Lemma 6.3), the lemma will then follow.The desired facts are proven by studying the action of the lie algebra of SL r on deformations.Set sl r equal to the deformation functor pro-represented by the completed local ring of SL r at theidentity and h equal to the deformation functor associated to the stabilizer H := Stab(˜ x ) ⊂ SL r .There is a natural map sl r / h → Def q given by the derivative of the orbit map. Concretely, this isdefined by the rule g g · v triv , where v triv is the trivial deformation (over an unspecified artin localalgebra). Because U admits a slice, there exists a morphism Def q → sl r / h that is a contractiononto the orbit in the sense that the derivative of the orbit map defines a section. Furthermore, thismorphism has the property that the preimage of the trivial element 0 ∈ sl r / h ( A ) is F ( A ) ⊂ Def q ( A ).The construction of the morphism is immediate: the scheme Z × H SL r admits a global contractionmorphism given by projection onto the second factor, and the desired infinitesimal contraction isobtained by choosing a local inverse of Z × H SL r → Quot( O ⊕ rX ).We can use the contraction morphism to deduce the second claim, that Def q ( A ) → Def I ( A ) and F ( A ) → Def I ( A ) have the same image. Indeed, if v ∈ Def q ( A ) maps to an element of sl r / h ( A )represented by g ∈ sl r ( A ), then g − · v lies in F ( A ). Because both v and g − · v map to the sameelement of Def I ( A ), we have proven the claim.We also need to verify that F ( k [ ǫ ]) → Def I ( k [ ǫ ]) is injective. This too can be proven using thecontraction map, but we must first relate the kernel of F ( k [ ǫ ]) → Def I ( k [ ǫ ]) to the contraction.Specifically, we claim the kernel equals the image of the orbit map. It is immediate that the imageis contained in the kernel, but the reverse inclusion requires more justification. Thus, suppose( q : O ⊕ rX → I , j ) is a 1st order deformation with the property that ( I , j ) is the trivial deformation.Because q induces an isomorphism on global sections, we can chose bases and use the identification j to represent q : H ( X, O ⊕ rX ) → H ( X, I ) by a matrix g that reduces to the identity modulo ǫ .The matrix g may not lie in sl r ( k [ ǫ ]), but if we set δ := det( g ), then the product δ · g − does. Onemay check that δ · g − maps to the deformation represented by ( q : O ⊕ rX ։ I , j ), establishing thereverse inclusion.We now prove injectivity by showing directly that the image of the orbit map has trivial inter-section with F ( k [ ǫ ]). Given v in this intersection, the image in sl r / h ( k [ ǫ ]) under the contractionmorphism is zero because v lies in F ( k [ ǫ ]). But, as v also lies in the image of the orbit map, theimage under the composition Def q ( k [ ǫ ]) → sl r / h ( k [ ǫ ]) → Def q ( k [ ǫ ]) of the contraction map withthe orbit map is v . Thus, v = 0, and the proof is complete. (cid:4) The following definition and lemma relate the stabilizer of a point of the Quot scheme to anautomorphism group.
Definition 6.5.
Let I be a rank 1, torsion-free sheaf on a nodal curve X ; q : O ⊕ rX ։ I a quotientmap corresponding to a point ˜ x belonging to some Quot scheme Quot( O ⊕ rX ). Assume • I is generated by global sections; • q : H ( X, O ⊕ rX ) → H ( X, I ) is an isomorphism.(i) If Stab(˜ x ) ⊂ SL r is the stabilizer under the natural action on Quot( O ⊕ rX ), then the naturalhomomorphism Stab(˜ x ) → Aut( I )is defined by sending g ∈ Stab(˜ x ) into the unique automorphism α ( g ) : I → I with the propertythat α ( g ) ◦ q = q ◦ g − (which exists since I is generated by the image of H ( X, O ⊕ rX )).Assume additionally that X is stable. Let p : X ֒ → P N be a 10-canonical embedding with ( p, q )corresponding to the point ˜ y of the relative Quot scheme Quot( O ⊕ rX g ) (as in the proof of Fact 2.11). ii) If Stab(˜ y ) ⊂ SL r × SL N +1 is the stabilizer under the natural action on Quot( O ⊕ rX g ), thenthe natural homomorphism Stab(˜ y ) → Aut(
X, I )is defined by sending g = ( g , g ) ∈ Stab(˜ y ) into the unique element α ( g ) = ( α ( g ) , α ( g )) ∈ Aut(
X, I ) such that p ◦ α ( g ) = g ◦ p and α ( g ) ◦ α ( g ) ∗ ( q ) = q ◦ g − . Lemma 6.6.
Same notation as in Definition 6.5.(i) The natural homomorphism
Stab(˜ x ) → Aut( I ) is injective with image equal to the subgroup Aut ( I ) ⊂ Aut( I ) consisting of the automorphisms τ ∈ Aut( I ) with the property that theinduced automorphism H ( X, I ) → H ( X, I ) has determinant +1 .(ii) The the natural homomorphism Stab(˜ y ) → Aut(
X, I ) is injective with image equal to thesubgroup Aut ( X, I ) ⊂ Aut(
X, I ) consisting of the automorphisms ( σ, τ ) ∈ Aut(
X, I ) withthe property that the induced automorphism H ( X, I ) can −→ H ( X, σ ∗ ( I )) τ −→ H ( X, I ) hasdeterminant +1 .Proof. As in the last proof, we only prove the statement for Stab(˜ x ) and leave the case of Stab(˜ y )to the interested reader. Set s , . . . , s r ∈ H ( X, I ) equal to the image of the standard basis for H ( X, O ⊕ rX ). We first show injectivity. Given g ∈ Stab(˜ x ), write ( a i,j ) := g − . Then α ( g ) satisfies(6.1) α ( g )( s i ) = a i, s + · · · + a i,r s r . If α ( g ) is the identity, then we must have α ( g )( s i ) = s i for all i . But the s i ’s form a basis, sothis is only possible if g = id r , showing injectivity. Similarly, given an α ∈ Aut( I ) that induces adeterminant +1 automorphism of H ( X, I ), define scalars a i,j as in Eqn. (6.1). Then g := ( a i,j ) − ∈ SL r is an element of Aut ( I ) with α ( g ) = α . This completes the proof. (cid:4) The last lemma we need asserts that the formation of the relevant group quotients commuteswith completion.
Lemma 6.7.
Let Z be an affine algebraic scheme, ˜ x ∈ Z a point, and H an algebraic group actingon Z that fixes ˜ x . Assume H is linearly reductive. Then the formation of H -invariants commuteswith completion, i.e. if we call x the image of ˜ x in Z/H , then we have b O HZ, ˜ x ∼ = b O Z/H,x . Proof.
This is an exercise in linear reductivity. The quotient map induces a local homomorphism b O Z/H,x → b O Z, ˜ x . Because ˜ x is a fixed point, H acts continuously on b O Z, ˜ x , and passing to invariants,we may replace the target of this map with b O HZ, ˜ x . Our goal is to show that the resulting map is anisomorphism.For injectivity, say r ∈ b O Z/H,x lies in the kernel. By picking a sequence { r i } ∞ i =1 in O Z/H,x converging to r and studying the valuation of r i , one can show that r = 0. Surjectivity requiresmore work.Given r ∈ b O HZ, ˜ x , consider the reduction map b O HZ, ˜ x → b O Z, ˜ x / m i +1˜ x . The element r maps to an H -invariant element ¯ r in the target, which is canonically isomorphic to O Z, ˜ x / m i +1˜ x . Fixing anequivariant splitting of O Z, ˜ x → O Z, ˜ x / m i +1˜ x (which exists by linear reductivity), we can lift ¯ r to aninvariant element r i of O Z, ˜ x . The collection of all these elements defines a sequence { r i } ∞ i =1 whoselimit is r . Furthermore, every term in the sequence lies in b O Z/H,x ; thus the limit must lie in thisring as well. This completes the proof. (cid:4)
Proof of Theorem 6.1.
The proof is an application of the Luna Slice Theorem, together with theprevious lemmas. As usual, we only give the proof for a compactified Jacobian of a fixed nodalcurve and leave the task of extending the argument to the universal compactified Jacobian to the nterested reader (replacing Lemma 6.4(i) with Lemma 6.4(ii) and Lemma 6.6(i) with Lemma 6.6(ii)in the argument that follows).According to Corollary 2.10, we can assume that ¯ J ( X ) ∼ = U// SL r = U ss / SL r , where U is theopen subset of the Quot scheme Quot( O ⊕ rX ) defined in loc. cit. Take now any lift of [ I ] ∈ ¯ J ( X ) to a point ˜ x ∈ U ⊆ Quot( O ⊕ rX ), corresponding to a quotientmap q : O ⊕ rX ։ I , and observe that the orbit of e x is closed in the semistable locus U ss since I is polystable. Lemma 6.6(i) identifies Stab( e x ) with the subgroup Aut ( I ) ⊂ Aut( I ) which is a(multiplicative) torus (since Aut( I ) is a torus by Lemma 4.2 and any subgroup of a torus is atorus), hence linearly reductive. Therefore, we can apply Luna Slice Theorem (see § Z of U at e x .Lemma 6.4(i) identifies the ring b O Z, ˜ x with the miniversal deformation ring R of Def I . Moreover,an exercise in unwinding the definitions shows that the natural transformation π : Spf b O Z, ˜ x → Def I is equivariant with respect to the natural homomorphism Stab(˜ x ) ֒ → Aut( I ) and the actions ofStab(˜ x ) on Spf b O Z, ˜ x and of Aut( I ) on Def I . Therefore, Fact 5.4 implies that the natural identifi-cation b O Z, ˜ x ∼ = R is Stab(˜ x ) ∼ = Aut ( I )-equivariant.Now, applying Eqn. (2.1) together with Lemma 6.7, we get(6.2) b O ¯ J ( X ) , [ I ] ∼ = b O Stab(˜ x ) Z, ˜ x ∼ = R Aut ( I )1 Now observe that the subgroup G m ⊂ Aut( I ) of scalar automorphism acts trivially on R , as itfollows by the explicit description of the action of Aut( I ) on R given in Theorem 5.10. Thus, thenatural action of Aut( I ) on R factors through the quotient Aut( I ) / G m . Because the natural mapAut ( I ) → Aut( I ) / G m is surjective, we get that(6.3) R Aut ( I )1 ∼ = R Aut( I )1 . Combining (6.2) and (6.3), we get the conclusion. (cid:4)
In the introduction, we asked if Theorem 6.1 remains valid when X is allowed to have an auto-morphism of order p . The condition on the automorphism group was only used to apply the LunaSlice Theorem, which applies to actions of linearly reductive groups. It is probably unreasonableto expect an analogue of the Slice Theorem to hold for actions of an arbitrary reductive group (see[25]), but we only need an analogue for actions of Aut( X, I ). This group is an extension of thefinite (reduced) group Aut( X ) by the multiplicative torus Aut( I ), and it is known that the SliceTheorem holds for both the action of a torus (it is linearly reductive) and for the action of a finitegroup (see e.g. [19, Prop. 2.2]). Perhaps there is a Slice Theorem for actions of an extension of atorus by an arbitrary finite group?Finally, we can prove Theorem B from the introduction. Proof of Theorem B.
Given Theorem A, this result follows from [13]. To establish Parts (i) and(ii) of Theorem B, it is enough to fix a point [ I ] ∈ ¯ J ( X ) with I polystable and prove the analogousstatement about the completed local ring b O ¯ J ( X ) , [ I ] . For the remainder of the proof, we will workexclusively with b O ¯ J ( X ) , [ I ] .Theorem A identifies b O ¯ J ( X ) , [ I ] with the T Γ -invariant subring of R I = b A (Γ)[[ W , . . . , W g Σ ]], where g Σ := g ( X ) − b (Γ). The ring b A (Γ) is the completion of the ring A (Γ) defined in [13, §
6] at themaximal ideal e m := ( U ← e , U → e : e ∈ Σ) and the action of T Γ on b A (Γ) is induced by the action on A (Γ) defined in loc. cit. By [13, Thm. 6.1], the invariant subring of A (Γ) is the cographic toric facering R (Γ) (from [13, Def. 1.4]). Thus, applying Lemma 6.7, we get(6.4) b O ¯ J ( X ) , [ I ] ∼ = b R (Γ)[[ W , . . . , W g Σ ]] here b R (Γ) is the completion of R (Γ) at the ideal m := e m ∩ R (Γ) (which appears in [13, Prop. 4.6]).We now prove Part (i) of the theorem. In [13], it is proven that R (Γ) is Gorenstein and has slcsingularities ([13, Thm. 5.7]), and these properties persist after passing to a completion and addingpower series variables.To establish Part (ii), it is enough to show that the multiplicity e m ( R (Γ)) is equal to 1 if and onlyif every element of Σ corresponds to a separating edge of the dual graph Γ X of X . The formulafor e m ( R (Γ)) given in [13, Thm. 5.7(vii)] shows that if e m ( R (Γ)) = 1 then Γ \ E (Γ) sep has a uniquetotally cyclic orientation and this can only happen if Γ \ E (Γ) sep is a disjoint union of points, i.e. ifΓ is a tree. As Γ is obtained from Γ X by contracting the edges not in Σ, Part (ii) follows. (cid:4) Examples
In this section we present some examples to further elucidate the connections between the resultsin this paper and those of [13].7.1.
Integral curves.
Suppose that X is an integral nodal curve of arithmetic genus g and with m nodes. From Definition 2.2, it follows that any compactified Jacobian of X is equal to the (fine)moduli space ¯ J d ( X ) of rank 1, torsion-free sheaves on X of degree d (for some d ∈ Z ).Consider now a point [ I ] ∈ ¯ J d ( X ) such that I is not locally free at all the nodes of X (note that I is stable). Then Theorem A(i) gives that b O ¯ J d ( X ) , [ I ] ∼ = dO mi =1 k [[ X i , Y i ]] X i Y i dO k [[ T , . . . , T g − m ]]We recover the well-know fact (see [11, Prop. 2.7]) that ¯ J d ( X ) is isomorphic, formal (indeed ´etale)locally at I , to the product of m nodes and a smooth factor.7.2. Two irreducible components.
Let X n be a nodal curve consisting of two smooth irreduciblecomponents C and C of genera, respectively, g and g , intersecting in n ≥ X n is g = g + g + n − n = 1 is easy: the curve X is of compacttype, hence all compactified Jacobians are smooth and isomorphic to the generalized Jacobian).The dual graph Γ X n of X n is depicted in Figure 7.2 below together with an orientation of it.... • o o e r +1 o o e n / / e r / / e v v • ... Figure 1.
The orientation φ r on Γ X n .Let ¯ J ( X n ) be a compactified Jacobian of X n and suppose that there exists a polystable sheaf[ I ] ∈ ¯ J ( X n ) that fails to be locally free at the n nodes of the curve. Therefore, Eqn. (6.4) gives b O ¯ J d ( X ) , [ I ] ∼ = b R (Γ X n )[[ W , . . . , W g + g ]] . Using this presentation of the complete local ring and the results of [13], we can prove thefollowing properties: ´Etale locally at [ I ], ¯ J ( X n ) has n − X r =1 (cid:18) nr (cid:19) irreducible components, which are in bijection by [13,Thm. 5.7(i)] with the totally cyclic orientations of Γ n , all of which look like the orientation φ r (for1 ≤ r ≤ n −
1) depicted in the figure above. • The dimension of the Zariski tangent space T [ I ] ¯ J ( X n ) (i.e. the embedded dimension of ¯ J ( X n )at [ I ]) is equal to g + g + 2 (cid:0) n (cid:1) , as it follows from the fact (proved in [13, Thm. 5.7(vi)]) that theembedded dimension of R (Γ X n ) at the maximal ideal m is equal to the number of oriented circuitsof Γ X n , which is 2 (cid:0) n (cid:1) . • Finally, it can be proved using [13, Thm. 5.7(vii)] that the multiplicity of ¯ J ( X n ) at [ I ] ismult [ I ] ¯ J = n − X r =1 (cid:18) nr (cid:19)(cid:18) n − r − (cid:19) . References
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E-mail address : [email protected] University of South Carolina, Department of Mathematics, South Carolina (USA)
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E-mail address : [email protected]@mat.uniroma3.it