Moduli of Generalized Line Bundles on a Ribbon
aa r X i v : . [ m a t h . AG ] J un MODULI OF GENERALIZED LINE BUNDLES ON ARIBBON
DAWEI CHEN AND JESSE LEO KASS
Abstract.
A ribbon is a first-order thickening of a non-singular curve.Motivated by a question of Eisenbud and Green, we show that a com-pactification of the moduli space of line bundles on a ribbon is given bythe moduli space of semi-stable sheaves. We then describe the geometryof this space, determining the irreducible components, the connectedcomponents, and the smooth locus.
Contents
1. Introduction 1Conventions 42. Coherent Sheaves on a Ribbon 53. Stable Sheaves on a Ribbon 104. Moduli of Sheaves on a Ribbon 144.1. Global Structure 154.2. Local Structure 22References 251.
Introduction
This paper describes the moduli space of slope semi-stable sheaves on aribbon. A ribbon is a first-order thickening of a non-singular curve, and inthe context of this paper, their study began with work of Bayer–Eisenbud([BE95]) and Eisenbud–Green ([EG95]), where ribbons were used to studyGreen’s conjecture on linear series using degeneration techniques. Morerecently, the first named author used these techniques to study Brill–Noetherloci ([Che10]). Given a linear series on a non-singular curve, the approach isto specialize the curve to a ribbon and then specialize the linear series to aseries on the ribbon. Here one encounters a difficulty: it is only possible tospecialize the linear series if the specialization is allowed to be a generalizedlinear series. Recall that a linear series is a line bundle I together witha subspace of the space of global sections. Eisenbud and Green defined a Mathematics Subject Classification.
Primary 14D20, Secondary 14C20.
Key words and phrases.
Ribbon, Semi-stable sheaf, Generalized line bundle.The second author was supported by NSF grant DMS-0502170. generalized linear series on a ribbon by allowing I to be a more generalcoherent sheaf, termed a generalized line bundle. In [EG95], the authorsraised the question: does there exist a moduli space of generalizedline bundles? Eisenbud and Green observed that the set of all generalized line bundlesof fixed degree cannot be parameterized by an algebraic k -scheme becausethe class of such sheaves is unbounded. Let X be a ribbon. Given a finitebirational morphism f : X ′ → X of ribbons, the direct image I := f ∗ ( I ′ )of a line bundle I ′ is a generalized line bundle, and varying over all f ,we obtain all generalized line bundles. The genus of X ′ is an importantinvariant of I , so following Dr´ezet ([Dr´e08, § b ( I )of I by b ( I ) := g ( X ) − g ( X ′ ). The index of a generalized line bundle can bearbitrarily negative, and unboundedness follows. The exact question posedby Eisenbud and Green ([EG95, pg. 758, midpage]) is: if X is a rationalribbon (i.e. X red ∼ = P ), then is it possible to compactify the modulispace of degree line bundles on X by a moduli space of degree generalized line bundles with non-negative index? Requiring that the index is non-negative is one way to recover bounded-ness. Another way is to impose the condition of slope semi-stability. In thegeneral theory of moduli of sheaves, it has long been recognized that manynatural classes of sheaves are unbounded, but one can recover boundednessby considering sheaves that satisfy slope semi-stability. In great generality,Simpson ([Sim94]) has constructed moduli spaces M( O X , P ) of semi-stablesheaves with Hilbert polynomial P on a polarized scheme ( X, L ). M( O X , P )is a coarse moduli space in the sense that non-isomorphic sheaves may cor-respond to the same point of M( O X , P ). Sheaves satisfying the strongercondition of stability sweep out an open locus M s ( O X , P ) ⊂ M( O X , P ), andthis locus is the fine moduli space of stable sheaves. The goal of this paper isto describe the Simpson moduli spaces M( O X , P ) and M s ( O X , P ) when X is a ribbon, with an eye towards addressing the question posed by Eisenbudand Green.Given a polarized ribbon ( X, L ), we study the Simpson moduli space pa-rameterizing semi-stable sheaves with Hilbert polynomial P d ( t ) := deg( L ) t + d +1 − g , the Hilbert polynomial of a degree d line bundle. Here g is the genusof X . There are two types of sheaves that are parameterized by M( O X , P d ):generalized line bundles on X and rank 2 vector bundles on X red . Moreprecisely, we have Theorem A.
Let ( X, L ) be a polarized ribbon. Set g equal to the genus of X and ¯ g equal to the genus of X red . If F is a coherent sheaf on X , then F corresponds to a point of M( O X , P d ) (resp. M s ( O X , P d ) ) if and only if F isisomorphic to one of the following sheaves: • a degree d generalized line bundle I of index less than or equal to(resp. strictly less than) g − g ; ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 3 • the direct image i ∗ E of a rank , slope semi-stable (resp. slope stable)vector bundle on X red of degree d + 2¯ g − − g . Here i : X red ֒ → X is the inclusion map. This is a reformation of Theorem 3.6.Consider the special case where X is a rational ribbon (i.e. ¯ g = 0), whichwas the case considered in [BE95]. Theorem A then asserts that the stablelocus M s ( O X , P ) of the Simpson moduli space is the fine moduli spaceparameterizing generalized line bundles of index at most g . When g is even,every semi-stable sheaf is stable, and M s ( O X , P ) is projective. Otherwise,M s ( O X , P ) is not projective, and its complement in the projective schemeM( O X , P ) consists of a single point. In other words, the Simpson modulispace satisfies the properties Eisenbud and Green ask for preciselywhen g is even . This result is restated as Corollary 4.1 in the body of thetext.We prove several results about the geometry of M( O X , P d ). The followingtheorem enumerates the irreducible components of M( O X , P d ). Theorem B.
Let X be a ribbon. Denote by g the genus of X , ¯ g the genusof X red , and set n := ( ⌊ ( g + 2) / ⌋ − ¯ g if d is even; ⌊ ( g + 1) / ⌋ − ¯ g if d is odd.Assume there exists a stable generalized line bundle of degree d (i.e. g > g − holds). Then M( O X , P d ) has exactly n irreducible components ofdimension g whose general element corresponds to a generalized line bundle.There is at most one additional component. When it exists, this compo-nent is of dimension g − and the general element corresponds to a stablerank vector bundle on X red . This additional component does not exist when ¯ g = 0 , but does exist when the two conditions ¯ g ≥ and g − ≥ g aresatisfied. This is Theorem 4.7. That theorem, together with Theorem 4.6, providesa more detailed description of the components. Theorem B says nothingwhen g ≤ g −
1, but this case is also discussed in Theorem 4.6,.We also compute the connected components of M( O X , P d ). The statementbelow is a restatement of Theorem 4.10. Theorem C.
For a ribbon X , the moduli space M( O X , P d ) is connected. Finally, we determine the smooth locus of M( O X , P d ). Theorem D.
Let X be a ribbon. Set g equal to the genus of X and ¯ g equalto the genus of X red . If ¯ g ≥ and g ≥ g − , then the smooth locus of M( O X , P d ) is equal to the open subset of line bundles on X . This is Corollary 4.14, which is a consequence of the computation of thetangent space to M s ( O X , P d ) at a point (Proposition 4.11). That computa-tion may be of independent interest. DAWEI CHEN AND JESSE LEO KASS
How do these results compare with results in the literature? In [Ina04],Michi-Aki Inaba studied the moduli space of stable sheaves on a non-reducedscheme and, in particular, proved results about the local structure of themoduli space of slope stable sheaves on a ribbon ([Ina04, Thm. 2.6]). Closestto our results is [Ina04, Rmk 2.7], which contains tangent space computa-tions similar to Lemma 4.13.Beginning with [Dr´e06], Jean-Marc Dr´ezet has written several papers([Dr´e06], [Dr´e08], [Dr´e09], [Dr´e11]) studying slope semi-stable sheaves ona multiple curve, the n -th order analogue of a ribbon. Much of this work fo-cuses on “quasi-locally free sheaves,” a class of sheaves that includes line bun-dles, but not generalized line bundles. Most relevant to this paper is [Dr´e11](esp. Thm. 5.4.2), which the authors became aware of while preparing thecurrent document. In that paper, Dr´ezet provides sufficient conditions for apure sheaf of dimension 1 (there called a “torsion-free sheaf”) on a primitivemultiple curve to be (semi-)stable. Also relevant are Dr´ezet’s classificationof pure sheaves with Hilbert polynomial P d ([Dr´e08, § K N of X is isomorphic to the anti-canonical bundle K − X red of X red (so g = 4¯ g − O X , P d ) is studied inSection 4. The main results proven in that section are Theorems B, C andD. Acknowledgements.
The authors would like to thank Robert Lazars-feld for informing them of his paper [DEL97] with Donagi and Ein and forproviding helpful expository suggestions. The authors would also like tothank Daniel Erman for expository feedback, Yusuf Mustopa for enlighten-ing conversations about vector bundles on curves, Matt Satriano for helpfuldiscussions on homological algebra.
Conventions k is an algebraically closed field . A curve is an irreducible, projective k -scheme of dimension . A non-singular curve is a curve that is smooth over k . ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 5 A ribbon X is a curve such that the reduced subscheme X red is anon-singular curve and the nilradical N is locally generated by a non-zero,square-zero element. A ribbon X is a rational ribbon if X red is isomorphic to P . The degree deg( I ) of a coherent sheaf I on X is deg( I ) := χ ( I ) − χ ( O X ) . The genus g ( X ) of a curve X is g := 1 − χ ( O X ) . η is the generic point of X . Coherent Sheaves on a Ribbon
Here we collect the facts about coherent sheaves on ribbons that areneeded to describe the Simpson moduli space. In this section, let X be afixed ribbon with generic point η .Recall that, by definition, X is a curve with the property that the reducedsubcurve X red is non-singular and the nilradical N is locally generated by asingle square-zero, but non-zero element. The nilradical is then square-zero,and hence may be considered as a line bundle on X red .The degree of this line bundle can easily be computed from the exactsequence(2.1) N ֒ → O X ։ O X red . We have: deg( N ) = 2¯ g − − g, where g is the genus of X and ¯ g the genus of X red . This quantity plays animportant role in many of the results of this paper, appearing, for example,as a bound in Theorem A.Notice that we have defined the genus g by g := 1 − χ ( O X ). The genusof a reduced curve X always equals h ( X, O X ), but this equality may failto hold when X is a ribbon. Indeed, the two numbers are equal preciselywhen N has no global sections, which holds when g is sufficiently large butnot in general.We now begin our review of coherent sheaves on a ribbon. First, somedefinitions from [EG95]. Definition 2.1. If F is a coherent sheaf on X , then we write d ( F ) for thedimension of the support of F . We say that F is pure if d ( F ) = d ( G ) forall non-zero subsheaves G ⊂ F .If F is a coherent sheaf, then we say that a regular function f ∈ H ( U, O X )is a non-zero divisor on F if the multiplication map f · : F | U → F | U isinjective. The sheaf F is said to be torsion-free if every non-zero divisoron O X is a non-zero divisor on F . The sheaf F is said to be rank DAWEI CHEN AND JESSE LEO KASS generic stalk F η is isomorphic to O X,η . A generalized line bundle is acoherent sheaf that is rank 1 and torsion-free.
Lemma 2.2.
Let I be a coherent sheaf satisfying d ( I ) = 1 . Then I istorsion-free if and only if I is pure.Proof. First, we assume I is pure and prove that it is torsion-free. Given anon-empty open subset U ⊂ X , a non-zero divisor f ∈ H ( U, O X ) and a lo-cal section s ∈ H ( U, I ) satisfying f s = 0, we will show s = 0. Consider theannihilator ann( s ). The quotient O U / ann( s ) is isomorphic the submoduleof I| U generated by s , so by purity, there are only two possibilities for thequotient: either its support equals X or it is zero. We can eliminate the firstcase. Indeed, as ann( s ) contains a non-zero divisor, it is not contained in thenilradical, hence is not contained in some maximal ideal. The correspondingpoint of X does not lie in the support of O X / ann( s ), so this quotient mustbe zero. Equivalently, ann( s ) is the unit ideal and s = 0. This proves that I is torsion-free.Now we show that torsion-free implies pure. Given a non-zero submodule J ⊂ I of a torsion-free module I , we will show d ( J ) = 1. First, pick anaffine open subset U ⊂ X that meets the support of J . The integers d ( J )and d ( J | U ) coincide, and d ( · ) does not increase if we replace J | U with asubmodule. Thus, by replacing J | U with a non-zero cyclic submodule, wecan assume J | U is cyclic. Say s ∈ H ( U, J ) generates J | U .What are the possibilities for Supp( s )? We claim that Supp( s ) = U .This support is non-empty because s is non-zero, so it is enough to showthat the only irreducible component of Supp( s ) is U . Suppose p is a primeideal corresponding to such an irreducible component. Then p = ann( f s )for some f ∈ H ( U, O X ). But I is torsion-free, so p is a union of zero-divisors and hence contained in the nilradical. The only such prime ideal isthe nilradical itself, so we may conclude that Supp( s ) equals U . This provesthat d ( J ) = 1 and thus that I is pure. (cid:4) One application of Lemma 2.2 is the following result, which provides analternative characterization of generalized line bundles.
Lemma 2.3.
Let X be a ribbon and I a pure sheaf whose generic stalk isan O X,η -module of length . Then I is a generalized line bundle if and onlyif the natural map (2.2) O X → End( I ) is injective.Proof. Let X and I be given. We have just shown that torsion-free is equiv-alent to pure (Lemma 2.2), so the content of the lemma is that I is rank 1 ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 7 if and only if the map (2.2) is injective. This map factors as O X −−−−→ End( I ) y y O X,η −−−−→
End( I η ) , where η is the generic point.When I is a generalized line bundle, the vertical maps are injective (be-cause O X and I are pure) and O X,η → End( I η ) is an isomorphism (because I is rank 1). We may conclude that (2.2) is injective.Conversely, assume this map is injective. Pick a generator ǫ of the genericstalk N η . By hypothesis, multiplication by ǫ on I η is not the zero map, sothere exists an element s ∈ I η with ǫs = 0.We claim that the natural map O X,η → I η given by f f s is anisomorphism. The kernel of this map is ann( s ), which is contained in N η (because I is pure). In fact, this containment is proper because ǫs = 0.The only ideal properly contained in the nilradical is the zero ideal, provinginjectivity. Surjectivity also follows: both the module I η and the submodulegenerated by s have length 2, so they must coincide. We may conclude that I η is isomorphic to O X,η , and the poof is complete. (cid:4)
The previous two results allow us to easily classify the pure sheaves whosegeneric stalk has length 2. The classification result below can be found inthe work of Dr´ezet, but we include a proof for the sake of completeness.
Proposition 2.4 ([Dr´e06, § . Let X be a ribbon and I a pure sheafwhose generic stalk is an O X,η -module of length . Then I is isomorphic toone of the following:(1) a generalized line bundle;(2) the direct image i ∗ ( E ) of a rank vector bundle E on X red under theinclusion map i : X red ֒ → X .Proof. Given I , consider the natural map O X → End( I ). The kernel ofthis map consists of zero-divisors (Lemma 2.2), hence is contained in thenilradical N . There are only two ideals with this property: the zero idealand the nilradical itself. When the kernel is the zero ideal, Lemma 2.3 statesthat I is a generalized line bundle. Thus, we focus on the case where thekernel is N .In this case, we can consider I as a module E over O X / N = O X red .Certainly, E then satisfies I = i ∗ ( E ). The module E is pure, hence locallyfree (by, say, the Auslander–Buchsbaum formula) and the generic rank is 2by hypothesis. In other words, E is a rank 2 vector bundle, completing theproof. (cid:4) We conclude this section with a discussion of the sheaves appearing inProposition 2.4. At this level of generality, there is not much to be said aboutthe classification of rank 2 vector bundles on X red . When X red has genus 0, DAWEI CHEN AND JESSE LEO KASS every vector bundle is a direct sum of line bundles. But in general, there arenon-split rank 2 vector bundles and the classification is more complicated.In Section 3, we introduce stability conditions and the semi-stable vectorbundles on X red are coarsely parameterized by the corresponding Simpsonmoduli space.There is a classification of generalized line bundles on X , which can befound in [EG95, Thm 1.1]. Given a Cartier divisor D on X red , we can con-sider D as a closed subcheme of X and form the blow-up f : X ′ := Bl D ( X ) → X . The blow-up X ′ is itself a ribbon and f is a finite morphism. In fact,the quotient f ∗ ( O X ′ ) / O X is isomorphic to O D . If L ′ is a line bundle on X ′ , then the direct image I := f ∗ ( L ′ ) is a generalized line bundle. Theo-rem 1.1 of [EG95] states that every generalized line bundle is of this formfor a unique blow-up f : X ′ → X and a unique line bundle L ′ . Given I , wecall f : X ′ → X the associated blow-up . Following Dr´ezet ([Dr´e08, § I . Definition 2.5.
Let I be a generalized line bundle on X . Say I := f ∗ ( I ′ )for the blow-up f : X ′ = Bl D ( X ) → X and the line bundle I ′ on X ′ . Asa divisor on X red , write D = P n p p . Given a point p ∈ X , we define the local index of I at p , written b p ( I ), to be b p ( I ) := n p . The localindex sequence is the collection { b p ( I ) : b p ( I ) = 0 } , and the index of I is defined by b ( I ) := P b p ( I ).The integers b ( I ) and b p ( I ) can be defined more intrinsically. The index b ( I ) is the length of End( I ) / O X , while b p ( I ) is the length of the localizationEnd( I p ) / O X,p . The index of a degree d generalized line bundle cannot bean arbitrary integer. Fact 2.6. If I is a generalized line bundle, then deg( I ) − b ( I ) is even.Proof. If I is a line bundle, by definition b ( I ) = 0. Tensoring the sequence N ֒ → O X ։ O X red with I and taking Euler characteristics, we see thatdeg( I ) = 2 · deg( I ⊗ O X red ) , which is an even number. For the general case, write I = f ∗ ( I ′ ) for someline bundle I ′ on a blow-up f : X ′ → X . The Euler characteristics χ ( I ) and χ ( I ′ ) are equal. Writing out these numbers, we seedeg( I ) = deg( I ′ ) + g − ¯ g = deg( I ′ ) + b ( I ) . In particular, deg( I ) − b ( I ) is even. (cid:4) One consequence of the classification theorem of Green–Eisenbud is thatthe generalized Jacobian, or moduli space of degree 0 line bundles, actstransitively on the set of generalized line bundles with fixed local indexsequence. This fact will be used later, so we record it.
ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 9
Lemma 2.7.
Let X be a ribbon. If I and I are two generalized linebundles that have the same local index at p for all p ∈ X , then there exists aline bundle L on X such that I is isomorphic to L ⊗ I . Furthermore, anytwo line bundles with this property differ by an element of ker( f ∗ : Pic( X ) → Pic( X ′ )) , where f : X ′ → X is the blow-up associated to I i .Proof. If f : X ′ → X is the blow-up as in Definition 2.5, then we can write I i = f ∗ ( L i ) for line bundles L and L on X ′ . The map f ∗ : Pic( X ) → Pic( X ′ ) is surjective, so we can find a line bundle L such that f ∗ ( L ) = L ⊗ L − . An application of the projection formula shows that this linebundle satisfies the desired conditions, and that any other line bundle M with this property must satisfy f ∗ ( M ) ∼ = f ∗ ( L ). (cid:4) The structure of the generalized Jacobian J ( X ) of X can be describedexplicitly: the pullback map i ∗ : J ( X ) → J ( X red ) from the generalizedJacobian of X to the Jacobian of X red is surjective with kernel equal to thevector space group associated to H ( X red , N ).This description of J ( X ) can be found in [Dr´e08, § J ( X ) as the identitycomponent of the moduli space Pic( X ) of line bundles on X . In [BLR90],Pic( X ) is described as the scheme that represents the ´etale sheaf R g ∗ ( O ∗ X ),where g : X → Spec( k ) is the structure morphism, and then computed usinglong exact sequence associated to the “exponential sequence” N ֒ → O ∗ X ։ O ∗ X red . In any case, we see that the dimension of J ( X ) equals h ( X, O X ) and J ( X ) is not proper once g ≥ ¯ g + 1 (as then H ( X red , N ) = 0). These twofacts will be used several times in Section 4.If we pass from X to its completed local ring at a point, then we candescribe generalized line bundles more explicitly. Definition 2.8.
Set O := k [[ s, ǫ ]] / ( ǫ ). Define the n -th blow-up algebra to be O n := k [[˜ s, ˜ ǫ ]] / (˜ ǫ ), considered as an O -algebra via s ˜ s , ǫ ˜ ǫ ˜ s n .We write I n for O n , considered as an O -module.One may easily compute that O → O n is the algebra extension cor-responding to the blow-up of O along ( s n , ǫ ). Given a blow-up f : X ′ :=Bl D ( X ) → X as in Definition 2.5 and a point q ∈ X ′ mapping to p ∈ X ′ , theinduced map b O X,p → b O X ′ ,q can be identified with O → O n p . Combiningthis observation with [EG95, Thm. 1.1], we can deduce the following lemma. Lemma 2.9.
Let X be a ribbon, p ∈ X a point and I a generalized linebundle. Fix an isomorphism between O and b O X,p . Then under this iso-morphism, I ⊗ b O X,p is identified with a module isomorphic to I n , where n = b p ( I ) .Proof. Write I = f ∗ ( L ′ ) for the blow-up f : X ′ := Bl D ( X ) → X and a linebundle L ′ on X ′ . If q ∈ X ′ is the unique point mapping to p , then the identification of b O X,p with O extends to an identification of the ˆ f : b O X,p → b O X ′ ,q map on completed local rings with O → O n . In particular, I ⊗ b O X,p is identified with the direct image of a line bundle on O n , and such a moduleis isomorphic to O n itself. (cid:4) For later computations, it is convenient to have an alternative descriptionof I n . This module can also be described as the ideal ( s n , ǫ ) of O . Oneisomorphism from I n to ( ǫ, s n ) is given by the map sending 1 ∈ I n to s n and˜ ǫ ∈ I n to ǫ . This common module admits the following presentation: h e, f : ǫf = 0 , s n f = ǫe i . Here the element e corresponds to 1 ∈ I n and f corresponds to ˜ ǫ . Thispresentation in fact extends to the period resolution:(2.3) · · · −→ O ǫ s n − ǫ ! −→ O ǫ s n − ǫ ! −→ O −→ I n −→ . Later, we shall use this presentation to describe how I n deforms.3. Stable Sheaves on a Ribbon
Here we study the stability of generalized line bundles. In general, thestability of a coherent sheaf on a projective scheme is a condition definedin terms of an auxiliary ample line bundle L . However, on a ribbon thestability condition is independent of L , as we will see. To fix ideas, let usfirst work with a polarized ribbon ( X, L ). As before, we write g for thegenus of X , ¯ g for the genus of X red and N for the nilradical of X . We nowrecall the definition of the Hilbert polynomial.Given a coherent sheaf I on X , the Hilbert Polynomial P ( I , t ) of I with respect to L is the unique polynomial satisfying P ( I , n ) = χ ( I ⊗ L ⊗ n ) for all n ∈ N .The leading term of P ( I , t ) is particularly significant. Write P ( I , t ) = a t d /d ! + a t d − / ( d − · · · + a d . We have d = d ( I ) , the dimension of Supp( I ); a = deg( L ) len( I η ) , where len( I η ) is the length of I η as an O X,η -module.If I is a sheaf on X with d ( I ) = 1, then the slope µ ( I ) of I with respectto L is defined to be µ ( I ) := a /a . We say that I is slope semi-stable with respect to L if I is pure and forall non-zero pure subsheaves J ⊂ I we have µ ( J ) ≤ µ ( I ) . ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 11
If this inequality is always strict, we say I is slope stable . A semi-stablesheaf that is not stable is said to be strictly semi-stable . An equivalentformulation of semi-stability is that µ ( I ) ≤ µ ( J ) for all quotients I ։ J with J pure of dimension d ( J ) = d ( I ), and similarly with stability.We have just given the general definition of semi-stability, but observethat, on a ribbon, the condition is independent of L . Indeed, the slope µ ( I ) depends on L , but replacing L with a different ample line bundle M modifies the slope by a factor of deg( L ) / deg( M ), so the slope semi-stabilitycondition is unchanged.Given a semi-stable sheaf I , there exists a filtration 0 = I ⊂ · · · ⊂ I n = I with the property that the successive quotients I k / I k − are slope stable ofdimension d ( I ) and the slopes µ ( I k / I k − ) are all equal. This filtration isnot unique, but the associated direct sumGr( I ) := M k I k / I k − is unique up to isomorphism. Two coherent sheaves I and I ′ are said tobe Gr-equivalent if there is an isomorphism Gr( I ) ∼ = Gr( I ′ ). Given asemi-stable sheaf I , observe that Gr( I ) is also a semi-stable sheaf whoseassociated graded is Gr( I ).We now specialize to the case of semi-stable sheaves on X with Hilbertpolynomial(3.1) P d ( t ) := deg( L ) t + d + 1 − g. This is the Hilbert polynomial of a degree d generalized line bundle. We willshow that the stability condition on a generalized line bundle is controlled bya distinguished quotient. Following [EG95], we make the following definition. Definition 3.1.
The sheaf ¯ I associated to a generalized line bundle I isdefined to be the maximal torsion-free quotient of I ⊗ O X red .The sheaf ¯ I is a line bundle on X red of degree (deg( I ) − b ( I )) /
2. Bythe projection formula, the Hilbert polynomial of i ∗ (¯ I ) with respect to L isequal to the Hilbert polynomial of ¯ I with respect to i ∗ ( L ). This commonpolynomial is(3.2) deg( L )2 t + (deg( I ) − b ( I )) / − ¯ g. Using ¯ I as a test quotient sheaf, we conclude that if I is semi-stable, thenits index must satisfy(3.3) b ( I ) ≤ g − g. In fact, this inequality is sufficient to characterize slope semi-stability.
Lemma 3.2.
Let ( X, L ) be a polarized ribbon and I a generalized line bun-dle of degree d . Then I is slope semi-stable with respect to L if and onlyif Inequality (3.3) holds. Similarly, I is slope stable if and only if Inequal-ity (3.3) is strict. Proof.
Inequality (3.3) is equivalent to the slope inequality µ ( I ) ≤ µ (¯ I ),so one implication is clear. For the converse, we assume µ ( I ) ≤ µ (¯ I ) andthen prove µ ( I ) ≤ µ ( J ) for all pure quotients q : I ։ J with d ( J ) = 1.There are two separate cases to consider: the case where the leading termof P ( J , t ) is deg( L ) · t and the case where it is deg( L ) / · t .First, suppose the leading term of P ( J , t ) is deg( L ) · t . We claim that q is in fact an isomorphism, and thus there is no slope inequality to check.We begin by showing that J is a generalized line bundle. The condition onthe Hilbert polynomial is equivalent to the condition that the generic stalk J η has length 2, so by Proposition 2.4 it is enough to show that J is notthe direct image of a rank 2 vector bundle on X red . This, however, is clear: J η is generated by a single element (the image of a local generator of I ),but the direct image of a rank 2 vector bundle does not have this property.Having shown that J is a generalized line bundle, the result follows easily.Consider the kernel of q . Because both I and J are generalized line bundles, q is generically an isomorphism, hence ker( q ) is generically zero. But ker( q )is a subsheaf of the pure sheaf I , so this forces ker( q ) = 0. We can concludethat q is an isomorphism.Next we consider the case where the leading term of P ( J , t ) is deg( L ) / · t .To begin, we claim that the kernel K of O X → End( J ) equals N . The kernelis certainly contained in N because J is pure. Furthermore, the only proper O X -submodule of N is the zero ideal, so it is enough to show that K 6 = 0.Consider the generic stalk J η . The assumption on the Hilbert polynomialimplies that the length of J η , and hence of every non-zero cyclic submodule,is 1. In particular, if s ∈ J η is non-zero, then O X,η · s = J η / ann( s )has length 1. Since I has length 2, we may conclude that K η = ann( s ) isnon-zero. This establishes the claim K = N .Because K = N , we may factor q as I → i ∗ (¯ I ) ¯ q → J . Furthermore,consider J as a module over O X / N = O X red . If we write F for this module,then F satisfies J = i ∗ ( F ). This module is also a line bundle on X red because it is locally free (as it is pure) of generic rank 1. Thus, ¯ q is asurjection between line bundles on X red , so it must be an isomorphism.In particular, the inequality µ ( I ) ≤ µ ( J ) is exactly the hypothesis of thelemma. This completes the proof. (cid:4) The lemma also shows that the strictly semi-stable generalized line bun-dles are exactly the generalized line bundles of index b ( I ) = 1 + g − g . Itis natural to ask what their associated graded modules are. Lemma 3.3.
Let ( X, L ) be a polarized ribbon and I a generalized line bundleof index b ( I ) = 1 + g − g . Set F ( I ) to be the kernel of I → ¯ I . Then ⊂ F ( I ) ⊂ I is filtration whose successive quotients are stable with thesame Hilbert polynomial, and Gr( I ) = F ( I ) ⊕ ¯ I . ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 13
Proof.
Both F ( I ) and ¯ I are pure of dimension 1 (in fact, direct images ofthe line bundles on X red ), so it is enough to show µ (¯ I ) = µ (F ( I )). Thisfollows from the work we have already shown. One computes µ (¯ I ) = (deg( I ) − b ( I ) + 2 − g ) / deg( L ) ,µ (F ( I )) = (deg( I ) + b ( I ) + 2¯ g − g ) / deg( L ) . It is easy to check that these two numbers are equal precisely when b ( I ) =1 + g − g . (cid:4) The submodule F ( I ) can be described more explicitly. Say the blow-up f : X ′ → X associated to I is given by blowing up the divisor D ⊂ X red .Then F ( I ) = N ⊗ ¯ I ( D ). (This is [EG95, pg. 759, bottom of page]).F ( I ) ⊕ ¯ I is, of course, the direct image of a split rank 2 vector bundle on X red . For the sake of completeness, we should also discuss the slope stabilitycondition on the direct image i ∗ ( E ) of a vector bundle, but there is little tosay. Slope stability of i ∗ ( E ) is equivalent to slope stability of E . This willbe used in a later section, so let us record it as a fact. Fact 3.4.
Let X be a ribbon. Then the direct image i ∗ E of a rank vectorbundle E on X red is semi-stable if and only if for every subbundle F ⊂ E ,we have (3.4) deg( F ) / rank( F ) ≤ deg( E ) / rank( E ) . In particular, if X is a rational ribbon (i.e. X red ∼ = P ), then there are nostable sheaves of this form and the only semi-stable sheaves are i ∗ ( O ( e ) ⊕O ( e )) . We summarize the results of this section with the following definition andtheorem.
Definition 3.5.
Let X be a ribbon. A coherent sheaf I is said to be length and semi-stable if it is one of the following sheaves:(1) a genefralized line bundle whose index satisfies Inequality (3.3);(2) the direct image i ∗ ( E ) of a semi-stable rank 2 vector bundle E on X red . I is said to be length 2 and stable if it is one of the following sheaves:(1) a generalized line bundle such that (3.3) is a strict inequality;(2) the direct image i ∗ ( E ) of a stable rank 2 vector bundle E on X red . Theorem 3.6.
Let ( X, L ) be a polarized ribbon. If I is a pure sheaf withHilbert polynomial P d ( t ) = deg( L ) t + d + 1 − g , then I is slope semi-stable(resp. stable) with respect to L if and only if it is a length , semi-stable(resp. stable) sheaf. The special case of rational ribbons is particularly nice.
Corollary 3.7.
Let ( X, L ) be a polarized rational ribbon. If I is a slopesemi-stable sheaf with Hilbert polynomial P d ( t ) , then it is one of the followingsheaves: (1) a degree d generalized line bundle whose index satisfies b ( I ) ≤ g ;(2) the direct image i ∗ ( E ) , where E = O (( d − − g ) / ⊕ O (( d − − g ) / .The second case can only occur when d − − g is even.If I is strictly semi-stable, then it is one of the following sheaves:(1) a degree d generalized line bundle whose index satisfies b ( I ) = g + 1 ;(2) the direct image i ∗ ( E ) , where E = O (( d − − g ) / ⊕ O (( d − − g ) / .In particular, any two strictly semi-stable sheaves are Gr -equivalent, and if d − − g is odd, then slope semi-stability is equivalent to slope stability. Moduli of Sheaves on a Ribbon
Semi-stable generalized line bundles were described in Section 3. Thesignificance of these sheaves is that they are coarsely parameterized by amoduli space. Simpson [Sim94] has constructed the coarse moduli space ofsemi-stable sheaves on an arbitrary polarized projective k -scheme, in partic-ular on a ribbon. In his paper, Simpson works over the complex numbers,but his work has since been generalized to positive characteristic ([Mar96,Thm. 0.6] or [Lan04, Thm. 0.2]). Here we describe the geometry of the Simp-son moduli space. As before, X will be a ribbon of genus g with nilradical N . We write ¯ g for the genus of X red .On a ribbon, we have seen that the slope stability condition is independentof the choice of polarization. But to fix ideas, let us temporarily work witha polarized ribbon ( X, L ). Given a polynomial P ( t ), the Simpson modulifunctor M ♯ ( O X , P ) : k -Sch → Sets is defined by letting the set of T -valuedpoints equal the set of isomorphism classes of O T -flat, finitely presented O X T -modules that are fiber-wise slope semi-stable with Hilbert polynomial P ( t ). Inside of the Simpson moduli functor, we can consider the subfunctorM ♯ s ( O X , P ) that parameterizes slope stable sheaves.The basic existence theorem [Sim94, Thm. 1.21] states that there ex-ists a pair (M( O X , P ) , p ) consisting of a projective scheme M( O X , P ) anda natural transformation p : M ♯ ( O X , P ) → M( O X , P ) that universally co-represents M ♯ ( O X , P ). In other words, p is universal with respect to naturaltransformations from M ♯ ( O X , P ) to a k -scheme and this property persistsunder base change by an arbitrary morphism T → M( O X , P ). Furthermore, p induces a bijection between the k -valued points of M( O X , P ) and the set ofGr-equivalence classes of semi-stable sheaves on X with Hilbert polynomial P ( t ). We call M( O X , P ) the Simpson moduli space .We can obtain stronger results by restricting to stable sheaves. Thereis an open subscheme M s ( O X , P ) of M( O X , P ) whose pre-image under p isM ♯ s ( O X , P ). The restriction M ♯ s ( O X , P ) → M s ( O X , P ) realizes M s ( O X , P )as the scheme that represents the ´etale sheaf associated to M s ( O X , P ). (Thisis a restatement of [Sim94, Thm. 1.21(4)].) The scheme M s ( O X , P ) is calledthe stable Simpson moduli space .If we specialize to the case where P ( t ) equals P d ( t ) := deg( L ) t + d + 1 − g ,the Simpson moduli space can be described using the results from Section 3: ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 15 it is the coarse moduli space of length 2, semi-stable sheaves of degree d .The stable locus is the fine moduli space parameterizing those sheaves thatare stable. In [EG95], Eisenbud and Green asked if it is possible to com-pactify the Jacobian of a rational ribbon by a moduli space parameterizinggeneralized line bundles of index at most g . The Simpson moduli space isa natural candidate for such a compactification; perhaps surprisingly it hasthe desired properties precisely when g is even. Corollary 4.1 (Reformulation of Corollary 3.7) . Let X be a rational rib-bon. Then the stable Simpson moduli space M s ( O X , P ) parameterizes gen-eralized line bundles of degree and index at most g . If g is even, then M s ( O X , P ) is projective. Otherwise, the complement of the stable locus in M( O X , P ) consists of a single point that represents the Gr -equivalence classof the polystable sheaf i ∗ ( O (( − − g ) / ⊕ ( − − g ) / . This equivalence class consists of the above sheaf and every generalized linebundle of degree and index g + 1 . We now turn our attention to describing the global and local geometry ofthe Simpson moduli space.4.1.
Global Structure.
Having defined M( O X , P d ), we study the globalgeometry of this space. First, we show that it is impossible for a rank 2vector bundle on X red to specialize to a generalized line bundle on X . Lemma 4.2.
Suppose T is a k -scheme and I a sheaf that represents a T -valued point of M ♯ ( O X , P ) . Define T ⊂ T to be the locus of points t ∈ T with the property that the restriction of I to the fiber X t := X T × T Spec( k ( t )) is a generalized line bundle. Then T ⊂ T is open.Proof. The property of being a generalized line bundle is equivalent to theproperty that N acts non-trivially. A standard argument [Gro65, 12.0.2]lets us reduce to showing that the property in question is stable under gen-eralization, which can be seen directly. We now make this sketch precise.Fix a non-empty, open affine subset U ⊂ X with the property thatthe restriction of the nilradical is generated by a single element ǫ and set I U := I| U × T . Using the classification of length 2 semi-stable sheaves (Propo-sition 2.4), we may assert that T is the locus of points t with the propertythat the restriction of ǫ · : I U → I U to X t is non-zero.A finite presentation argument ([Gro65, 8.9.1, 11.2.6]) allows us to reduceto the case that T is an affine, noetherian scheme. Having made this reduc-tion, we can cite [Gro65, Cor. 9.4.6] to assert that T is constructible. Tocomplete the proof, it is enough to show that the complement of T is stableunder specialization.Thus, let us assume that T = Spec( A ) is the spectrum of a discretevaluation ring with uniformizer π and that ǫ acts trivially on the genericfiber of I U . By flatness, we may deduce that ǫ acts trivially on I U (because I U injects into its generic fiber) and this property persists upon passingto the special fiber. In other words, the complement of T is closed underspecialization, hence the proof is complete. (cid:4) Next, we compute the dimension of various loci of generalized line bundlesin M( O X , P ). Definition 4.3.
Let X be a ribbon of genus g and b := ( b , . . . , b k ) a (pos-sibly empty) sequence of positive integers satisfying b + · · · + b k ≤ g − g .Define Z b ⊂ M s ( O X , P d ) to be the subset of stable generalized line bundleswhose local index sequence equals b . Lemma 4.4. If d − b − · · · − b k is odd, then Z b is empty. Otherwise, it isa locally closed, irreducible subset of dimension dim( Z b ) = g − ( b − − · · · − ( b k − . Proof.
Set b := b + · · · + b k , the index of a generalized line bundle withlocal index sequence b . When d − b is odd, the claim is just a restatement ofFact 2.6. Thus, assume d − b is even. We prove the lemma by parameterizing Z b by an irreducible variety of the appropriate dimension.Let U ⊂ Hilb b ( X red ) denote the subset parameterizing closed subschemesΣ supported at k distinct points with multiplicities b , . . . , b k . This subsetis locally closed and irreducible of dimension k , because it is isomorphic toan open subset of the k -th symmetric power of X red . The hypothesis on b implies that a generalized line bundle of index b is stable, so we can define amorphism U × Pic d − b ( X ) → M( O X , P d ) by the rule (Σ , L )
7→ I Σ ⊗ L . Here I Σ is the ideal sheaf of Σ.Observe that U has been chosen so that if p , . . . , p k is a collection of k distinct points, then there is a unique closed subscheme Σ correspondingto a point of U with the property that b p i ( I Σ ) = b i . By Lemma 2.7, wemay deduce that the image of U × Pic d − b ( X ) → M( O X , P d ) is Z b . Notethat the fiber over a point is an irreducible variety of dimension b , given byker( f ∗ : Pic( X ) → Pic( X ′ )) where f : X ′ → X is the associated blow-up.We can immediately conclude that Z b is irreducible and constructible.Furthermore, using the formula b ( I ) = len(End( I ) / O X ) for index, it iseasy to see that the index of a generalized line bundle can only increaseunder specialization. Because Z b is constructible, it follows that this subsetis locally closed. To complete the proof, we compute the dimension of Z b :dim( Z b ) = dim( U × Pic d − bX ) − dim(Fiber)= ( g + k ) − b. (cid:4) Lemma 4.5.
Let X be a ribbon and I a generalized line bundle. Say that p ∈ X is a point such that b p ( I ) = b + 2 for some b ≥ . Then there isa Spec( k [[ α ]]) -flat, finitely presented O X × Spec( k [[ α ]]) -module I α whose special ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 17 fiber is isomorphic to I and whose generic fiber I α [ α − ] is a generalized linebundle with local index b p ( I α [ α − ]) = ( b p ( I ) if p = p ; b if p = p .Proof. Given I , we first exhibit another generalized line bundle I ′ withthe same local index as I at every point p ∈ X that admits a suitabledeformation. Then we use Lemma 2.7 to deform I .Say p , . . . , p k are the points p with b p ( I ) = 0, labeled so p = p . Define Z to be the (unique) closed subscheme of X red that is supported at p , . . . , p k and has length b p i ( I ) at p i . If we consider Z as a closed subscheme of X ,then the ideal sheaf I ′ := I Z is a generalized line bundle with the propertythat b p ( I ) = b p ( I ′ ) for all p ∈ X . We deform I ′ by deforming Z as a closedsubscheme.Fix an open affine neighborhood U of p with the property that p i / ∈ U for i = 1 and Z | U (considered as a subscheme of X ) is defined by ( ǫ, s b +2 )for regular functions ǫ, s with ǫ in the nilradical. A deformation of ( ǫ, s b +2 )over k [[ α ]] is given by the ideal generated by ǫ ( s − α ) , (4.1) s b ( ǫ − α b +1 ( s − α )) ,s b ( s − α ) ,ǫ − αs b +1 + α s b . One may verify that the restriction of this ideal to the generic fiber equals( ǫ, s b ) ∩ ( ǫ − α b +1 ( s − α ) , ( s − α ) ), which geometrically is the union ofa degree 2 Cartier divisor supported at { s = α } and the degree b closedsubscheme contained in the reduced subcurve and supported at p . In par-ticular, the degree of the generic fiber equals the degree of the special fiber,so this family of closed subschemes is flat. We can extend this deformationof Z U to a deformation Z α of Z by defining Z α to be the constant deforma-tion away from U . The associated ideal I ′ α := I Z α is a deformation of I ′ satisfying the conditions of the lemma.This proves the lemma when I = I ′ . To deduce the general case, byLemma 2.7 there exists a line bundle L such that I = I ′ ⊗ L . If we define I α to be the tensor product of I ′ α with the constant family with fiber L ,then I α satisfies the conditions of the lemma. This completes the proof. (cid:4) Theorem 4.6.
Let X be a ribbon and i an integer satisfying ≤ i ≤ ( ( g + 2) / − ¯ g if d is even; ( g + 1) / − ¯ g if d is odd.Define ¯ Z i ⊂ M( O X , P d ) to be the Zariski closure of Z b , where b = (1 , . . . , is the sequence of ’s with length equal to i − if d is even and i − if d is odd. Then ¯ Z i is a g -dimensional irreducible component of M( O X , P d ) . Fur-thermore, if ¯ Z ⊂ M( O X , P d ) is any irreducible component that contains astable generalized line bundle, then ¯ Z = ¯ Z i for some i .Proof. The theorem follows from Lemmas 4.2, 4.4, and 4.5. Observe thata repeated application of Lemma 4.5 shows that ∪ ¯ Z i contains the locus ofgeneralized line bundles. Furthermore, the subsets ¯ Z i are all irreducibleand of dimension g by Lemma 4.4. We now use these facts to prove theproposition.First, let us prove that every irreducible component that contains a sta-ble generalized line bundle is of the form ¯ Z i for some i . Say ¯ Z is an ir-reducible component that contains a stable generalized line bundle. Thesubset of ¯ Z consisting of stable generalized line bundles is non-empty andopen (Lemma 4.2), hence dense. We may conclude that ¯ Z is contained inthe union ∪ ¯ Z j , but each of the ¯ Z j ’s is irreducible, so this is only possible if¯ Z = ¯ Z i for some i .This shows that some of the ¯ Z i ’s are irreducible components of M( O X , P d ).To complete the proof, we must show that every ¯ Z i is a component, so let¯ Z i be given. Certainly this subset is contained in some component, whichwe have shown must be of the form ¯ Z j for some j . In other words, we have¯ Z i ⊂ ¯ Z j . As both are irreducible of dimension g , this is only possible if¯ Z i = ¯ Z j , showing that ¯ Z i is a component. This completes the proof. (cid:4) Theorem 4.6 describes all the irreducible components of M( O X , P d ) thatcontain a stable generalized line bundle. What about the locus parameter-izing sheaves of the form i ∗ E for E a semi-stable rank 2 vector bundle on X red ? We prove the following result. Theorem 4.7.
Let X be a ribbon. If g ≤ g − , then no componentof M( O X , P d ) contains a stable generalized line bundle. Furthermore, themoduli space is empty when ¯ g = 0 and d − g is even. Otherwise M( O X , P d ) is irreducible of dimension (4.2) dim M( O X , P d ) = if ¯ g = 0 ; if ¯ g = 1 , d − g is even; if ¯ g = 1 , d − g is odd; g − if ¯ g ≥ .If g > g − , then M( O X , P d ) has at most one irreducible componentthat does not contain a stable generalized line bundle. When this componentexists, it has dimension g − . This component does not exist when ¯ g ≤ ,but does exist when ¯ g ≥ and g − ≥ g . Theorem 4.7 follows from known results about moduli of vector bundleson non-singular curves, so before giving the proof, we recall the relevantfacts from [LP95].Computing Euler characteristics, we see that the direct image of a rank2 semi-stable vector bundle E on X red corresponds to a point of M( O X , P d ) ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 19 precisely when deg( E ) = d + 2¯ g − g −
1. Thus, if we set e := d + 2¯ g − g − , e ) equal to the coarse moduli space of semi-stable rank 2 vectorbundles on X red of degree e (which exists by, say, Simpson’s work), then therule E 7→ i ∗ E defines a closed embedding M(2 , e ) → M( O X , P d ) with imageequal to the subset parameterizing sheaves of the form i ∗ E . The geometryof M(2 , e ) depends on the genus of X red . There are three cases to consider:¯ g = 0, ¯ g = 1 and ¯ g ≥ g = 0, every rank 2 vector bundle splits as a direct sum of linebundles, so there are no stable vector bundles and the strictly semi-stablevector bundles are of the form O P ( e/ ⊕ O P ( e/
2) for e even. Thus, M(2 , e )is empty when e is odd (i.e. d − g is even) and equal to a point when e iseven (i.e. d − g is odd).The next case to consider is ¯ g = 1. Again, the geometry of M(2 , e ) de-pends on the parity of e . For e odd (i.e. d − g even), every semi-stable rank2 vector bundle of degree e is stable and the determinant map defines an iso-morphism M(2 , e ) → Pic e ( X red ) ([LP95, Thm. 8.29]). In particular, M(2 , e )is irreducible of dimension 1. By contrast, there are no stable vector bundlesof odd degree on X red ([LP95, pg. 160-161]), and M(2 , e ) is irreducible ofdimension 2 ([LP95, Thm. 8.29]).The final case is when ¯ g ≥
2. In this case, there always exists a stable rank2 vector bundle of degree e ([LP95, Thm. 8.28]), and M(2 , e ) is irreducibleof dimension 4¯ g − Proof of Theorem 4.7.
The case where g ≤ g − , e ) → M( O X , P d ) is a set-theoretic bijection, and the claimfollows from the results about M(2 , e ) just reviewed. For the remainder ofthe proof, we assume g > g − g = 0 can be treated without difficulty. By the classificationof vector bundles on P , it is enough to show that if d − g is odd, then thepoint of M( O X , P d ) corresponding to the direct image of O ( e/ ⊕ O ( e/ g + 1, and every such generalized line bundle is the specialization of astable generalized line bundle by Lemma 4.5, combined with Corollary 3.7.We now turn our attention to the case where ¯ g = 1. When d − g is odd,the claim follows from essentially the same argument as in the ¯ g = 0: thereare no stable vector bundles of degree e on X red and every strictly semi-stable rank 2 vector bundle is Gr-equivalent to a generalized line bundle ofindex g −
1. However, when d − g is even, a different argument is needed. Consider first the special case where we not only assume ¯ g = 1 and d − g is even, but also g = 2. Then there are exactly two types of semi-stablesheaves with degree d : line bundles on X and sheaves of the form i ∗ E for E a rank 2 stable vector bundle on X red . The moduli space Pic d ( X ) is notprojective, hence its image in M( O X , P d ) is not closed (because M( O X , P d )is projective). Therefore, its closure in M( O X , P d ) must contain some stablevector bundle. But the natural action of Pic ( X ) on the set of stable rank2 vector bundles of degree e is transitive, so if the closure of Pic d ( X ) con-tains one such vector bundle, then it contains all such vector bundles. Thiscompletes the proof when g = 2.The case where g > g = 2 to g ≥
2. Blowing-up a length g − X red , we obtain a finite morphism f : X ′ → X with g ( X ′ ) = 2. Now suppose that we are given a sheaf of the form i ∗ E for E a stable rank 2 vector bundle on X red . We need to show that i ∗ E is thespecialization of a stable generalized line bundle. By what we have justshown, there exists a family of sheaves I ′ on X ′ × Spec( k [[ α ]]) whose genericfiber is a stable line bundle and whose special fiber is isomorphic to i ′∗ E .The direct image of this family under f realizes i ∗ E as the specialization ofa stable generalized line bundle. This completes the proof in the case that¯ g = 1.The remaining case is where ¯ g ≥ g − ≥ g . Under these conditions,we wish to show that the closure of the image of M(2 , e ) → M( O X , P d ) is anirreducible component. Because M(2 , e ) is itself irreducible, it is enough toshow that the image is not contained in ∪ ¯ Z j . This follows from a dimensioncount: ∪ ¯ Z j has dimension equal to g , which is smaller than or equal to thedimension of M(2 , e ) by assumption. (cid:4) Observe that the proposition shows that there are ribbons with the prop-erty that every semi-stable rank 2 vector bundle is the specialization of astable generalized line bundle, and there are ribbons that do not have thisproperty. However, the proposition does exhaustively analyze this phenome-non: the proposition says nothing when the inequalities ¯ g ≥ g > g − Question 4.8.
Let X be a ribbon such that ¯ g ≥ and g > g − . Doesthere exist an irreducible component of M( O X , P d ) whose general element isa rank vector bundle on X red ? We now prove that M( O X , P d ) is connected. To establish this, we needanother lemma about deformations of generalized line bundles. Lemma 4.9.
Let X be a ribbon and I a generalized line bundle. If the localindex sequence of I is ( b + 1 , b , . . . , b k ) , then I is the specialization of ageneralized line bundle with local index sequence (1 , b , b , . . . , b k ) . ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 21
Proof.
The proof is similar to the proof of Lemma 4.5. The essential pointis to show that the ideal ( ǫ, s b +1 ) can be deformed to an ideal with localindex b at { ǫ = s = 0 } and local index 1 at a second point. One suchdeformation over k [[ α ]] is given by the ideal generated by ǫ, (4.3) s b ( s − α ) . (cid:4) Theorem 4.10.
For a ribbon X , the moduli space M( O X , P d ) is connected.Proof. The case where g − g + 1 ≤ O X , P d ) is, in fact, irreducible. Thus, for the remainderof the proof we assume g − g + 1 > e := d − g + 2¯ g −
1. There are two separate casesto consider: e even and e odd. When e is even, there exist strictly semi-stable generalized line bundles with local index sequence ( d − e ). Fix onesuch sheaf I . We claim that the corresponding point x of M( O X , P d ) liesin every irreducible component of M( O X , P d ). Using Theorem 4.6, repeatedapplications of Lemmas 4.5 and 4.9 show that x lies in every irreduciblecomponent containing a stable generalized line bundle. There is at mostone other irreducible component, which, when it exists, parameterizes rank2 vector bundles on X red . In any case, x must lie in the locus of rank 2 vectorbundles on X red because I is Gr-equivalent to such a sheaf (Lemma 3.3).This proves the proposition when e is even.Now suppose that e is odd. If we replace the strictly semi-stable general-ized line bundle with local index sequence ( d − e ) with a stable generalizedline bundle of local index ( d − e − e even case shows that there is a point x lying inevery irreducible component that contains a stable generalized line bundle.To complete the proof, we must show that the locus of stable rank 2 vectorbundles in M( X, P d ) has non-empty intersection with the locus of stable gen-eralized line bundle. Because there are no semi-stable rank 2 vector bundleson P , we may assume ¯ g ≥ g − g +1 = 1. Then there are at most two types ofsheaves corresponding to points of M( O X , P d ): stable line bundles of degree d and stable rank 2 vector bundles on X red . The locus of line bundles is notclosed in M( O X , P d ) because the Simpson moduli space is projective, butthe line bundle locus is not proper. Thus, the closure of the locus of linebundles must contain at least one point corresponding to a rank 2 vectorbundle on X red , which is what we wished to show.When g − g +1 >
1, we can reduce to the previous case. Indeed, a suitableblow-up f : X ′ → X of X has the property that g ( X ′ ) − g ( X ′ ) + 1 = 1. Wehave just shown that there is a family of stable line bundles on X ′ specializing to a stable rank 2 bundle on X ′ red = X red , and the direct image of this familyunder f is a family of stable generalized line bundles specializing to a rank2 vector bundle on X red . In other words, the intersection of the locus ofgeneralized line bundles in M( O X , P d ) has non-trivial intersection with thelocus of vector bundles on X red , completing the proof. (cid:4) Local Structure.
We now turn our attention to the local structureof the Simpson moduli space. We compute the tangent space dimensionof M( O X , P d ) at a point corresponding to a stable sheaf and then applythis result to determine the smooth locus of the moduli space. The specifictangent space computation we give is the following proposition: Proposition 4.11.
Let x be a point of M s ( O X , P d ) . If x corresponds to astable generalized line bundle I , then we have dim T x M( O X , P d ) = g + b ( I ) . If x corresponds to the direct image i ∗ E of a stable rank vector bundle E on X red , then dim T x M( O X , P d ) = 4¯ g − h ( X red , End( E ) ⊗ N − )= 4 g + 5 − g if g ≥ g − . The proof of the proposition is broken up into two separate lemmas: onecomputing the dimension when x is a generalized line bundle (Lemma 4.12)and one when x corresponds to a rank 2 vector bundle on X red (Lemma 4.13).In both cases, the starting point is the identification of the tangent spacewith the cohomology group Ext ( F , F ). Let us begin by recalling how thisidentification works.Elements of the tangent space are in natural bijection with morphismsSpec( k [ α ] / ( α )) → M( O X , P d ) sending the closed point to x . Because x lies in the stable locus, these morphisms in turn are in natural bijectionwith first-order deformations of F (i.e. deformations over k [ α ] / ( α )). Theisomophism T x M( O X , P d ) ∼ = Ext ( F , F ) is constructed by exhibiting a bi-jection between first-order deformations of F and elements of Ext ( F , F ).If F is a deformation of F , then tensoring the short exact sequence k ∼ = ( α ) ֒ → k [ α ] / ( α ) ։ k with F yields the sequence(4.4) F ∼ = α · F ֒ → F ։ F /α · F ∼ = F . This is an extension of F by F , and hence defines an element c ( F ) ofExt ( F , F ). One shows that the rule F c ( F ) is well-defined and theinduced map T x M( O X , P d ) → Ext ( F , F )is a bijection. As an aside, we remark that when x is strictly semi-stable,there is no longer a canonical identification T x M( O X , P d ) ∼ = Ext ( F , F ).Rather, there is a more complicated description of the tangent space involv-ing a natural action of Aut( F ) on Ext ( F , F ).We now compute Ext ( I , I ) for I a generalized line bundle. ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 23
Lemma 4.12. If I is a generalized line bundle, then we have (4.5) dim(Ext ( I , I )) = g + b ( I ) + h ( X ′ , O X ′ ) − , where X ′ is the associated blow-up of I .If we additionally assume g ≤ g , then this formula simplifies to (4.6) dim(Ext ( I , I )) = g + b ( I ) . Proof.
An inspection of the local-to-global spectral sequence H p ( X, Ext q ( I , I )) ⇒ Ext p + q ( F , F )computing Ext shows that there is a short exact sequence H ( X, End( I )) ֒ → Ext ( I , I ) ։ H ( X, Ext ( I , I )) . We prove the proposition by computing the right-most term and the left-most term in the above sequence. If we write I = f ∗ ( I ′ ) for a line bundle I ′ on a blow-up f : X ′ → X , then the canonical map f ∗ ( O X ′ ) → End( I ) isan isomorphism. As a consequence, H ( X, End( I )) = H ( X ′ , O X ′ ), whichis of dimension g − b ( I ) + h ( X ′ , O X ′ ) − H ( X, Ext ( I , I )). The sheaf Ext ( I , I ) issupported on the points p satisfying b p ( I ) >
0. If we label these points as p , . . . , p n , then the space of global sections decomposes as H ( X, Ext ( I , I )) = ⊕ nj =1 Ext ( b I p j , b I p j ) , where b I p j is the restriction of I to the completed local ring b O X,p j . Using thefree resolution (2.3), one computes dim Ext ( b I p j , b I p j ) = 2 b p j ( I ). Summingover all j , we havedim Ext ( I , I ) = ( g − b ( I ) + h ( X ′ , O X ′ ) −
1) + (2 b p ( I ) + · · · + 2 b p n ( I ))= g + b ( I ) + h ( X ′ , O X ′ ) − . This establishes (4.5). When 2¯ g ≤ g , the nilradical has negative degree,and hence no non-zero global sections. In particular, h ( X ′ , O X ′ ) − (cid:4) We now turn our attention to sheaves of the form F = i ∗ E . Lemma 4.13.
Let E be a stable rank vector bundle on X red . Then wehave (4.7) dim Ext ( i ∗ E , i ∗ E ) = 4¯ g − h ( X red , N − ⊗ End( E )) . If we additionally assume g ≥ g − , then this formula simplifies to (4.8) dim Ext ( i ∗ E , i ∗ E ) = 4 g + 5 − g. Proof.
Like the previous lemma, this is proven using a spectral sequenceargument. For any two O X red -modules A and B , adjunction provides acanonical identification Hom O X ( i ∗ A , i ∗ B ) = Hom O X red ( i ∗ i ∗ A , B ), hence thegroups Ext n ( i ∗ A , i ∗ B ) and Ext n ( i ∗ i ∗ A , B ) are isomorphic. We computeExt ( i ∗ E , i ∗ E ) by working with a spectral sequence describing Ext ( i ∗ i ∗ E , E ). The functor Hom( i ∗ i ∗ , E ) is the composition of the functors G := i ∗ i ∗ and F := Hom O X red ( , E ), so there is a Grothendieck spectral sequence:Ext p (Tor q ( i ∗ ( O X red ) , i ∗ E ) , E ) ⇒ Ext p + q ( i ∗ i ∗ E , E ) . The first four terms of the associated exact sequence of low degree termsare:(4.9)Ext ( E , E ) ֒ → Ext ( i ∗ i ∗ E , E ) → Hom(Tor ( i ∗ O X red ) , E ) → Ext ( E , E ) . Because X red is a non-singular curve, the last term Ext ( E , E ) vanishes. Wecan also compute the second-to-last term. Associated to the short exactsequence N ֒ → O X ։ O X red is a long exact sequence of Tor( , i ∗ E )-groupswhose connecting map ∂ : Hom( N ⊗ E , E ) → Hom(Tor ( i ∗ O X red , i ∗ E ))is an isomorphism. We now compute the dimension of Ext ( i ∗ E , i ∗ E ) usingSequence (4.9):dim Ext ( i ∗ E , i ∗ E ) = dim Ext ( i ∗ i ∗ E , E )= dim Ext ( E , E ) + dim Hom( N ⊗ E , E )= 4¯ g − h ( X red , N − ⊗ End( E )) . The group Ext ( E , E ) is computed by the Riemann–Roch formula. (Note:Hom( E , E ) is 1-dimensional as E is stable.) This proves the first part of theproposition.Now assume g ≥ g −
2. Because E is stable, the rank 4 vector bundleEnd( E ) ⊗ N − is semi-stable ([Laz04, Cor. 6.4.14]) of degree 4 + 4 g − g .Using Serre duality on X red , one checks that the higher cohomology of thisbundle vanishes. Formula (4.8) now follows from the Riemann-Roch formula,completing the proof. (cid:4) One immediate corollary of the proposition is the following:
Corollary 4.14.
Let X be a ribbon. If ¯ g ≥ and g ≥ g − , then thesmooth locus of M( O X , P d ) is equal to the open subset parameterizing linebundles on X .Proof. First, let us prove the weaker claim concerning the smooth locusof the space M s ( O X , P d ) of stable sheaves. It is enough to show that if x ∈ M s ( O X , P d ), then the tangent space dimension of M( O X , P d ) equals thelocal (topological) dimension if and only if x corresponds to a line bundle.Theorems 4.6, 4.7 together with Proposition 4.11 show that equality mustfail except possibly in the following cases: when x corresponds to a stableline bundle and when x corresponds to a stable rank 2 vector bundle on X red . We must show that the second case cannot occur. ODULI OF GENERALIZED LINE BUNDLES ON A RIBBON 25 If x corresponds to a stable rank 2 vector bundle on X red , then the tangentspace dimension is: dim T x M s ( O X , P d ) = 4 g + 5 − g. We have not computed the local dimension of M( O X , P d ) at x , but Theo-rems 4.6 and 4.7 show this local dimension at x is either g or 4¯ g −
3. A directcomputation shows that both numbers are strictly smaller than 4 g + 5 − g ,proving the claim concerning the locus of stable sheaves.What about the strictly semi-stable locus? Because ¯ g ≥
2, every strictlysemi-stable point is the specialization of a stable point that is not a linebundle. Indeed, the strictly semi-stable locus is contained in the image ofM(2 , e ), and the stable locus is dense in M(2 , e ). Because the singular locusis closed, we can conclude that the strictly semi-stable locus is contained inthe singular locus, completing the proof. (cid:4)
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