On Brill-Noether loci over Quot schemes and a Torelli theorem
aa r X i v : . [ m a t h . AG ] O c t ON BRILL-NOETHER LOCI OVER QUOT SCHEMES ANDA TORELLI THEOREM
CRISTINA MART´INEZ
Abstract.
We prove a non abelian Torelli type result for smooth pro-jective curves by working in the derived category of some associatedpolarized Quot schemes and defining Brill-Noether loci and Abel-Jacobimaps on it. Introduction
The derived category of coherent sheaves on a variety X defined overan algebraically closed field k , is a triangulated category whose objects arebounded and coherent cohomology sheaves on X . Due to a result of Orlov([Orl]), an equivalence F : D ( X ) → D ( X ′ ) between derived categories ofcoherent sheaves on smooth projective varieties X, X ′ is always of Fourier-Mukai type, that is, there exists a unique (up to isomorphism) object P ∈D ( X × X ′ ) such that the functor F is isomorphic to the derived functor ofthe pushforward map: Ξ P ( − ) := R q ∗ ( P ⊗ p ∗ ( − )) , where p and q are the projections of X × X ′ onto X and X ′ respectively.For smooth projective curves, a derived equivalence always corresponds toan isomorphism. In particular this implies the classical Torelli Theorem.If there is an equivalence between the derived categories of two smoothprojective curves, then there is an isomorphism between the Jacobians ofthe curves that preserves the principal polarisation, [Be].Let C be a projective irreducible non-singular curve over an algebraicclosed field k . We consider the Jacobian J ( C ) of C which is a projectiveabelian variety parametrizing line bundles of degree 0 on C . The Brill-Noether locus as defined in [ACGH] is the subvariety of J ( C ) parametrizingvarieties of special divisors. The classical Torelli Theorem allows us to re-cover the curve from its Jacobian as a polarized abelian variety.In general, it is of interest to know how much information about a space X can be recovered from Hodge data on X . In the language of motives inthe sense of D. Arapura in [Ar], the classical Torelli theorem implies that theJacobian of a curve C is motivated by the curve, that is, the motive of C iscontained in the category generated from J ( C ) by taking sums, summandsand products. This generalizes to other moduli spaces. For the moduli spaceof vector bundles over a curve, this result was first proved by S. del Ba˜no in[Ba]. Date : October 2011.2000
Mathematics Subject Classification.
Key words and phrases.
Quot schemes, Torelli Theorem, derived categories.
The space of matrix divisors (of given rank and degree) studied by Bifet-Ghione-Letizia in [BGL], is closely related to the moduli space of stable rank n and degree d bundles on C , which since work of Weil has been considereda non-abelian variant of the Jacobian.Here we show that a smooth projective curve C over k is determined by acertain Quot scheme compactification of the scheme of degree d morphismsfrom the curve to the Grassmannian G (2 ,
4) with a certain polarization onthis Quot scheme. The proof uses a Fourier-Mukai functor along the linesof the Beilinson-Polishchuk proof of the classical Torelli theorem, [BP]. Inthe genus 0 case, these spaces are considered in [Mar1] as parameter spacesfor rational ruled surfaces in order to solve a certain enumerative problem.However the Fourier-Mukai functor is defined on a general Quot schemeparametrizing quotient sheaves of a trivial bundle on C . This method wasalso applied to Prym varieties by J. C. Naranjo in [Nar]. Conventions.
Points of a scheme are always closed points. By a sheaf ona scheme X we mean a coherent O X − module. By D ( X ) we denote thederived category of cohomology sheaves on a smooth projective variety X over an algebraically closed field k , (the case of primary interest for us iswhen k is of characteristic 0), and by D b ( X ) its full subcategory of boundedcomplexes. We write H i ( X ) for its cohomology with rational coefficients.For E ∈ D ( X ) a rank n vector bundle on X , we denote by det ( E ) itsdeterminant line bundle.2. Geometry of the Jacobian and Quot schemes
Non abelian Torelli problem for smooth projective curves.
Wefix a complete non-singular genus g curve C over an algebraically closed field k . Let P ic ( C ) be the Picard group of C parametrizing all line bundles over C , and P ic d ( C ) the degree d subset in it. The Jacobian J = J ( C ) of C is an abelian variety such that the group of k − points of J is isomorphic to P ic ( C ) (resp. J d ( C ) is isomorphic to P ic d ( C )). There is a canonical map C → P ic ( C ) (see § I. 3.3.1 of [ACGH]), and upon choice of a point P ∈ C ,we get a map e : C → J ( C ), normalized such that e ( P ) = 0.For every d >
0, we denote by
Sym d C the d th symmetric power of acurve C . By definition, Sym d C is the quotient of C d by the action of thesymmetric group S d . We can identify the set of effective divisors of degree d on C with the set of k − rational points of the symmetric power Sym d C , thatis, Sym d C represents the functor of families of effective divisors of degree d on C . Theorem 2.1. (Torelli) Let C and C be two smooth projective curves ofgenus g > over k . If there is an isomorphism between the Jacobians J ( C ) and J ( C ) preserving the principal polarization then C ∼ = C . The subset in
P ic g ( C ) consisting of line bundles L with h ( L ) = 1 cor-responds to the set of k − points of an open subset in Sym g C . Translatingthis subset by various line bundles of degree − g we obtain algebraic chartsfor P ic ( C ). The Jacobian variety J is constructed by gluing together theseopen charts. It is a consequence of Torelli’s theorem that if Sym g − C ∼ = Sym g − C , then C ∼ = C . N BRILL-NOETHER LOCI OVER QUOT SCHEMES AND A TORELLI THEOREM 3
The next theorem states that the same result continues to hold for all d ≥ Theorem 2.2. [Fak]
Let C and C be two smooth projective curves of genus g ≥ over an algebraically closed field k . If Sym d C ∼ = Sym d C for some d ≥ , then C ∼ = C unless g = d = 2 . It is well known that there exist non-isomorphic curves of genus 2 withisomorphic Jacobians. Everett W. Howe provides some examples in [Ho].2.2.
Varieties of special divisors.
There is a fundamental relation be-tween linear series on C and maps of C to projective spaces expressed inthe language of line bundles. Given a non-zero holomorphic section s of aline bundle L , and denoting by D the divisor of s , there is an isormophismbetween L and O ( D ). The complete linear series L = P H ( C, L ) is the setof effective divisors equivalent to D . A linear series P V , where V is a vectorspace of L is said to be a g rd if deg ( D ) = d and dim ( V ) = r + 1.There exists an algebraic variety G rd parametrizing the series g rd . Weobserve that as g = 0, the space of g rd is precisely the Grassmannian G ( r, d )of r − planes in a d − dimensional vector space. This allows us to see G rd as ageneralized Grassmannian, (see § IV. 3 of [ACGH]).The variety of special divisors W rd ⊂ J d ( C ) = J ( C ) parametrizes linebundles L of degree d such that h ( L ) > r .One has a canonical scheme structure on W rd , since it can be described asthe degeneration locus of some morphism of vector bundles on J d , (see § IV.1 of [ACGH]). The subscheme W g − is exactly the theta divisor Θ ⊂ J g − .All theta divisors in the Jacobian are translations of the natural divisor Θ ⊂ J g − . We have a canonical involution corresponding to the map ν : Θ → Θ ,L → K C ⊗ L − , where K C denotes the canonical line bundle over the curve C and L − is the dual line bundle of L . There is a canonical identificationof P ic ( J g − ) with P ic ( J ) = b J induced by any standard isomorphism J → J g − given by some line bundle of degree g −
1. The correspondingFourier transform F on the derived categories of coherent sheaves on b J and J g − is an equivalence.Denote by Θ ns the open subset of smooth points of Θ. We can identifyΘ ns with an open subset of Sym g − C consisting of effective divisors D ofdegree g −
1, such that h ( D ) = 1.For sufficiently large degree d , the morphism σ d Sym d C σ d → J d D → isomorphism class of O C ( D )is a projective bundle. The fiber of σ d over L is the variety of effectivedivisors D such that O C ( D ) ∼ = L . Further, ( σ d ) − ( L ) ∼ = P H ( C, L ). Letus identify J d with J using the line bundle O C ( dp ), where p ∈ C is a fixedpoint. Then we can consider σ d as a morphism Sym d C → J sending D to O C ( D − dp ). In more invariant terms: Let L be a line bundle of degree d > g − C . Then the morphism σ L : Sym d C → J sending D to O C ( D ) ⊗ L − can be identified with the projective bundle associated with F ( L ), the Fourier transform of L . CRISTINA MART´INEZ
Higher rank divisors.
Let O C be the structure sheaf of the curve C and let K be its field of rational functions, considered as a constant O C module. Following [BGL], we define a divisor of rank r and degree d or( r, d ) divisor as a coherent sub O C -module of K r = K ⊕ r , having rank r anddegree d .This set can be identified with the set of rational points of an algebraicvariety Div r,dC/k which may be described as follows. For any effective ordinarydivisor D , set: Div r,dC/k ( D ) = { E ∈ Div r,dC,k | E ⊂ O C ( D ) r } , where O C ( D ) is considered as a submodule of K r .The space of all matrix divisors of rank r and degree d can be identi-fied with the set of rational points of Quot m O C ( D ) r /C/k parametrizing tor-sion quotients of O C ( D ) r and having degree m = r · deg D − d . It is asmooth projective irreducible variety. As in the Jacobian case, tensoring by O C ( − D ) defines an isomorphism between Q r,d ( D ) = Quot m O C ( D ) r /C/k and Quot m O rC /C/k . Since the whole construction is algebraic, it can be performedover any complete valued field, for example, a p − adic field.Let Q d,r,n ( C ) be the Quot scheme parametrizing rank r coherent sheafquotients of O nC of degree d . It is a fine moduli space that comes equippedwith a universal exact sequence over Q d,r,n ( C ) × C :0 → K → O nQ d,r,n ( C ) × C → E → E is flat over the Q d,r,n ( C ) Quot scheme, that is, foreach q ∈ Q d,r,n ( C ), E q := E |{ q }× C is a coherent sheaf over C and h ( E q ) − h ( E q ) = d + 2 (1 − g ) , is constant by Riemann-Roch. That is, it does not depend on q . Theuniversal subbundle K is a locally free sheaf of rank 2, therefore ϕ : K →O is a morphism of locally free sheaves. Observe that the sheaf E is notlocally free in the points ( p, t ) ∈ Q d,r,n ( C ) × C , where the rank of the map ϕ : K| r,t → O | r,t is 0.By analogy with Sym d C , it is natural to define maps v : Quot m O rC /C/k → J ( C ) , of Abel-Jacobi type. The geometry of the curve C interacts with the ge-ometry of Q d,r,n ( C ) and J ( C ) via these maps. Note that J ( C ) is identifiedwith J d by means of a degree d line bundle. Proposition 2.3.
For d sufficiently large and coprime with r , there is amorphism from the Quot scheme Q d,r,n ( C ) to the Jacobian of the curve J d .Proof. Let M ( r, d ) be the coarse moduli scheme of stable rank r anddegree d vector bundles on C and let U be the universal bundle over C × M ( r, d ). We consider the projective bundle ρ : P d,r,n ( C ) → M ( r, d ) whosefiber over a stable bundle [ F ] ∈ M ( r, d ) is P ( H ( C, F ) ⊕ n ). We take thedegree sufficiently large to ensure that the dimension of P ( H ( C, F ) ⊕ n ) isconstant. Globalizing, we have P d,r,n ( C ) = P ( U ⊕ n ) . N BRILL-NOETHER LOCI OVER QUOT SCHEMES AND A TORELLI THEOREM 5
Alternatively, P d,r,n ( C ) may be thought of as a fine moduli space for n +1 − pairs ( F ; φ , . . . , φ n ) of a stable rank r , degree d bundle F together witha non-zero n − tuple of holomorphic sections φ = ( φ , . . . , φ n ) : O n → F considered projectively. When φ is surjective, it defines a point of the Quotscheme Q d,r,n ( C ), 0 → N → O n → E → N = F ∨ . The induced map ϕ : Q d,r,n ( C ) → P d,r,n ( C ) is a birationalmorphism, so that Q d,r,n ( C ) and P d,r,n ( C ) coincide on an open subschemeand also the universal structures coincide.From the universal quotient O nQ d,r,n ( C ) × C → E Q d,r,n ( C ) × C for all q ∈ Q d,r,n ( C ), we have a surjective morphism O nC → E → . We now consider the canonical morphism to the Jacobian of the curve: det : M ( r, d ) → J d . Then the composition of the morphisms, ρ = det ◦ ρ ◦ ϕ gives a morphismfrom Q d,r,n to the Jacobian J d . (cid:3) Remark 2.4.
For L a degree d line bundle on C , the fiber ρ − ([ L ]) at L = V r F where [ F ] ∈ M ( r, d ) , is isomorphic to P ( H ( C, L ) ⊕ n ) . In particular, if r = 0 then ρ − ([ L ]) corresponds to the variety of higher rank (n,d)-divisors E ⊆ O C ( D ) n . Remark 2.5.
The morphism ρ : Q d,r,n → J d has good functorial proper-ties, that is, it is compatible with pull-backs, pushforwards and products. Inparticular, if we consider a cycle class [ α ] ∈ H i ( J d ) , the pull-back ρ ∗ ([ α ]) defines a cycle class in H i ( Q d,r,n ) . Brill-Noether theory on the Quot scheme.
Recalling the nota-tion of [Mar2], let R C,d be the Quot scheme compactifying the variety ofmorphisms
M or d ( C, G (2 , r and n in Q d,r,n to be 2 and 4 respectively. The image of a curve C by f is ageometric curve in G (2 ,
4) or equivalently a ruled surface in P . For each f : C → G (2 ,
4) there exists a unique corresponding quotient O C → f ∗ Q → R C,d , where Q is the universal quotient over the Grassmannian. Werestrict here to the case of morphisms to the Grassmannian G (2 , f ∗ Q of the universal quotient bundle, is a rank 2vector bundle over C and thus there is a well defined invariant called theSegre invariant.Let us denote by s the Segre invariant s of the bundle f ∗ Q , which isdefined as the minimal degree of f ∗ Q ∨ ⊗ L having a non-zero section andwhich satisfies s ≡ d ( mod
2) and 2 − g ≤ s ≤ g . In particular, when theminimum value s of deg ( f ∗ Q ∨ ⊗ L ) is achieved for some line subbundle L ,then deg ( L ) = d + s . In other words, if L is a line subbundle of f ∗ Q of degree d + s , we say it is a maximal line subbundle of f ∗ Q .Since the universal quotient E is flat over R C,d , for each q ∈ R C,d , E q := E| { q }× C is a coherent sheaf over C and it is isomorphic to the pull-back CRISTINA MART´INEZ f ∗ q Q of the universal quotient bundle by the corresponding morphism f q .The variety Mor d ( C, G (2 , R C,d as the opensubscheme of locally free quotients of O C .Analogously to the case of the Jacobian, we can consider the followingBrill-Noether loci associated with a line bundle L of degree d + s on C for afixed integer k : R kC,d,s = n q ∈ R C,d | h ( C, E ∨ q ⊗ L ) ≥ k, deg L = d + s o = n q ∈ R C,d | h ( C, E q ⊗ K C ⊗ L − ) ≥ k, deg L = d + s o = n q ∈ R C,d | h ( C, E q ⊗ K C ⊗ L − ) ≥ k + 2 g − − s, deg L = d + s o . The subset R kC,d,s has a canonical scheme structure on R C,d , since it canbe described as the degeneration locus of some morphism of vector bundleson R C,d . These sets are analogous to the varieties of special divisors in theJacobian of a curve. Note that in the case k = 1, this scheme correspondsexactly to the points q ∈ R C,d such that f ∗ q Q has Segre invariant s , (this casehas been studied in detail in [Mar2]). Next proposition shows that when s takes the value 2 ( g − R C,d which we will later take as a polarization for R C,d . Proposition 2.6.
If the Segre invariant s takes the value g − , then R C,d,s is a divisor in R C,d .Proof.
By the Krull-Schmidt Theorem, the rank 2 bundle f ∗ Q has aunique decomposition into a direct sum of indecomposable bundles f ∗ Q ∼ = M ⊕ M . There are two possibilities:(1) If f ∗ Q decomposes into a sum of line bundles, then M is a linebundle of degree s and M of degree d − s (resp. M of degree d − s , M of degree s ). The condition for the non-emptiness of the Brill-Noether locus implies s ≤ g (see Proposition 3.1 of [Mar1]). As weare asumming s = 2 g −
2, this case only occurs when the genus of C is 0, 1 or 2.(2) If f ∗ Q is indecomposable M ∼ = Ø C and M ∼ = f ∗ Q (resp. M ∼ = Ø C and M ∼ = f ∗ Q ) and since 2 − g ≤ s ≤ g (see Theorem 2.2 of § V. 2of [Har]) and we are assuming that s = 2 ( g − s = 0. Thus being s ≤ g , the locus R C,d,s is non-empty as was provedin [Mar1].We just need to prove that the stratum R C,d,s is different from the ambientvariety R C,d . Then by Theorem 3.2 of [Mar2], it would follow that R C,d,s isirreducible and has the right codimension 1. But the strata R C,d,s as definedin [Mar2] are dense in R C,d , just when the subbundles M and M are ofmaximal degree d + s (resp. minimal degree d − s ), in which the dimensionof R C,d,s is the maximal one, that is the dimension of R C,d . But as we areassuming s = 2 g −
2, we have s ≤ g , and thus by a dimensional argument itfollows that R C,d defines a divisor in
P ic ( R C,d ). (cid:3) N BRILL-NOETHER LOCI OVER QUOT SCHEMES AND A TORELLI THEOREM 7
Tangent spaces.
Let 0 → N q → O C → E q → q ∈ Q d,r,n ( C ). We consider the tangent space to thatpoint, T q Q d,r,n ( C ) ∼ = Hom ( N q , E q ) ∼ = H ( N ∗ q ⊗ E q ) . If H ( C, N ∗ q ⊗ E q ) ∼ = Ext ( N q , E q ) is trivial, then q is a smooth point in Q d,r,n ( C ) . In that case, we compute using the Riemann-Roch theorem thedimension of T q Q d,r,n ( C ) to be deg ( N ∗ q ⊗ E q ) + r · (1 − g ). Lemma 2.7.
The singular locus of R kC,d,s contains strictly R k +1 C,d,s .Proof . Since the schemes R kC,d,s are determinantal varieties, locally thereexists a morphism g from R kC,d,s to the variety of matrices M k ( n, m ) of rankless or equal than k , such that R kC,d,s is the pull-back by g of the varietyof matrices M k ( n, m ) of rank equal or less than k . Then from PropositionII.2 of [ACGH] on determinantal varieties of matrices, the singular locusof M k ( n, m ) is exactly equal to M k +1 ( n, m ). Thus the image of R kC,d,s isa linear space contained in M k ( n, m ) not meeting M k +1 ( n, m ) which is itssingular locus and then the result follows. (cid:3) Lemma 2.8.
For d sufficiently large depending on s , the expected codimen-sion of R kC,d,s as a determinantal variety is (2 g − s − k ) · k .Proof . R kC,d,s is exactly the locus of degeneration of the morphism ofvector bundles: φ : R π ∗ ( K ⊗ π ∗ ( L − ⊗ K C )) → R π ∗ ( O nR C,d × C ⊗ π ∗ ( L − ⊗ K C )) , where L is a line bundle of degree d + s and d ≡ s mod
2, (see Theorem 3.2 of[Mar2]). Then by the theory of determinantal varieties, a simple dimensioncomputation gives the result. (cid:3)
The Segre invariant admits an obvious generalization to higher rank vec-tor bundles. If we consider any Grassmannian G ( r, n ) of r − planes in an k − dimensional vector space V over k , to any morphism f : C → G ( r, n ) weassociate the universal exact sequence over the Grassmannian:0 → K → O nG → Q → . The pull-back f ∗ Q of the universal quotient bundle by f is now a vectorbundle over C of rank r . Its Segre invariant s ( r ) is the maximal degreeof f ∗ Q ∨ ⊗ L having a non-zero section and satisfying s ≡ d ( mod r ) and2 − g ≤ s ≤ g .The variety of morphisms Mor d ( C, G ( r, k )) has the expected dimension nd − r ( n − r )( g −
1) if d >> Q d,r,n of the variety of morphisms M or d ( C, G ( r, n )) is endowed with a universalexact sequence on C × Q d,r,n :0 → N → V ⊗ O C × Q d,r,n → E → . We can consider similarly to the case of the Grassmannian G (2 , Q d,r,n : Q kd,r,n,s ( r ) = n q ∈ Q d,r,n | h ( C, E ∨ q ⊗ L ) ≥ k, deg L = d + s ( r )2 o . CRISTINA MART´INEZ A polarization and Torelli-type result for the variety R C,d
Let F be a flat family of coherent sheaves on a relative smooth projectivecurve π : C → S , such that for each member of the family, the Euler char-acteristic χ ( C s , F s ) vanishes. We associate to F a line bundle det − R π ∗ ( F )(up to isomorphism) equipped with a section θ F . We define the theta linebundle on J associated with a line bundle L of degree g − C by applyingthis construction to the family p ∗ L ⊗ P on C × J , where P is the Poincareline bundle. Zeroes of the corresponding theta function θ L constitute thetheta divisor Θ L = { ξ ∈ J : h ( L ( ξ )) > } , which gives rise to a polarizationfor the Jacobian J . This means that isomorphism classes of theta line bun-dles have the form det − S ( L ), where S ( L ) is the Fourier-Mukai transformof a line bundle L of degree g − b J supported on C .Beilinson and Polishchuk gave a proof of the Torelli theorem for the Jaco-bian J of a curve in [BP], based on the observation that the Fourier-Mukaitransform of a line bundle of degree g − C , is a coherent sheaf (up toshift), supported on the corresponding theta divisor in J . Here we presentan analogue of the Torelli theorem for the variety R C,d of ruled surfaces,defining a polarization or theta divisor of R C,d . Let C be an algebraic curveof genus g ≥ ν : P ic d + s ( C ) → P ic g − d − s − ( C ) by ν ( L ) = K C ⊗ L − .In this section we construct an exact functor over the Quot scheme, and wedescribe the image of a line bundle on C under this functor, as a consequencewe derive a Torelli theorem for the Quot scheme.Let K C be the canonical bundle over C and π , π be the projection mapsof Q d,r,n ( C ) × C over the first and second factors respectively. Tensoringthe sequence0 → K → O nQ d,r,n ( C ) × C → E → Q d,r,n ( C ) × C with the linear bundle π ∗ ( K C ⊗ L − ), yields the exact sequence:(1)0 → K⊗ π ∗ ( K C ⊗ L − ) → O nQ d,r,n × C ⊗ π ∗ ( K C ⊗ L − ) → E ⊗ π ∗ ( K C ⊗ L − ) → L is a line bundle of fixed degree. The π ∗ direct image of the abovesequence yields the following long exact sequence on Q d,r,n ( C ):0 → π ∗ ( K ⊗ π ∗ ( K C ⊗ L − )) → π ∗ ( O nQ d,r,n ⊗ π ∗ ( K C ⊗ L − )) →→ π ∗ ( E ⊗ π ∗ ( K C ⊗ L − )) → R π ∗ ( K ⊗ π ∗ ( K C ⊗ L − )) →→ R π ∗ ( O nQ d,r,n ⊗ π ∗ ( K C ⊗ L − )) → R π ∗ ( E ⊗ π ∗ ( K C ⊗ L − )) → . The universal sheaf E considered as an object in the derived category D b ( Q d,r,n ( C ) × C ) of the product by viewing it as a complex situated indegree 0, determines the exact functor of tensor product: L ⊗ E : D b ( Q d,r,n ( C ) × C ) → D b ( Q d,r,n ( C ) × C ) . N BRILL-NOETHER LOCI OVER QUOT SCHEMES AND A TORELLI THEOREM 9
The morphism of projection π induces the direct image functor: R ( π ∗ ) : D b ( C ) → D b ( Q d,r,n ( C )) . These two functors combined define the exact functor φ E ( − ) with kernel E to be the full derived functor χ E ( − ) : D b ( C ) → D b ( Q d,r,n ( C ))(2) L R π ∗ ( E L ⊗ π ∗ ( L )) , (3)between the bounded derived categories of coherent sheaves over Q d,r,n ( C )and C respectively. The same object determines another functor: ψ E ( − ) : D b ( Q d,r,n ( C )) → D b ( C )(4) L R π ∗ ( E L ⊗ π ∗ ( L )) . (5) Remark 3.1.
The definition of the functor χ E uses the universal quotientsheaf E which is defined over the product Q d,r,n ( C ) × C . However, since theintegral functor χ E pushes forward the corresponding coherent sheaf on theproduct to Q d,r,n ( C ), we can take any E ⊗ π ∗ N , where N is a line bundleover Quot d,r,n ( C ), to define an integral exact functor on D b ( Quot d,r,n ( C )). Remark 3.2.
Since Q d,r,n ( C ) and C have different dimension, the functor χ E cannot be an equivalence between the corresponding derived categoriesof coherent sheaves. However it is a representable functor or a functor ofFourier-Mukai type of kernel E . Lemma 3.3.
The composition of two full and faithful functors is always fulland faithful.Proof.
We recall that a functor F : C → D is said to be fully faithful, iffor every pair of objects X and Y in C , the morphism of sets: F : Hom C ( X, Y ) → Hom D ( F ( X ) , F ( Y ))is bijective. Now it is clear that if F : C → D and G : D → J are fullyfaithful functors, the composition G ◦ F : C → J satisfies
Hom C ( X, Y ) ∼ = Hom D ( F ( X ) , F ( Y )) ∼ = Hom E ( GF ( X ) , GF ( Y )) , and by the definition of composition of functors, the above composite mapis G ◦ F . Thus G ◦ F is fully faithful. (cid:3) Proposition 3.4.
The compostion functor ψ E ◦ χ E : D b ( C ) → D b ( C ) is aFourier-Mukai equivalence.Proof. Given an object L ∈ D b ( C ), we can see by applying the composi-tion ψ E ◦ ξ E that ψ ( χ ( L )) := R π ∗ ( E ⊗ π ∗ ( R π ∗ ( E ⊗ π ∗ ( L )))) . Now the functor π ∗ π ∗ : D ( Q d,r,n × C ) → D ( Q d,r,n × C ) being the com-position of two full and faithful functors, the pull-back functor π ∗ and theimage direct functor π ∗ , is also fully faithful. Then by Orlov’s result (The-orem 2.2 of [Orl]), it is represented by an object L ∈ D ( Q × C ). This object is constructed by considering a closed embeding u : Q ֒ → P N in a projectivespace given by a very ample invertible sheaf on Q d,r,n , and constructing anobject L ′ ∈ D ( P N × C ) such that L ′ = ( u × id ) ∗ L .Thus π ∗ π ∗ ( . ) = R π ∗ ( L ⊗ π ∗ ( . )), and by the projection formula applied tothe morphism π , we have ψ E ◦ ξ E ( L ) = R π ∗ ( E ⊗ π ∗ R π ∗ E ⊗ L ⊗ π ∗ ( L )) . Observe that π ∗ R π ∗ E is a well defined object in D ( Q × C ) with co-homology sheaves H i ( π ∗ R π ∗ E ) = H i ( E ) ⊗ O Q d,r,n . Finally by recallingthe object N = E ⊗ π ∗ R π ∗ E ⊗ L in D ( Q × C ), we see that the functor ψ E ◦ ξ E : D ( C ) → D ( C ) is a full and faithful functor represented by N . BySerre duality, we have N ∼ = N ∨ ⊗ π ∗ K C . Thus ψ E ◦ χ E , has the left (right)adjoint functor ( ψ E ◦ χ E ) ∗ with kernel ( N ∨ ) ⊗ π ∗ K C , defined on objects by L π ∗ ( N ∨ ⊗ π ∗ ( K C ) ⊗ π ∗ ( L )) where, N ∨ := R H om • ( N , O Q d,r,n ( C ) × C ) , and K C is the canonical sheaf on C .( ψ E ◦ χ E ) ∗ ◦ ( ψ E ◦ χ E ) is the identity functor, id D ( C ) : Ob D ( C ) → Ob D ( C ) , M or D ( C ) → M or D ( C ) , and the right adjoint functor ( ψ E ◦ χ E ) ! N ( . ) = K Q [ dim Q ] ⊗ π ∗ ( N ∨ ⊗ ( . )),where K Q is the canonical sheaf on Q , gives the identity functor by com-posing on the right ( ψ E ◦ χ E ) ◦ ( ψ E ◦ χ E ) ! . (cid:3) Lemma 3.5.
There is an embedding i : C ֒ → Q d,r,n from the curve C to theQuot scheme Q d,r,n .Proof. Consider effective divisors D , . . . , D r on C and positive inte-gers d . . . , d r . For each point p ∈ C , we construct the bundle E p = L ri =1 O C ( − d i [ p ] − D i ).Then E p naturally embeds into C r ⊗ O C , E p ֒ → C r ⊗ O C and E p ֒ → C n ⊗ O C by picking an r − plane in C n .Hence dually, we have a morphism C n ⊗ O C → E ∨ p , thus a point in the Quot scheme. Observe that when C n ⊗ O C → E ∨ p , is notsurjective, it produces points in the boundary of the Quot scheme. (cid:3) Remark 3.6.
Note that the morphism given in Lemma 3.5 is not canonicaland there can be several embeddings C → Q d,r,n depending of the choice ofdivisors. One can think of the Quot scheme as a space of rank r and degree d matrix of divisors as Bifet in ([Bif]), where it is proved that the fixed pointloci for the standard action of the multiplicative group G rn on the space ofdivisors are products of symmetric powers of the curve C , and sometimesone of this factors will be a copy of C .From now on, we assume that r = 2 and n = 4, that is, Q d, , is the Quotscheme R C,d , parametrizing rank 2 quotient locally free sheaves of the rank4 trivial bundle O C . Proposition 3.7.
There is a representable Fourier-Mukai functor F : D ( R C,d ) / / D ( R C,d ) . N BRILL-NOETHER LOCI OVER QUOT SCHEMES AND A TORELLI THEOREM 11
Proof.
Let L i ∗ be the left derived functor of the embedding i : C → R C,d of Lemma 3.5. It is a Fourier-Mukai functor with kernel the structure sheaf O Γ i of the graph Γ i of i : C ֒ → R C,d . Choose a base point p ∈ C , and takethe direct product of C and R C,d over this base point. Then the graph Γ i isdefined by: Γ i : C → R C,d × p Cx ( i ( x ) , x ) . (6)If we apply the following result due to Mukai (see Proposition 5.10 of[Huy]) to the objects O Γ i ∈ D ( R C,d × C ) and E ∈ D ( R C,d × C ), and themorphisms R C,d × C × R C,dp v v mmmmmmmmmmmm p ( ( QQQQQQQQQQQQ p (cid:15) (cid:15) R C,d × C R
C,d × R C,d C × R C,d we have that the composition F := χ E ◦ L i ∗ : D ( R C,d ) → D ( R C,d ) D ( C ) χ E / / D ( R C,d ) D ( R C,d ) χ E ◦ L i ∗ rrrrrrrrrr L i ∗ d d JJJJJJJJJ , is represented by the object R = ( R p ∗ )( p ∗ O Γ i ⊗ p ∗ E ).It has the right adjoint functor F ! with kernel R ∨ ⊗ K R C,d [ dimR C,d ],where K R C,d is the canonical sheaf on R C,d . Thus there is a map id R C,d → F ! ◦ F . (cid:3) Remark 3.8.
Observe that the functor F is not fully faithful since thefunctor χ E is not fully faithful and thus is not an equivalence. Remark 3.9.
Note that the existence of the right adjoint functor impliesthe existence of the left adjoint functor F ∗ (and reciprocally) by means ofthe formula: F ! = S R C ◦ L F ∗ ◦ S − R C , where S R C = ( . ) ⊗ K R C,d [ dim R C,d ], is the Serre functor on D b ( R C ).Let us denote by f the composition functor ψ E ◦ χ E : D b ( C ) → D b ( C )of proposition 3.4 and by f ∗ its left adjoint functor. As we showed thecomposition on the left by the adjoint functor gives the identity functor andthus we can talk of the involutive property of f which allows to recover thecurve C as we see in the next theorem.By Serre duality, the functor ν ( − ) : D ( C ) / / D ( C )(7) L L − ⊗ K C , (8)defines a Fourier-Mukai involution. Theorem 3.10.
For d > g − , and L ∈ P ic d + s ( C ) , the module χ E ( ν ( L ))) in D b ( R C,d ) is a coherent sheaf F supported by the divisor R C,d, g − on R C,d . Moreover, the restriction of F to the non-singular part of this divi-sor (understood as a polarization for R C,d ) is a line bundle and L can berecovered from this line bundle.Proof. Let L be a line bundle of degree d + s on C ( d ≡ s mod 2). Itis an object in D b ( C ) and we can consider the module F := χ E ( ν ( L )) = R π ∗ ( E ⊗ π ∗ ( K C ⊗ L − )). We observe that F is a coherent sheaf supportedon the divisor R C,d, g − in R C,d , that is, on the locus of points q ∈ R C,d such that h ( C, E ∨ q ⊗ L ) ≥
1, or dually h ( C, E q ⊗ K C ⊗ L − ) ≥ F is the derived pushforward of E ⊗ π ∗ ( K C ⊗ L − ), so thatwe can represent it as the cone of the morphism of vector bundles on R C,d ,[Mar2]:(9) φ : L := R π ∗ ( K⊗ π ∗ ( L − ⊗ K C )) → L := R π ∗ ( O nR C,d × C ⊗ π ∗ ( L − ⊗ K C )) , that is, by the complex V . = [ L → L ] in D b ( R C,d ).Since χ E is an isomorphism outside of R C,d, g − , it is injective and F = cokerφ . Moreover, when s = 2 ( g − R C,d, g − is a divisor and it is apolarization for R C,d . We see that detφ = det R π ∗ ( E ⊗ π ∗ ( K C ⊗ L − )) = det (cid:0) R π ∗ ( K ⊗ π ∗ ( L − ⊗ K C )) (cid:1) ⊗ det (cid:16) R π ∗ ( O nR C,d × C ⊗ π ∗ ( L − ⊗ K C ) (cid:17) − . The line bundle det ( χ E ( ν ( L )) − on R C,d has a canonical (up to multipli-cation by a non-zero scalar) global section θ φ ∈ H ( R C,d , det ( χ E ( ν ( L ))) − )whose zero locus constitutes a divisor which is supported on R C,d, g − , thatis, det F is an equation of R C,d, g − .If we change the resolution L → L for F , the pair ( det F, θ φ ) gets replacedby an isomorphic one.There is a natural locally closed embedding j : R C,d, g − ֒ → R C,d .Let R nsC,d,s be the set R C,d, g − \ R C,d, g − . This set corresponds to thenon-singular part of R C,d,s by Lemma 2.7. Let us denote by j ns the inducedembedding R nsC,d,s ֒ → R C,d .The singular locus of R C,d, g − contains R C,d, g − and by Lemma 2.8 ithas codimension in R C,d greater than 2 so that M := j ns ∗ F .Next we prove that F is the pushforward F = j ns ∗ M by j ns of a linebundle M on the non singular part of R C,d, g − .The restriction of F to the non singular part R nsC,d, g − of the divisor R C,d, g − is a line bundle from which F can be recovered by taking thepush-forward with respect to the induced embedding j ns . By the basechange theorem for a flat morphism: L j ns ∗ F ∼ = R π ∗ ( E ⊗ π ∗ ( K C ⊗ L − )) | C × R nsC,d, g − . N BRILL-NOETHER LOCI OVER QUOT SCHEMES AND A TORELLI THEOREM 13
Since h ( C, E q ⊗ K C ⊗ L − ) = 1 for every q ∈ R nsC,d, g − , by applying thebase change theorem again, we deduce that rk M | q = 1 for every q ∈ R nsC,d, g − . Since R nsC,d, g − is reduced, M is a line bundle on R nsC,d, g − .We can characterize the set M of all line bundles on R nsC,d, g − in termsof ( R C,d , R nsC,d, g − ). The set M has two properties:(1) For every M ∈ M , M ⊗ ν ∗ M ∼ = K R nsC,d, g − , where ν is the map L → K C ⊗ L − .(2) The class of M generates the cokernel of the map(10) P ic ( R C,d ) → P ic ( R nsC,d, g − ) . Since the Picard variety has the structure of a polarized abelian variety,the morphism (10) is a homomorphism of abelian groups induced by theinclusion j : R nsC,d,s ֒ → R C,d . Its cokernel coker ( P ic ( R C,d ) → P ic ( R nsC,d,s )) isa group. Moreover it is isomorphic to Z for d sufficiently large, (Prop. 2.3).Thus to recover the curve C from ( R C,d , R
C,d, g − ), by the involutiveproperty of the integral transform f , we can recover the curve C by taking aline bundle M on M , extending it to R C,d, g − by taking the image underthe pushforward map j ∗ , and then applying again the integral functor ψ E .Since F = j ∗ ns M , if we apply the adjoint functor f ∗ of f to f ( ν ( L )) := ψ E ( j ns ∗ ( M )), ( f ∗ ◦ f )( ν ( L )), we recover the line bundle L and thus we getthe Torelli result.We just need to show that M ∈ M . First, we apply duality theory to theprojection π : R C,d × C → R C,d to prove that(11) R Hom ( F, O R C,d ) ∼ = ν ∗ F [ − , where ν : P ic ( C ) → P ic ( C ) corresponds to the involution L → L − ⊗ K C .Applying the left derived functor L j ns ∗ to the isomorphism (11), we obtain(12) R Hom ( L j ns ∗ F, O R nsC,d, g − ) ∼ = ν ∗ L j ns ∗ F [ − . Since L j ns ∗ F has locally free cohomology sheaves, this implies that M − ∼ = ν ∗ L j ns ∗ F [ − L − j ns ∗ F ∼ = L − j ns ∗ j ns ∗ M ∼ = M ⊗ O R nsC,d, g − ( − R C,d, g − ) , and that ν ∗ M − ∼ = M ( − R C,d, g − ), which proves condition (1) of M .In order to prove the second condition, we consider the universal quotient E| { p }× R C,d, g − restricted to { p } × R C,d, g − . The line bundle M − on R nsC,d, g − = { q ∈ R C,d | h ( C, E ∨ q ⊗ L ) = 1 , L ∈ P ic d + s ( C ) } is isomorphic to j ns ∗ p ∗ ( O ( R C,d, g − ))( − R p ), where p : C × R C,d → R C,d , j ns is the embedding j ns : R nsC,d,s ֒ → R C,d and R p := R C,d, g − ∩ { p } × R C,d .Therefore M − ∼ = α ∗ ( O R C,d ( − R p )) which generates the cokernel of the map P ic ( R C,d ) → P ic ( R nsC,d, g − ). (cid:3) Remark 3.11.
Observe that we can get the same kind of result by usingthe Fourier-Mukai transform of Proposition 3.7. We see that F ( i ∗ ( ν ( L ))) := χ E ( i ∗ ( i ∗ ( ν ( L ))), by the definition of the functor F . Remark 3.12.
Note that once we recover the line bundle L supported on C , the Torelli result follows from the classical Torelli theorem from whichone can recover the curve from the Jacobian J ( C ). Consider the morphism ρ of Proposition 2.3. Then the functor χ E extends naturally to a functordefined on D ( J ), composing χ E by the right with the pull-back functor ρ ∗ : D ( J ) → D ( Q ). Corollary 3.13.
Given two smooth projective curves C and C , if thereexists an isomorphism f : ( R C ,d , θ ) ∼ → ( R C ,d , θ ) of polarized Quot-schemes, then C ≃ C .Proof. By Theorem 3.10, the restriction of F = χ E ( ν ∗ ( L )) to the non-singular part of θ i ( L ∈ P ic d + s ), is a line bundle j ns ∗ M and L can be recov-ered from this line bundle since M := j ns ∗ F . Therefore f | supp ( j ns ∗ ) : supp ( j ns ∗ M ) ∼ → supp ( j ns ∗ M ) , and C ∼ = C , where j : θ ns ֒ → R C ,d , and j : θ ns ֒ → R C ,d . (cid:3) Acknowledgments.
The idea to use a Fourier-Mukai functor applied to thiskind of problem, was inspired by the work of A. Polishchuk and A. Beilinson,[BP].
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