Koszul modules with vanishing resonance in algebraic geometry

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the stability of sufficiently positive rank 2 vector bundles on curves is governed by resonance.


INTRODUCTION
Given a suitably nice space (for instance a compact Kähler manifold) X, one can view its cohomology ring H • (X, C) as a module over its exterior algebra E := H 1 (X, C).Multiplication with a class a ∈ H 1 (X, C) defines a complex on H • (X, C) and the jump loci for the cohomology of these complexes lead to the definition of the resonance variety R(X) of X, which turned out to be instrumental in several investigations involving generic vanishing on varieties, see for instance [8], [15], [22], [26].This definition has then been extended by Suciu and Papadima [32] first to the case of finitely generated groups and then in [33] to an entirely algebraic context.For important applications of these techniques to Torelli groups we refer to [14] and references therein.Closely related to the concept of resonance is the definition of a Koszul module, initially introduced in [32] and [33] to explain via the BGG correspondence homological properties of Alexander invariants (and more generally of quadratic algebras), then further studied in a purely algebraic context in [1] and [2].We recall now this set-up.
where the map in question is the projection 2 V ⊗ S → ( 2 V /K) ⊗ S composed with the Koszul differential δ 3 .The grading is inherited from Sym V under the convention that 2 V /K is placed in degree 0. It is straightforward to see that the graded piece W q (V, K) of the Koszul module can be identified with the cohomology of the complex It is shown in [33, Lemma 2.4] that the support of the Koszul module W (V, K) in the affine space V ∨ , if non-empty, coincides with the resonance variety In particular, W (V, K) has finite length if and only R(V, K) = {0}.In [2] (see also [1,Theorem 1.3]), we found an optimal characterization of those subpaces K ⊆ 2 V having trivial resonance and established the following equivalence: (1) R(V, K) = {0} ⇐⇒ W q (V, K) = 0 for q ≥ dim(V ) − 3.
We refer to Theorem 2.2 for a precise formulation of this result.The paper [2] presents applications of the equivalence (1) to geometric group theory in the case G is a finitely generated group, V = H 1 (G, C) and K ⊥ = Ker ∪ G : 2 H 1 (G, C) → H 2 (G, C) .On the other hand, we explained in [1] how by specializing to the tangent developable of a rational normal curve in P g , one can prove Green's Conjecture [21] on syzygies of generic canonical curves of genus g by applying the equivalence (1) to the case of the Weyman module, which is a particular Koszul module corresponding to the choice V = Sym n−1 (U ) and K = Sym 2n−4 (U ), with U being a 2-dimensional vector space.This has led to an alternate approach to Green's Conjecture (including an essentially optimal result in positive characteristic) different from the one of Voisin's [38], [39].This paper is devoted to the study of other important classes of Koszul modules with vanishing resonance that appear naturally in algebraic geometry.First, recalling that V is an n-dimensional complex vector space, we note that if R(V, K) = {0} then dim(K) ≥ 2n−3.We provide a refinement involving multiplicities of the equivalence (1) in the case of (2n−3)-dimensional subspaces K ⊆ 2 V as an equality of two particular divisor on the Grassmannian G := Gr 2n−3 2 V parametrizing such subspaces, see Theorem 3.4.Denoting by D Kosz the divisor consisting of subspaces [K] ∈ G such that W n−3 (V, K) = 0 (with its natural scheme structure) and by D Res the divisor consisting of those [K] ∈ G with R(V, K) = {0}, we have an equality of divisors (2) D Kosz = (n − 2) • D Res on the Grassmannian G.An immediate application of the equality ( 2) is then the calculation of what we call the resonance divisor of a morphism of vector bundles φ : where E and F are vector bundles on a stack X with rk(E) = e and rk(F) = 2e − 3. We denote by Res(φ) the locus of points x ∈ X such that the map φ(x) : 2 E(x) → F(x) contains a pure tensor 0 = s 1 ∧ s 2 in its kernel.A parameter count quickly shows that when φ is sufficiently general, Res(φ) is a divisor on X.
Theorem 1.1.Given a morphism φ : 2 E → F of vector bundles over X with rk(E) = e and rk(F) = 2e − 3, assuming Res(φ) is a divisor on X, its class is given by the formula Theorem 1.1 has numerous applications in moduli theory, one of them on the Kodaira dimension of the moduli space of Prym varieties having been presented in [18].While referring to Theorem 1.5 for further applications to K3 surfaces, we discuss one consequence of Theorem 1.1 to the geometry of the moduli space M g,n of n-pointed stable curves of genus g.For a smooth curve C, a canonical pencil is the degree 2g − 2 cover C → P 1 induced by two canonical forms without common zeroes.Since C has a (2g − 4)-dimensional family of canonical pencils each of them having finitely many ramification points, imposing the condition that 2g − 3 marked points are ramification points of such a pencil yields a divisorial condition in moduli.
Theorem 1.2.The class of the divisor Cp g of pointed curves [C, x 1 , . . ., x 2g−3 ] ∈ M g,2g−3 such that x 1 , . . ., x 2g−3 are ramification points of a canonical pencil on C is equal to Here λ is the Hodge class, whereas ψ i denotes the cotangent class on M g,2g−3 corresponding to the i-th marked point.Theorem 1.2 follows directly from Theorem 1.1 by letting E to be the Hodge bundle on M g,2g−3 , whereas F is the vector bundle having as fibre over a point . Koszul modules associated to vector bundles.One can naturally associate a Koszul module to any vector bundle as we shall describe next.Suppose E is a vector bundle on an algebraic variety X and consider the determinant map This gives rise to the following Koszul module If we let R(X, E) := R(V, K) for V and K as above, then the non-triviality of the resonance amounts to the vector bundle E carrying a subpencil, that is, a line subbundle L with h 0 (X, L) ≥ 2. We show in §4, that the equivalence (1) can be reformulated in this context as follows: Theorem 1.3.Let X be a projective variety with H 1 (X, O X ) = 0 and let E be a globally generated vector bundle on X.Then one has an isomorphism where M E denotes the kernel of the evaluation morphism Theorem 1.3 is particularly interesting for a polarized K3 surface (X, L), where L is an ample line bundle on X with L 2 = 2g − 2. Recall that the Mukai vector of a sheaf E on X is defined as v(E) := rk(E), det(E), χ(E) − rk(E) ∈ H • (X) and that M L (v) denotes the moduli space of L-semistable sheaves on X having Mukai vector v, see §4.1 for further details.A Lazarsfeld-Mukai bundle is a globally generated vector bundle E is also a Lazarsfeld-Mukai bundle with vector v(M ∨ E ) = (s, L, r).Lazarsfeld-Mukai bundles have been instrumental in Voisin's proof of the Generic Green Conjecture [38], [39], Lazarsfeld's proof of the Petri Theorem [27], or in the recent proof of the Mercat Conjecture [7].In the case of K3 surfaces, Theorem 1.3 implies the following result: Theorem 1.4.Let X be a polarized K3 surface with Pic(X) = Z • L and let E be a Lazarsfeld-Mukai bundle on X with v(E) = (r, L, s).Then for all b ≥ r + s − 1 one has In connection with Green's Conjecture, of particular relevance is the case g = 2r for r ≥ 2, when E is the unique Lazarsfeld-Mukai bundle on X having Mukai vector v(E) = (r, L, 2).Theorem 1.4 reads in this case Remarkably, an independent geometric proof of the vanishing H 1 X, Sym r+1 E = 0 (whose failure is a divisorial condition on the moduli space F g of polarized K3 surfaces of genus g) would give yet another proof, different from Voisin's [38], [39] or from that of [1] of the Generic Green Conjecture.Note that in this case The Voisin curve of a polarized K3 surface of odd genus.Assume now that (X, L) is a polarized K3 surface of odd genus g = 2r + 1 ≥ 11, with Pic(X) = Z • L. The moduli space X := M L (2, L, r) turns out to be a smooth K3 surface, called the Fourier-Mukai partner of X.Furthermore, as explained in [30], there is a canonical way to endow X with a genus g polarization h.
We fix a general curve C ∈ |L|, thus via Lazarsfeld's result [27], the curve C is Petri general of genus 2r + 1 and W 1 r+2 (C) is a smooth curve.Voisin [37] associated to any pencil [4].This assignment induces a map W 1 r+2 (C) → X.Since for a general [E] ∈ X, the restriction E A has canonical determinant and h 0 (C, ω C ) = 2h 0 (X, E) − 3, we observe that the locus of vector bundles [E] ∈ X whose restriction to C has non-trivial resonance is a curve on X, which we call the Voisin curve of the pair (X, C).We have the following application of Theorem 1.1 concerning the class of this curve: Furthermore, the map

Gaussian Koszul modules.
Gaussian maps provide another context where Koszul modules appear naturally.Suppose L is a very ample line bundle on a complex projective variety X and denote by ϕ L : X ֒→ P r = P H 0 (X, L) ∨ the corresponding embedding.Let I ⊆ O P r be the ideal sheaf of X.We then consider the Gaussian map The cokernel of ψ L parametrizes deformations of the cone over the embedded variety X ⊆ P r inside P r+1 .Wahl showed [40] that for a curve C lying on a K3 surface, the map ψ ω C is not surjective.A remarkable converse of this result has been recently established by Arbarello-Bruno-Sernesi [5].
We fix (X, L) and set V := H 0 (X, L) ∨ and K ⊥ := Ker(ψ L ), to obtain a Koszul module whose resonance is always trivial.We have the following result, indirectly concerning the Koszul module G(X, L): Theorem 1.6.Let X ⊆ P r be a smooth variety satisfying q(X) = 0 and H 0 X, Ω 1 X (1) = 0.If X b ⊆ P r is the b-th infinitesimal neighborhood of X defined by the ideal I b+1 , then the maps are isomorphisms for all b ≥ a ≥ r.
A more general version of Theorem 1.6, without any assumptions on X, is provided by Theorem 5.2.To place this result into context, we recall that Hartshorne [23] showed that for a vector bundle F on P r and a closed subvariety X ⊆ P r , for all j ≥ 0 the maps A quantitative version of these results for F = O P r (a) has been recently obtained in [10]: The restriction maps are isomorphisms for all j ≥ 0 as long as b ≥ dim(X) + a + 1, see [10,Remark 2.18].Our Theorem 1.6 can be viewed as a significant improvement (under certain assumptions) of this result at the level of global sections.Concerning the hypothesis of Theorem 1.6, they are satisfied for most Fano varieties (for instance for all Hermitian Symmetric Spaces of type A, B, C or D, see [36]).Also, if X is a Fano threefold then always q(X) = 0, whereas from the Iskosvskikh-Mukai classification it follows that the condition H 0 (X, Ω 1 X (1)) = 0 implies that X is of index one and has genus 10 or 12, see [25].The hypothesis of Theorem 1.6 are also satisfied for many varieties of Kodaira dimension zero.For instance, if (X, L) is a polarized K3 surface of degree L 2 = 2g − 2, the condition H 0 X, Ω 1 X ⊗ L = 0 is equivalent to the statement that a general curve of genus g lies on a K3 surface and is thus satisfied if and only g ≤ 9 or g = 11, see [9].
Acknowledgment.Above all, we acknowledge the important contribution of S ¸tefan Papadima.This paper, which is a natural continuation of [1] and [2] is part of a project that was initiated by him.We also profited from numerous discussions with Alex Suciu related to this circle of ideas.

BASICS ON KOSZUL MODULES
We recall the basic definitions of Koszul modules following [1], [2], [33].For simplicity, we stick to characteristic zero and let V be a complex vector space of dimension n ≥ 2 and denote by S := Sym V the symmetric algebra of V .We consider the standard grading on S, where the elements in V are of degree 1.We fix a linear subspace K ⊆ 2 V of dimension m and denote by ι : K → 2 V the inclusion and let K ⊥ := Ker(ι ∨ ) ⊆ 2 V ∨ .We introduce the Koszul differentials We have a decomposition δ p = q δ p,q into graded pieces, where (4) δ p,q : The Koszul module W (V, K) defined in the Introduction is a graded S-module, whose degree q component has the following description. (5) Since the Koszul complex is exact, it is often convenient to realize W q (V, K) as the middle cohomology of the following complex of vector spaces: As pointed out in [33] and further explained in [1], [2], the construction of Koszul modules displays good functoriality properties.For instance, if K ⊆ K ′ ⊆ 2 V are linear susbspaces, one has an induced surjective morphism of graded S-modules 2.1.Resonance varieties.Building on work of Green-Lazarsfeld [22], Dimca-Papadima-Suciu [15] and others, Papadima and Suciu [33] gave an algebraic definition of the resonance variety associated to a pair (V, K) as above, which we now recall.
Definition 2.1.The resonance variety associated to the pair (V, K) is the locus The resonance variety R(V, K) is the union of 2-dimensional subspaces of V ∨ parameterized by the intersection PK ⊥ ∩ Gr 2 (V ∨ ), where Gr 2 (V ∨ ) ⊆ P 2 V ∨ is the Pl ücker embedding.Setting up the diagram where Ξ ⊆ PV ∨ ×Gr 2 (V ∨ ) is the incidence variety, we observe that R(V, K) is the affine cone over the following projective variety which we refer to as the projectivized resonance variety of (V, K).Note that the correspondence )).It was showed in [33,Lemma 2.4] that away from 0, the support of the graded Smodule W (V, K) inside V ∨ coincides with the resonance variety R(V, K).In particular, ( 9) In [1] we provide a sharp vanishing result for Koszul modules with vanishing resonance.This is the starting point for many of the geometric applications in this paper.Theorem 2.2.Let V be a complex n-dimensional vector space and let The connection between resonance and Koszul modules shows that the resonance carries a natural scheme structure which, in some cases might be non-reduced.In the forthcoming paper [3] we shall have a close look at this phenomenon.
2.2.Isotropy and separability.(see [3]) In geometric situations (like those when the resonance variety parametrizes complexes with jumping cohomology in the spirit of [15]), the resonance variety R(V, K) often enjoys further properties, which we summarize in a definition.Before formulating it, let E := V ∨ be the exterior algebra on the vector space V ∨ , and write U E for the ideal in E generated by a subset U ⊆ E.
• Strongly isotropic if it is separable and isotropic, that is, if Similar definitions can be given for the projective subspaces of PV ∨ .
Definition 2.4.We say that the resonance variety • Isotropic, separable, or strongly isotropic if it is linear and each component For the relevance of these conditions in the case of resonance varieties associated to hyperplane arrangements we refer to [12].In the paper [3] we relate separability to the reduceness of the projectived resonance scheme and establish an optimal effective version of Chen's rank conjecture for Koszul modules with strongly isotropic resonance.
Note that if two lines contained in R(V, K) intersect, then the whole plane they generate is contained in R(V, K).If [a] ∈ R(V, K), then the projectivization of the subspace Proof.Consider F the kernel of the composed sheaf morphism on Then F(a) as defined above can be identified with the fibre of F at [a] and we apply Grauert's Theorem.
Proof.We prove first that the line spanned by a and b.Denote this subspace by V ∨ .By semicontinuity, for each a ′ in a neighborhood of Without loss of generality, we assume a n → a in R(V, K) and dim(F(a n )) = 2 for all n.Hence we obtain a sequence of lines (ℓ n ) n contained in R(V, K) and different from ℓ ab , converging to the limit ℓ ab .This corresponds to a sequence of points in which is true by hypothesis.
Proposition 4.10 provides one application of Proposition 2.6 in geometric setting.

THE CHOW FORM OF THE GRASSMANNIAN OF LINES AND ALTERNATING DEGENERACY LOCI
We begin by recording a well-known sufficient conditions for the supports of two Cartier divisors on an algebraic variety to be equal.Let X be a smooth quasi-projective variety and A and B vector bundles on X of the same rank r and let ϕ : A → B be a vector bundle morphism.Assume its degeneracy locus is a Cartier divisor on X, that is, D(ϕ) = X and that for any point x in an irreducible component Z of D(ϕ), we have dim Ker(ϕ(x)) ≥ k.Then Z enters with multiplicity at least k in D(ϕ).We shall use the following well-known fact, presented here for the convenience of the reader.
Proof.The hypotheses imply that the only effective divisor D on U whose rational class is zero is the zero-divisor itself.Indeed, if D = 0, then its closure D in Y satisfies D • H dim(Y )−1 > 0 for any ample divisor H on Y , a contradiction.We apply this to the divisor D := D 1 − kD 2 , which is effective, for, as explained, D 2 enters in D 1 with multiplicity at least k.
Throughout this section let V be an n-dimensional complex vector space and set G := Gr 2n−3 2 V .Theorem 2.2 offers a set-theoretic description of the Koszul divisor The fact that D Kosz is a divisor on G follows once we observe that if U is the universal rank-(2n − 3) subbundle on G, then D Kosz is the degeneracy locus of the morphism which in the fiber over a point [K] ∈ G is given by the Koszul differential δ 2,n−3 .Theorem 2.2 implies that γ is non-degenerate; for instance if we write V = Sym n−1 (U ), with U being a 2-dimensional vector space, then we have established in [2] that the point and therefore γ is non-degenerate and D Kosz is a genuine divisor on G.
On the other hand, we can consider the Cayley-Chow form of the Grassmannian Gr 2 (V ∨ ) ⊆ P 2 V ∨ .Explicitly, this divisor is the locus and comes with an induced scheme structure.Theorem 2.2 (see Theorem 1.3 from [1] for a version in positive characteristic) can then be formulated as a set-theoretic equality: (11) Supp(D Res ) = Supp(D Kosz ).
The divisor classes of D Res and D Kosz are easy to describe in terms of the generator L = det(U ∨ ) of the Picard group Pic(G), which is the hyperplane section bundle coming from the Pl ücker embedding of G.It follows from (10) that the degree of D Kosz equals the dimension of Sym n−3 (V ), which proves that the divisor class of D Kosz equals (12) [ To compute the class of D Res we recall that the degree of a Cayley-Chow form equals the degree of the variety to which it is associated [13, Corollary 2.1], which in our case is equal to the Catalan number [17,Proposition 4.12].Hence, we have that the divisor classes of D Kosz and D Res are related by (13) [ Lemma 3.2.We have a set-theoretical inclusion Supp(D Res ) ⊆ Supp(D Kosz ).
Since it is generated in degree zero, it follows that W q (V, K) = 0 for all q ≥ 0, and in particular 8) and ( 9) that we may find a basis {v V is the codimension one subspace with basis v i ∧v j with 1 ≤ i < j ≤ n and (i, j) = (1, 2).A direct calculation shows that the Hilbert series of W (V, K ′ ) equals q≥0 (q + 1)t q , while (7) proves that the graded module W (V, K ′ ) is a quotient of W (V, K), concluding our proof.
The following result is a refinement of Theorem 2.2 and provides an explicit description, including multiplicities, of the Chow form of the Grassmannian Gr 2 (V ∨ ) in its Pl ücker embedding.Proof.If n = 3 then m = 3 and therefore K = 2 V , which implies that W (V, K) = 0. Assume from now on n ≥ 4 and we take D 1 := D Kosz and D 2 := D Res , for which we apply Lemma 3.1: we know by Lemma 3.2 that Supp(D 2 ) ⊆ Supp(D 1 ), while (13) shows that ; by Lemma 3.3 with q = n − 3, it follows that over the point [K] ∈ Supp(D 2 ) the fiber of the map (10) is the locus of curves with a non-trivial (n − 2)nd syzygy in their canonical embedding.
It would be highly interesting to establish a direct geometric connection between these equalities and also explain the occurrence of the same multiplicity n − 2.

3.1.
The resonance divisor of a skew-symmetric degeneracy locus.We present now an application of Theorem 3.4 to a situation appearing frequently in moduli theory.Assume we are given two vector bundles E and F over a stack X such that rk(E) = e and rk(F) = 2e − 3 where e ≥ 3, and a generically surjective morphism of vector bundles Identifying the Grassmannian Gr 2 E(x) ⊆ P 2 E(x) of lines in the fibre E(x) over a point x ∈ X with the (projectivization of the) space of rank 2 exterior tensors on E(x), the numerical conditions at hand imply that the locus is a virtual divisor on X.We assign a divisor structure to this locus as follows.
Let Σ be the variety consisting of pairs (ϕ, K), where ϕ ∈ Hom 2 C e , C 2e−3 , and which immediately leads to the claimed formula.

KOSZUL MODULES ASSOCIATED TO VECTOR BUNDLES
We now discuss a class of Koszul modules naturally associated to vector bundles.For a vector bundle E on a projective variety X, we consider the determinant map Definition 4.1.The Koszul module associated to the pair (X, E) as above is defined as The triviality of the resonance variety R(X, E) associated to the Koszul module W (X, E) has a transparent geometric interpretation.Proposition 4.2.One has R(X, E) = {0} if and only if E has no locally free subsheaf of rank one L with h 0 (X, L) ≥ 2.
Proof.Indeed, via (9), the resonance R(X, E) is non-trivial if and only if we can find sections s 1 , s 2 ∈ H 0 (X, E) with 0 = s 1 ∧ s 2 ∈ K ⊥ = Ker(d), which in turn is equivalent to the fact that s 1 and s 2 generate a rank-one subsheaf whose double dual is a locally free subsheaf of E.
If the vector bundle E in Definition 4.1 is globally generated, then the corresponding Koszul module can be given a geometric description in terms of kernel bundles: Theorem 4.3.Let X be a projective variety and let E be a globally generated vector bundle on X such that the determinant map is not identically zero.If we denote by M E the kernel of the evaluation map then we have an isomorphism In particular, if Proof.Based on (6), we know that W q (X, E) is the middle cohomology of the complex where V = H 0 (X, E) ∨ and K = Ker(d) ⊥ .Dualizing this complex and replacing K ∨ = Im(d) by the ambient space H 0 (X, 2 E) (which does not affect the middle cohomology), we realize W q (X, E) ∨ as the middle cohomology of a complex which arises from an alternative construction as follows.Since M E is resolved by the 2-term complex ( 14), Sym q+2 M E is resolved by the (q + 2)-nd symmetric power of ( 14) and the previous description of W q (X, E) ∨ shows that it coincides with the first cohomology group of the complex obtained from (15) by taking global sections.
Since (15) resolves Sym q+2 M E , its hypercohomology coincides with the sheaf cohomology of Sym q+2 M E , so we get a spectral sequence Since E i,j 2 = 0 for i < 0 or j < 0, it follows that we have an exact sequence where ι denotes the natural inclusion of is the complex obtained from (15) by taking global sections, we conclude that E 1,0 2 = W q (X, E) ∨ is the kernel of H 1 (X, ι), and that it is moreover isomorphic to H 1 (X, Sym q+2 M E ) when H 1 (X, O X ) = 0, as desired.
4.1.Koszul modules associated to K3 surfaces.An important application of Theorem 4.3 is provided by Lazarfeld-Mukai bundles on K3 surfaces.Let (X, L) be a polarized K3 surface of genus g ≥ 2, where L is an ample line bundle of degree L 2 = 2g − 2. We set H • (X) := H 0 (X, Z) ⊕ H 2 (X, Z) ⊕ H 4 (X, Z).Following [29], we define the Mukai pairing on H • (X) by For a sheaf E on X, its Mukai vector is defined following [29, Definition 2.1], by setting Note that we have −χ(F, F ) = v(F ) 2 .We denote by M L (v) the moduli space of Sequivalence classes of L-semistable sheaves E on X and having prescribed Mukai vec- consists of a single point.Definition 4.4.A globally generated vector bundle E on a polarized K3 surface (X, L) is said to be a Lazarsfeld-Mukai bundle if det(E) ∼ = L and H 1 (X, E) = H 2 (X, E) = 0.
The Lazarsfeld-Mukai bundles were introduced in [27], [28], [29].They can be constructed by choosing a smooth curve C ∈ |L| and a linear system A ∈ W r−1 d (C) such that both A and ω C ⊗ A ∨ are globally generated, where r ≥ 2. The dual Lazarsfeld-Mukai bundle sits in the following exact sequence on where ι : C ֒→ X is the inclusion.Dualizing, we obtain the short exact sequence (16) 0 Then E is a globally generated L-stable bundle with det(E) ∼ = L and . We refer to [27] for all these properties.To (X, L) and E as above, we consider the Koszul module of the associated Lazarsfeld-Mukai bundle W (X, E) := W H 0 (X, E) ∨ , K , where K ⊥ is the kernel of the determinant map d : 2 H 0 (X, E) → H 0 X, 2 E .Proof.Two non-proportional sections s 1 and s 2 of E such that d(s 1 ∧ s 2 ) = 0 correspond to a locally free subsheaf of rank one Tensoring the sequence ( 16) with L ∨ and taking cohomology we obtain a contradiction.
Let E be a Lazarsfeld-Mukai bundle with Mukai vector v(E) = (r, L, s).Since E is globally generated, we consider the kernel bundle M E sitting in the exact sequence In particular, M ∨ E is also a Lazarsfeld-Mukai bundle.
Proof of Theorem 1.4.We start with a Lazarsfeld-Mukai bundle E with Mukai vector v(E) = (r, L, s).Then M ∨ E is also a Lazarsfeld-Mukai bundle with v(M ∨ E ) = (s, L, r) which has vanishing resonance.Since h 0 (X, M ∨ E ) = h 0 (X, E) = r + s, the conclusion follows by applying Theorem 1.3.
If v(E) = (r, L, s), a rather lengthy but elementary calculation with Chern classes shows that the symmetric powers of E have Mukai vector When E is a spherical object, that is v 2 (E) = −2, in which case the moduli space M L (v) consists only of E, then g = rs and the above formula becomes more manageable: In particular, Theorem 1.4 shows that a general vector bundle F ∈ M L (v), where v is the Mukai vector given by ( 17), satisfies H 1 (X, F ) = 0. Theorem 1.4 is optimal when Theorem 2.2 is applied in the divisorial case.We record this result: Proof.Apply directly Theorem 1.4 coupled with the estimate provided by Theorem 2.2.
Remark 4.7.Inside the moduli space F g of polarized K3 surfaces of genus g, the locus NL 1 of those polarized K3 surfaces [X, L] for which H 1 (X, Sym r+1 E) = 0 for a vector bundle is via Theorem 4.6 of Noether-Lefschetz type and its class can be computed in terms of the Hodge classes on F g .Understanding the relative position of the classes NL b , in particularly deciding when these loci are empty will thus lead to non-trivial relations among tautological classes in CH • (F g ) in the spirit of [19] or [31].
Keeping the set-up as above, we fix a general curve C ∈ |L|, therefore C is smooth of genus 2r and W 1 r+1 (C) consists of (2r)! r!•(r+1)!reduced points, see [27].The restriction E C of the Lazarsfeld-Mukai bundle E ∈ M L (r, L, 2) is a stable rank 2 vector bundle with det(E C ) ∼ = ω C and h 0 (C, E C ) = h 0 (X, E) = r + 2. Since h 0 (C, ω C ) < 2h 0 (C, E C ) − 3, the vector bundle E C has non-trivial resonance which we describe below.
corresponds to a globally generated subpencil of E C .Without loss of generality, we may assume that the quotient is locally free.We can prove even more: Since E C is globally generated, ω C ⊗ A ∨ is also globally generated and hence either the latter case is ruled out.In particular, A contributes to the Clifford index.On the other hand, Hence either A or its residual ω C ⊗ A ∨ belong to W 1 r+1 (C).However, the latter case contradicts the stability of E C , hence it does not appear.Lemma 4.8 shows that Gr 2 (V ∨ ) ∩ PK ⊥ ∼ = W 1 r+1 (C) and is finite and moreover Before stating the next result we recall the various properties of the resonance variety of a Koszul module given in Definition 2.4.In the case of a vector bundle over a curve, isotropy and separability are related to specific geometric properties.The following result will be used later: Lemma 4.9.Let F be a vector bundle of rank 2 over a smooth curve C and let V ∨ ⊆ H 0 (C, F ).
(i) V ∨ is isotropic if and only if it generates a rank-one subsheaf B inside F .
(ii) Suppose that E is given by an extension of line bundles with B globally generated, and denote We now return to the set-up when C ∈ |L| is a curve of genus 2r on a K3 surface X. Proof.From Proposition 2.6 we infer that that R(C, E C ) is a union of (2r)! r!•(r+1)!disjoint lines, all isotropic.In order to establish the separability of these components, ℓ ab be a component, corresponding to a point [a ∧ b] ∈ Gr 2 (V ∨ ) ∩ PK ⊥ and denote by V ∨ the subspace in V ∨ generated by a and b.If A denotes the subpencil of E C generated by a and b, then E C is presented as an extension ) is injective, we can apply Lemma 4.9 (ii) to conclude.

4.2.
Koszul modules associated to K3 surfaces of odd genus.Using a variation compared to the even genus case, one can also associate to a general K3 surface of odd genus a Koszul module W (V, K) in the divisorial case dim(K) = 2 dim(V ) − 3 as follows.
Fix a polarized K3 surface [X, L] of odd genus g = 2r + 1 such that Pic(X) = Z • L and choose a smooth curve C ∈ |L|.Recall that X := M L (2, X, r) is the Fourier-Mukai partner of X. Denoting by SU C (2, ω, r + 2) the moduli space of S-equivalence classes of semistable rank 2 vector bundles E C on C with det(E C ) ∼ = ω C and h 0 (C, E C ) ≥ r + 2, the restriction map induces an isomorphism, see [4], Moreover, it can be shown that X as the Fourier-Mukai dual of SU C (2, ω, r + 2) is the only K3 surface containing C as long as s is odd, see [4], [20].
The Brill-Noether locus W 1 r+2 (C) is a smooth curve (recall that C satisfies Petri's Theorem [27]) and we have the following formula for its genus [16]: Using [37], one has a map j : W 1 r+2 (C) → X which associates to A ∈ W 1 r+2 (C) the rank 2 Lazarsfeld-Mukai vector bundle E C,A defined by the sequence (16).Its restriction To a pair (C, E C ), where C ∈ |L| and E C ∈ SU C (2, ω, r + 2), using Definition 4.1 we associate the Koszul module W (C, E C ) := W (V, K) and its resonance variety R(C, E C ).Note that since h 0 (C, ω C ) = 2h 0 (C, E) − 3, we are in the divisorial case of Theorem 2.2.We denote by M E C the kernel of the evaluation map

Theorem 4.11. One has a canonical identification
The geometric meaning of the injectivity of the map ( 19) is mysterious and requires further study.In what follows we will prove Theorem 1.5.
Recall that [X, L] ∈ F 2r+1 with Pic(X) = Z • L and we consider the projections and denote by P a Poincaré bundle of rank 2 on X × X. 1 One writes for the first Chern class of P respectively the middle class in the K ünneth decomposition of c 2 (P).Following [30] we define the class ψ ∈ H 2 ( X) characterized by the property π , where [pt] is the fundamental class of X.It is also shown in [30] that if one sets (20) h then h is a polarization on X satisfying h 2 = h 2 = 2g − 2 = 4r.We now introduce the following vector bundles on X having as fibres over a point [E] the spaces H 0 (X, E) and Proposition 4.12.The following formulas hold in H 2 ( X): Proof.We apply Grothendieck-Riemann-Roch to the map π 2 and the sheaf P using that 1 A Poincaré bundle P exists only when g ≡ 3 mod 4, that is, when r is odd.When r is even, the class φ is divisible by two and P does not exist globally.As pointed out in [30], one has to take instead the universal P 1 -bundle corresponding to P(P) (which does exist) and carry out the calculation of the class of the curve R( X, C) at that level.Theorem 1.5 remains valid independent of the parity of r.
Remark 4.13.It is natural to conjecture that for a general C ∈ |L|, the singularities of the curve R X,C are nodal.Proving this seems challenging even for small r.

GAUSSIAN KOSZUL MODULES AND THICKENINGS OF ALGEBRAIC VARIETIES
An important class of Koszul modules where the triviality of resonance is automatically satisfied is given by the Gaussian maps [41] on projective varieties.Let L be a line bundle on a smooth complex projective variety X.The Gaussian Wahl map If X is a smooth curve and L = O X (1), the map ψ L is given by associating to a point p ∈ X the projectivized tangent line T p (X) ∈ Gr 2 H 0 (X, L) ∨ under the Pl ücker embedding of the Grassmannian of lines.
is called the Gaussian module of the pair (X, L).
Since ψ L (f ∧ g) = 0 if and only if d f g = 0, it follows that ψ L is injective on decomposable tensors, therefore R(V, K) = {0}.In particular, rk(ψ L ) ≥ 2h 0 (X, L) − 3.If X is a smooth curve, the equality rk(ψ L ) = 2h 0 (X, L) − 3 holds if and only if the image of X under the linear system |L| is a rational normal curve, see [11,Theorem 1.3].
We introduce the vector bundle R L defined by the exact sequence where J 1 (L) is the first jet bundle of L. The map r in ( 21) can be defined locally by r(w) = (dw, w).We consider the exact sequence and observe that p • r is the evaluation map H 0 (X, L) ⊗ O X → L. In particular, one also has the following exact sequence on X: In case L is very ample and we consider the embedding ϕ L : X ֒→ P(V ) defined by V = H 0 (X, L) and write I for the ideal sheaf of X in this embedding, we have 22) we obtain an induced exact sequence The induced map on global sections is the Gaussian map ψ L .Our goal is to give a cohomological interpretation of the graded components of the Koszul module G(V, K).
To that end, we fix q ≥ 0 and consider the composition (25) s : where the first one is the natural inclusion, and the second map is Sym q+1 (ι) ⊗ id R L .
Theorem 5.2.For each q ≥ 0, the components of the Gausssian module G(X, L) are given by where the map t = H 1 (X, s) is induced by (25).
To prove the theorem we first show that K ⊥ is also equal to Ker H 0 (X, 2 r) : Lemma 5.3.The restriction of the map α = H 0 (X, a) to the image of β = H 0 (X, 2 r) is injective.In particular we have that Ker(ψ L ) = Ker(β).
Proof.Since all the sheaves involved are locally free, it suffices to localize at the generic point of X and show that α is injective on the image of β there.In particular we may choose a local generator of L and identify elements of W with rational functions on X.
We have that β(w Differentiating this equality shows that i dw i ∧ dw ′ i = 0, so that β(x) = 0, that is, α is injective on Im(β), as desired.
Proof of Theorem 5.2.It follows from ( 21) that for q ≥ 0 we have an exact sequence (27) Dropping the first term and taking global sections we obtain the middle row of the commutative diagram where the first row of the diagram is given by ( 26), and δ is the connecting homomorphism associated with the long exact sequence in cohomology of (21).Since p • r is the evaluation map H 0 (X, L) ⊗ O X → L, we get that H 0 (X, r) is injective.
If we write H for the middle homology of the middle row of the above diagram, it follows from (26) that where the map u is induced by id ⊗ δ.Just as in Theorem 2.8, we shall realize H as so H can be thought of as a subgroup of H 1 X, Sym q+2 R L .Under this identification, we claim that u is the restriction of t to H.Moreover, v factors through t, therefore H = Ker(v) ⊇ Ker(t) and Ker(u) = Ker(t) as desired.
In order to see that v factors through t, we consider the diagram (commutative up to multiplication by a non-zero scalar) where the map o is induced by ι and the multiplication Sym q+1 H 0 (X, L) ⊗ H 0 (X, L) −→ Sym q+2 H 0 (X, L).Since v = H 1 (X, Sym q+2 (ι)) and t = H 1 (X, s), it follows that v factors through t.
To prove the assertion (28) and that u = t |H , we split ( 27) into short exact sequences By construction, H is the cokernel of the map Sym q+2 H 0 (X, L) → H 0 (X, M ), which is the same as the kernel of We consider the commutative diagram (where ∆ is the natural inclusion) which gives rise by taking cohomology to a commutative diagram Since u was induced by id ⊗ δ, it follows that after identifying H with a subgroup of H 1 (X, Sym q+2 R L ) we get that u is the restriction of t, concluding the proof.
Theorem 5.2 has a more transparent geometric interpretation under suitable assumptions.
One can reformulate these results in terms of stabilization of cohomology on the successive thickenings of the subvariety X ⊆ P r .For b ≥ 0, we denote by X b ⊆ P r the subscheme defined by the ideal I b+1 ⊆ O P r , thus we have a system of subschemes Proof of Theorem 1.6.Suppose the projective variety X ⊆ P r is embedded by the line bundle L := O X (1).Then for each b ≥ 1, one has the short exact sequence L ⊗ L, tensoring this exact sequence with L b and taking cohomology, we obtain from Corollary 5.4 that the map H , where we use that our assumptions force the map H 0 (r) : H 0 (X, J 1 (L)) → H 0 (X, L) induced by the sequence (21) to be an isomorphism.
We assume now that 0 ≤ a < b and set c := b − a ≥ 1.To complete the proof we have to show that H i (X, Sym b R L ⊗ L −c ) = 0, for i = 0, 1.To that end, we use the notation from the proof of Theorem 5.2.By Kodaira vanishing H 1 (X, L −c ) = 0, thus it follows that there is a surjection H In order to show that this last cohomology group vanishes, we use the sequence (22).Since H 0 (X, Ω 1 X ) = 0, clearly H 0 (X, J 1 (L) ⊗ L −c ) = 0, for c ≥ 2. For c = 1, the existence of a non-zero section in H 0 (X, J 1 (L)⊗L ∨ ) implies that the sequence ( 22) is split.But this is impossible, for it is known that the Atiyah class η X ) expressing J 1 (L) as an extension in the sequence (22) equals precisely the Chern class c 1 (L).Since L is very ample, this class cannot be zero.
Finally, in order to show that H 5.1.Ramification divisors of canonical pencils.We now prove Theorem 1.2.We use throughout the standard notation [6] for the tautological and boundary classes on M g,n .We consider the universal curve π : M g,n+1 → M g,n endowed with its n tautological sections whose images we identify with the boundary divisors ∆ 0:i,n+1 on M g,n for i = 1, . . ., n.We consider the Hodge bundle E := π * (ω π ) and the rank n vector bundle One has a morphism φ : 2 E → F which fibrewise is given by the composition in the ramification divisor of the cover C → P 1 induced by s 1 and s 2 .
For our next result, recall that ψ i denotes the class of the line bundle on M g,n having as fibre over a point [C, x 1 , . . ., x n ] the cotangent space T ∨ x i (C), for i = 1, . . ., n.

RESONANCE, STABILITY AND SPLIT BUNDLES
In this section, we prove that important intrinsic properties of bundles of sufficiently large degree on a curve, such as instability or being split, can be read off its resonance.We use the following notation, for a given vector bundle E on a curve C and an integer k By projectivization, these closed loci provide us with a stratification of the projectivized resonance.Indeed, R ≥k (C, E) ⊇ R ≥(k+1) (C, E), R ≥k (C, E) = ∅ for k ≫ 0, and R ≥d (C, E) = R(C, E) if d equals te gonality of the curve.We call this stratification the degree stratification.Theorem 6.1.Let E be a globally generated rank 2 vector bundle on a smooth curve C of genus g ≥ 1. Assume that deg(E) ≥ 4g + 1 and H 1 (C, E) = 0. Then (i) E is not stable (respectively unstable) if and only if H 0 (C, E) ∨ has an isotropic subspace of dimension at least h 0 (E)

2
(respectively > h 0 (E) 2 ).(ii) E splits as a sum of line bundles if and only if there exist an integer k and isotropic subspaces  (ii) Assume E = N ⊕ M splits and h 0 (N ⊗ M ∨ ) = h 0 (M ⊗ N ∨ ) = 0. Put k = min{deg(N ), deg(M )}.Assume, for simplicity, k = deg(N ).Since E is globally generated and h 1 (C, E) = 0, it follows that both N and M are globally generated and nonspecial.Put V 1 := H 0 (C, N ) and V 2 := H 0 (C, M ).These two subspaces are isotropic and H 0 (C, E) = V 1 ⊕ V 2 .We prove that R ≥k (C, E) = V ∨ 1 ∪ V ∨ 2 .Let a ∈ R ≥k (C, E) \ {0}, then there exists b such that 0 = a ∧ b ∈ K ⊥ and hence a and b span a line bundle L of degree at least k inside E = N ⊕ M .It the induced map L → N is non-zero, then L = N and, since h 0 (M ⊗ N ∨ ) = 0, it follows that the map L → M is zero, which implies a ∈ V ∨ 1 .If the map L → N is zero, then a ∈ V ∨ 2 .Conversely, assume we are given isotropic subspaces Let N and M be the line bundles of degree at least k contained in E generated by V ∨ 1 and V ∨ 2 respectively.By isotropy, it follows that N and M are globally generated with V ∨ 1 ⊆ H 0 (C, N ) and V ∨ 2 ⊆ H 0 (C, M ).Since H 0 (C, N ) and H 0 (C, M ) are also isotropic, and hence contained in the resonance, the are in fact contained in R ≥k (C, E).The assumption R ≥k (C, E) = V ∨ 1 ∪ V ∨ 2 implies V ∨ 1 = H 0 (C, N ) and V ∨ 2 = H 0 (C, M ).Claim 1.The natural map N ⊕ M → E is injective.Indeed, otherwise its image is a line bundle L of degree at least k.Passing to global sections we obtain ) and the composition is the identity.In conclusion the inclusion L ⊆ E yields an equality H 0 (C, L) = H 0 (C, E).Since H 0 (C, L) is isotropic, and hence contained in the resonance, we find R ≥k (C, E) = H 0 (C, E) ∨ , in contradiction with the hypothesis.Claim 3. Suppose h 1 (C, N ) = 0. Then L := E/N is torsion-free.Indeed, if it has torsion, then we consider the line bundle N ′ := Ker{E → L/tors(L)} ⊆ E which is also of degree at least k, and an inclusion N N ′ .Since N is non-special, N ′ is also nonspecial and by Riemann-Roch H 0 (C, N ) H 0 (C, N ′ ).Note however that H 0 (C, N ′ ) is isotropic, therefore contained in the resonance, contradicting that H 0 (C, N ) is a component of R ≥k (C, E).
Having proved these claims, we conclude.Denoting by α : M → L the composition, note that α = 0 for otherwise M ⊆ N , contradicting the hypothesis.Since N is non-special, the equality H 0 (C, E) = H 0 (C, N ) ⊕ H 0 (C, M ) and the long cohomology sequence of the exact sequence shows that H 0 (α) : H 0 (C, M ) → H 0 (C, L) is an isomorphism.Since E is globally generated, it follows that L is globally generated as well.We have the following situation: M and L are globally generated line bundles, and α : M → L is a morphism inducing an isomorphism on global sections.It implies that α is surjective, and hence it is an isomorphism, providing us with a splitting E ∼ = N ⊕ M .To prove that h 0 (M ⊗ N ∨ ) = 0 observe that any non-zero section in H 0 (M ⊗ N ∨ ) gives an embedding N ⊆ N ⊕ M with torsion-free quotient which yields to elements in R ≥k (C, E) that are neither in V ∨ 1 nor in V ∨ 2 .Remark 6.2.For (ii), the bound deg(E) ≥ 4g + 1 in the assumption of Theorem 6.1 can be improved to 4g if C is non-hyperelliptic.Indeed, in Claim 2, the inequalities resulting from Clifford's Theorem are strict.Remark 6.3.The resonance of split bundles is in general much more complicated than the union of two subspaces.The easiest example is obtained on the projective line for the bundle O(1) ⊕ O(1) whose resonance is a smooth quadric in the three-dimensional projective space.In this case, the stratification consists of only one stratum, the maximal one.
A more elaborate example is the following.Suppose C is an elliptic curve, p = q are two points on C, and which shows that the bound in the theorem above is not sharp.

Lemma 3 . 1 .
Let Y be an irreducible projective variety and U ⊆ Y an open subset with codim(Y \ U, Y ) ≥ 2. Assume A and B are vector bundles of the same rank on U , and we are given a morphism ϕ : A → B, whose degeneracy locus D 1 = D(ϕ) is a genuine divisor.Let D 2 be a reduced Cartier divisor on U such that Supp(D 2 ) ⊆ Supp(D 1 ) and

Theorem 3 . 4 .
One has the following equality of divisors on G D Kosz = (n − 2) • D Res .
as the restriction of the determinant map to 2 H 0 (C, B) vanishes identically.Conversely, let V∨ be an isotropic subspace of H 0 (C, F ). Then any vector 0 = a ∧ b ∈ 2 V ∨ generates a rank-one subsheaf of F .In particular, for a generic point x ∈ C the vectors a(x), b(x) ∈ F (x) are linearly dependent, hence the span of a

2 ≥
and let L ⊆ E be a maximal destabilizing line subbundle.Since deg(L) ≥ deg(E) 2g, the bundle L is non-special and globally generated.Therefore h 0 (C, L) ≥ h 0 (C,E)