GREEN’S CONJECTURE FOR CURVES ON ARBITRARY K3 SURFACES

MARIAN APRODU AND GAVRIL FARKAS

1. Introduction

Green’s Conjecture on syzygies of canonical curves asserts that one can recognize existence of special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. Precisely, if C is a smooth algebraic curve of genus g, K_{i,j}(C, K_{C}) denotes the (i, j)-th Koszul cohomology group of the canonical bundle K_{C} and Cliff(C) is the Clifford index of C, then M. Green [Gr84] predicted the vanishing statement

(1)  K_{p,2}(C, K_{C}) = 0, for all p < Cliff(C).

In recent years, Voisin [V02], [V05] achieved a major breakthrough by showing that Green’s Conjecture holds for smooth curves C lying on K3 surfaces S with Pic(S) = ℤ · C. In particular, this establishes Green’s Conjecture for general curves of every genus. Using Voisin’s work, as well as a degenerate form of [HR98], it has been proved in [Ap05] that Green’s Conjecture holds for any curve C of genus g of gonality gon(C) = k ≤ (g + 2)/2, which satisfies the linear growth condition

(2)  dim W_{k+n}^{1}(C) ≤ n, for 0 ≤ n ≤ g − 2k + 2.

Thus Green’s Conjecture becomes a question in Brill-Noether theory. In particular, one can check that condition (2) holds for a general curve [C] ∈ M_{g,k}^{1} in any gonality stratum of M_{g}, for all 2 ≤ k ≤ (g + 2)/2. Our main result is the following:

Theorem 1.1. Let S be a K3 surface and C ⊂ S be a smooth curve with g(C) = g and gon(C) = k. If k ≤ (g + 2)/2, then C satisfies Green’s conjecture.

Note that Theorem 1.1 has been established in [V02] when Pic(S) = ℤ · C. The proof relies (via [Ap05]) on the case of curves of odd genus of maximal gonality. Precisely, when g(C) = 2k − 3 and gon(C) = k, Green’s conjecture is due to Voisin

[V05] combined with results of Hirschowitz-Ramanan [HR98]. Putting together these results and Theorem 1.1, we conclude:

Theorem 1.2. Green’s Conjecture holds for every smooth curve C lying on an arbitrary K3 surface S.

In the proof of Theorem 1.1, we distinguish two cases. When Cliff(C) is computed by a pencil (that is, Cliff(C) = gon(C) − 2), we use a parameter count for spaces of Lazarsfeld-Mukai bundles [La86], [CP95], in order to find a smooth curve C′ ∈ |C|, such that C′ verifies condition (2). Since Koszul cohomology satisfies the Lefschetz hyperplane principle, one has that K_{p,2}(C, K_{C}) ≅ K_{p,2}(C′, K_{C′}). This proves Green’s Conjecture for C.

Research of the first author partly supported by a PN-II-ID-PCE-2008-2 grant (cod 1189, contract no 530) and by a resumption of a Humboldt Research Fellowship. Research of both authors partly supported by the Sonderforschungsbereich ”Raum-Zeit-Materie”. MA thanks HU Berlin for the kind hospitality and the excellent atmosphere during the preparation of this work.

When Cliff(C) is no longer computed by a pencil, it follows from [CP95], [Kn09] that either C is a smooth plane curve or else, a generalized ELMS example, in the sense that there exist smooth curves D, Γ ⊂ S, with Γ^{2} = −2, Γ · D = 1 and D^{2} ≥ 2, such that C ≡ 2D + Γ and Cliff(C) = Cliff(O_{C}(D)) = gon(C) − 3. This case requires a separate analysis since condition (2) is no longer satisfied, and we refer to Section 5 for details.

Theorem 1.1follows by combining results obtained by using the powerful techniques developed in [V02], [V05], with facts about the effective cone of divisors of \overline{M}_{g}. As pointed out in [Ap05], starting from a k-gonal smooth curve [C] ∈ M_{g} satisfying the Brill-Noether growth condition (2), by identifying pairs of general points x_{i}, y_{i} ∈ C for i = 1, …, g + 3 − 2k one creates a stable curve

[X := C/x_{1} ∼ y_{1}, …, x_{g+3−2k} ∼ y_{g+3−2k}] ∈ \overline{M}_{2g+3−2k}

having maximal gonality g +3−k, that is, lying outside the closure of the Hurwitz divisor M_{2g+3−2k,g+3−k}^{1} consisting of curves with a pencil 𝔤_{g+3−k}^{1}. Since the class of the virtual failure locus of Green’s Conjecture is a multiple of the Hurwitz divisor \overline{M}_{2g+3−2k,g+3−k}^{1} on \overline{M}_{2g+3−2k}, see [HR98], Voisin’s theorem can be extended to all irreducible stable curves of genus 2g+3−2k and having maximal gonality, in particular to X as well, and a posteriori to smooth curves of genus g sitting on K3 surfaces with arbitrary Picard lattice. On the other hand, showing that condition (2) is satisfied for a curve [C] ∈ M_{g}, is a question of pure Brill-Noether nature.

Theorem 1.1 has strong consequences on Koszul cohomology of K3 surfaces. It is known that for any globally generated line bundle L on a K3 surface S, the Clifford index of any smooth irreducible curve is constant, equal to, say c, [GL87]. Applying Theorem 1.1, Green’s hyperplane section theorem, the duality theorem and finally the Green-Lazarsfeld nonvanishing theorem [Gr84], we obtain a complete description of the distribution of zeros among the Koszul cohomology groups of S with values in L.

Theorem 1.3. Suppose L^{2} = 2g − 2 ≥ 2. The Koszul cohomology group K_{p,q}(S, L) is nonzero if and only if one of the following cases occur:

(1) q = 0 and p = 0, or

(2) q = 1, 1 ≤ p ≤ g − c − 2, or

(3) q = 2 and c ≤ p ≤ g − 1, or

(4) q = 3 and p = g − 2.

The analysis of the Brill-Noether loci implies also that the Green-Lazarsfeld Gonality Conjecture is satisfied for curves of Clifford dimension one on arbitrary K3 surfaces, general in their linear systems, see Section 4 for details.

2. Brill-Noether loci and their dimensions

Throughout this section we fix a K3 surface S, a globally generated line bundle L ∈ Pic(S) and denote by |L|_{s} the locus of smooth connected curves in |L|. For integers r, d ≥ 1, we consider the morphism π_{S} : W_{d}^{r}(|L|) → |L|_{s} with fibre over a point C ∈ |L|_{s} isomorphic to the Brill-Noether locus W_{d}^{r}(C). The analysis of the Brill-Noether loci W_{d}^{r}(C) for a general curve C ∈ |L| in its linear system, is equivalent to the analysis of the restricted maps π_{S} : W → |L| over irreducible components W of W_{d}^{r}(|L|) dominating the linear system. The main ingredient used to study W_{d}^{r}(|L|) is the Lazarsfeld-Mukai bundle [La86] associated to a complete linear series. To any pair (C, A) consisting of a curve C ∈ |L|_{s} and a base point free linear series A ∈ W_{d}^{r}(C) \ W_{d}^{r+1}(C), one associates the Lazarsfeld-Mukai bundle E_{C,A} := F_{C,A}^{∨} on S, via an elementary transformation along C ⊂ S:

(3)  0 → F_{C,A} → H^{0}(C, A) ⊗ O_{S} →^{ev} A → 0.

Dualizing the sequence (3), we obtain the short exact sequence

(4)  0 → H^{0}(C, A)^{∨} ⊗ O_{S} → E_{C,A} → K_{C} ⊗ A^{∨} → 0.

The bundle E_{C,A} comes equipped with a distinguished subspace of sections H^{0}(C, A)^{∨} ∈ G(r + 1, H^{0}(S, E_{C,A})). We summarize some characteristics of E_{C,A}:

Proposition 2.1. One has that (1) det(E_{C,A}) = L.

(2) c_{2}(E_{C,A}) = d.

(3) h^{0}(S, E_{C,A}) = h^{0}(C, A) + h^{1}(C, A), h^{1}(S, E_{C,A}) = h^{2}(S, E_{C,A}) = 0.

(4) χ(S, E_{C,A} ⊗ F_{C,A}) = 2(1 − ρ(g, r, d)).

(5) E_{C,A} is globally generated off the base locus of K_{C} ⊗ A^{∨}.

In particular, E_{C,A} is globally generated if K_{C} ⊗ A^{∨} is globally generated. Conversely, if E is a globally generated bundle on S with rk(E) = r+1 and det(E) = L, there is a rational map h_{E} : G(r + 1, H^{0}(S, E)) −−→ |L|. defined in the following way. A general subspace Λ ∈ G(r+1, H^{0}(S, E)) is mapped to the degeneracy locus of the evaluation map: ev_{Λ} : Λ ⊗ O_{S} → E; note that, generically, this degeneracy locus cannot be the whole surface. The image h_{E}(Λ) is a smooth curve C_{Λ} ∈ |L|, and we set Coker(ev_{Λ}) := K_{C_{Λ}} ⊗ A_{Λ}^{∨}, where A_{Λ} ∈ Pic(C_{Λ}) and deg(A_{Λ}) = c_{2}(E).

Remark 2.2. A rank-(r + 1) vector bundle E on S is a Lazarsfeld-Mukai bundle if and only if H^{1}(S, E) = H^{2}(S, E) = 0 and there exists an (r + 1)-dimensional subspace of sections Λ ⊂ H^{0}(S, E), such that the the degeneracy locus of the morphism ev_{Λ} is a smooth curve. In particular, being a Lazarsfeld-Mukai vector bundle is an open condition.

Coming back to the original situation when C ∈ |L|_{s} and A ∈ W_{d}^{r}(C)\W_{d}^{r+1}(C) is globally generated, we consider the Petri map

μ_{0,A} : H^{0}(C, A) ⊗ H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, K_{C}),

whose kernel can be described in terms of Lazarsfeld-Mukai bundles. Let M_{A} the vector bundle of rank r on C defined as the kernel of the evaluation map

(5)  0 → M_{A} → H^{0}(C, A) ⊗ O_{C} →^{ev} A → 0.

Twisting (5) with K_{C} ⊗ A^{∨}, we obtain that Ker(μ_{0,A}) = H^{0}(C, M_{A} ⊗ K_{C} ⊗ A^{∨}). Note also that there is an exact sequence sequence on C

0 → O_{C} → F_{C,A} ⊗ K_{C} ⊗ A^{∨} → M_{A} ⊗ K_{C} ⊗ A^{∨} → 0,

while from the defining sequence of E_{C,A} one obtains the exact sequence on S

0 → H^{0}(C, A)^{∨} ⊗ F_{C,A} → E_{C,A} ⊗ F_{C,A} → F_{C,A} ⊗ K_{C} ⊗ A^{∨} → 0.

Since h^{0}(C, F_{C,A}) = h^{1}(C, F_{C,A}) = 0, one writes that

H^{0}(C, E_{C,A} ⊗ F_{C,A}) = H^{0}(C, F_{C,A} ⊗ K_{C} ⊗ A^{∨}).

We shall use the following deformation-theoretic result [Pare95], which is a consequence of Sard’s theorem applied to the projection π_{S} : W_{d}^{r}(|L|) → |L|.

Lemma 2.3. Suppose W ⊂ W_{d}^{r}(|L|) is a dominating component, and (C, A) ∈ W is a general element such that A is globally generated and h^{0}(C, A) = r + 1. Then the coboundary map H^{0}(C, M_{A} ⊗ K_{C} ⊗ A^{∨}) → H^{1}(C, O_{C}) is zero.

The above analysis can be summarized as follows:

Proposition 2.4. If W ⊂ W_{d}^{r}(|L|) is a dominating component, and (C, A) ∈ W is a general element such that A is globally generated and h^{0}(C, A) = r +1, then dim_{A}W_{d}^{r}(C) ≤ ρ(g, r, d) + h^{0}(C, E_{C,A} ⊗ F_{C,A}) − 1. Moreover, equality holds if W is reduced at (C, A).

In particular, if E_{C,A} is a simple bundle, then μ_{0,A} is injective and W is reduced at (C, A) of dimension ρ(g, r, d) + g. Thus, the problem of estimating dim_{A}W_{d}^{r}(C), when (C, A) ∈ W is suitably general, can be reduced to the case when E_{C,A} is not a simple bundle.

3. Varieties of pencils on K3 sections

Throughout the remaining sections we mix the additive and the multiplicative notation for divisors and line bundles. If L is a line bundle on a smooth projective variety X and L ∈ Pic(X) is a line bundle, we write L ≥ 0 when H^{0}(X, L) ≠ 0. If E is a vector bundle on X and L ∈ Pic(X), we set E(−L) := E ⊗ L^{∨}.

As in the previous section, we fix a K3 surface S together with a globally generated line bundle L on S. We denote by k be the minimal gonality of smooth curves in the linear system |L|, and set g := 1 + L^{2}/2. Suppose that ρ(g, 1, k) ≤ 0 (this leaves out one single case, namely g = 2k − 3, when ρ(g, 1, k) = 1). Our aim is to prove the Koszul vanishing statement

K_{g−Cliff(C)−1,1}(C, K_{C}) = 0,

for any curve C ∈ |L|_{s}. By duality, this is equivalent to Green’s Conjecture for C.

It was proved in [Ap05] that any smooth curve C that satisfies the linear growth condition (2), verifies both Green’s and Green-Lazarsfeld Gonality Conjecture. By comments made in the previous section, a general curve C ∈ |L|_{s} satisfies (2), if and only if for any n = 0, …, g − 2k + 2, and any irreducible component W ⊂ W_{k+n}^{1}(C) such that a general element A ∈ W is globally generated, has h^{0}(C, A) = 2, and the corresponding Lazarsfeld-Mukai bundle E_{C,A} is not simple, the estimate dim W ≤ n, holds.

Condition (2) for curves which are general in their linear system, can be verified either by applying Proposition 2.4, or by estimating directly the dimension of the corresponding irreducible components of the scheme W_{k+n}^{1}(|L|). In our analysis, we need the following description [DM89] of non-simple Lazarsfeld-Mukai bundles, see also [CP95] Lemma 2.1:

Lemma 3.1. Let E_{C,A} be a non-simple Lazarsfeld-Mukai bundle. Then there exist line bundles M, N ∈ Pic(S) such that h^{0}(S, M), h^{0}(S, N) ≥ 2, N is globally generated, and there exists a zero-dimensional, locally complete intersection subscheme ξ of S such that E_{C,A} is expressed as an extension

(6)  0 → M → E_{C,A} → N ⊗ I_{ξ} → 0.

Moreover, if h^{0}(S, M ⊗ N^{∨}) = 0, then ξ = ∅ and the extension splits.

We say that (6) is the Donagi-Morrison (DM) extension associated to E_{C,A}.

Lemma 3.2. For any indecomposable non-simple Lazarsfeld-Mukai bundle E on S, the DM extension (6) is uniquely determined by E.

Proof. We assume that two DM extensions

0 → M_{j} → E → N_{j} ⊗ I_{ξ_{j}} → 0, j = 1, 2,

are given. Observe first that H^{0}(S, N_{1} ⊗ M_{2}^{∨}) = H^{0}(S, N_{2} ⊗ M_{1}^{∨}) = 0. Indeed, if N_{1} − M_{2} ≥ 0, we use M_{1} − N_{1} ≥ 0, M_{2} − N_{2} ≥ 0 (we are in the non-split case), and M_{1} + N_{1} = M_{2} + N_{2} = L to get a contradiction. Then H^{0}(S, (N_{1} ⊗ M_{2}^{∨}) ⊗ I_{ξ_{1}}) = H^{0}(S, (N_{2} ⊗M_{1}^{∨})⊗I_{ξ_{2}}) = 0, so we obtain non-zero maps M_{1} → M_{2} and M_{2} → M_{1}. This implies that M_{1} = M_{2}. □

Remark 3.3. Similarly, one can prove that a decomposable Lazarsfeld-Mukai bundle E cannot be expressed as an extension (6) with ξ ≠ ∅. Thus a DM extension is always unique, up to a permutation of factors in the decomposable case. Moreover, E is decomposable if and only if the corresponding DM extension is trivial.

The size of the space of endomorphisms of a non-simple Lazarsfeld-Mukai bundle can be explicitly computed from the corresponding DM extension:

Lemma 3.4. Let E be a non-simple Lazarfeld-Mukai bundle on S with det(E) = L, and M and N the corresponding line bundles from the DM extension. If E is indecomposable, then

h^{0}(S, E ⊗ E^{∨}) = 1 + h^{0}(S, M ⊗ N^{∨}).

If E = M ⊕ N, then h^{0}(S, E ⊗ E^{∨}) = 2 + h^{0}(S, M ⊗ N^{∨}) + h^{0}(S, N ⊗ M^{∨}).

Proof. The decomposable case being clear, we treat the indecomposable case. Twisting the DM extension by E^{∨} and taking cohomology, we obtain the exact sequence

0 → H^{0}(S, E^{∨}(M)) → H^{0}(S, E ⊗ E^{∨}) → H^{0}(S, E^{∨}(N) ⊗ I_{ξ}).

Since det(E) = L, it follows that E^{∨}(M) ≅ E(−N), and E^{∨}(N) ≅ E(−M). Therefore, h^{0}(S, E^{∨}(N) ⊗ I_{ξ}) = h^{0}(S, E(−M) ⊗ I_{ξ}). Using extension (6), we claim that h^{0}(S, M ⊗ N^{∨}) = h^{0}(S, E(−N)). Indeed, if ξ ≠ ∅, then h^{0}(S, I_{ξ}) = 0. If ξ = ∅, the image of 1 ∈ H^{0}(S, O_{S}) under the map H^{0}(S, O_{S}) → H^{1}(S, M ⊗ N^{∨}) is precisely the extension class, hence it is non-zero.

Observe that H^{0}(S, O_{S}) ≅ H^{0}(S, E(−M)), in particular, h^{0}(S, E^{∨}(N) ⊗ I_{ξ}) ≤ 1. On the other hand, the morphism H^{0}(S, E ⊗ E^{∨}) → H^{0}(S, E^{∨}(N) ⊗ I_{ξ}) maps id_{E} to the arrow E → N ⊗I_{ξ}, hence it is non-zero. It follows that h^{0}(S, E^{∨}(N)⊗I_{ξ}) = 1, and H^{0}(S, E ⊗ E^{∨}) → H^{0}(S, E^{∨}(N) ⊗ I_{ξ}) is surjective. □

In order to parameterize all pairs (C, A) with non-simple Lazarsfeld-Mukai bundles, we need a global construction. We fix a non-trivial globally generated line bundle N on S such that H^{0}(S, L(−2N)) ≠ 0, and an integer ℓ ≥ 0. We set M := L(−N) and g := 1 + L^{2}/2. Define P̃_{N,ℓ} to be the family of vector bundles of rank 2 on S given by non-trivial extensions

(7)  0 → M → E → N ⊗ I_{ξ} → 0,

where ξ is a zero-dimensional lci subscheme of S of length ℓ, and set

P_{N,ℓ} := {[E] ∈ P̃_{N,ℓ} : h^{1}(S, E) = h^{2}(S, E) = 0}.

Equivalently (by Riemann-Roch), [E] ∈ P_{N,ℓ} if and only if h^{0}(S, E) = g −c_{2}(E)+3 and h^{1}(S, E) = 0. Note that any non-simple Lazarsfeld-Mukai bundle on S with determinant L belongs to some family P_{N,ℓ}.

Remark 3.5. Using the Cayley-Bacharach property, we observe that P̃_{N,ℓ} ≠ ∅ whenever Ext_{S}^{1}(N ⊗ I_{ξ}, M) ≠ 0.

Remark 3.6. If P_{N,ℓ} ≠ ∅, then h^{1}(S, N) = 0 and h^{0}(S, N ⊗ I_{ξ}) = h^{0}(S, N) − ℓ. Indeed, we choose [E] ∈ P_{N,ℓ}. Then

h^{0}(S, E) = h^{0}(S, M) + h^{0}(S, N ⊗ I_{ξ}) − h^{1}(S, M)
≥ h^{0}(S, M) + h^{0}(S, N) − length(ξ) − h^{1}(S, M)
≥ χ(S, M) + χ(S, N) − ℓ = 2 + \frac{1}{2}M^{2} + 2 + \frac{1}{2}N^{2} − ℓ
= 2 + \frac{1}{2}L^{2} − M · N + 2 − ℓ = g + 3 − c_{2}(E).

Since h^{0}(S, E) = g + 3 − c_{2}(E), all the inequalities are actually equalities, hence h^{1}(S, N) = 0 and h^{0}(S, N ⊗ I_{ξ}) = h^{0}(S, N) − ℓ.

The family P_{N,ℓ}, which, a priori, might be the empty set, is an open Zariski subset of a projective bundle of the Hilbert scheme S^{[ℓ]}, as shown below:

Lemma 3.7. If ξ ∈ S^{[ℓ]} and Ext_{S}^{1}(N ⊗ I_{ξ}, M)) ≠ 0, then

dim Ext_{S}^{1}(N ⊗ I_{ξ}, M) = ℓ + h^{1}(S, M ⊗ N^{∨}) − h^{2}(S, M ⊗ N^{∨}).

Proof. Let E be a vector bundle given by a non-trivial extension

0 → M → E → N ⊗ I_{ξ} → 0.

Applying Hom_{S}( − , M) to this extension, we obtain the exact sequence

H^{0}(S, O_{S}) → Ext_{S}^{1}(N ⊗ I_{ξ}, M) → H^{1}(S, E^{∨}(M)) → H^{1}(S, O_{S}) = 0.

Since 1 ∈ H^{0}(S, O_{S}) is mapped to the extension class of E which is non-zero, it follows that dim Ext_{S}^{1}(N ⊗ I_{ξ}, M) = h^{1}(S, E^{∨}(M)) + 1 = h^{1}(S, E(−N)) + 1. We apply the identification E^{∨}(M) ≅ E(−N) as well as the Riemann-Roch theorem for E(−N) and M − N - note that c_{1}(E(−N)) = M − N and c_{2}(E(−N)) = ℓ (compute the Chern classes from the defining extension twisted with N^{∨}):

χ(S, E(−N)) = 4 + \frac{1}{2}(M − N)^{2} − ℓ = 2 + χ(S, M − N) − ℓ.

We note that h^{0}(S, E(−N)) = h^{0}(S, M − N). Indeed, if ℓ ≥ 1 then h^{0}(S, I_{ξ}) = 0, and if ℓ = 0 use that 1 ∈ H^{0}(S, O_{S}) is mapped to the extension class through H^{0}(S, O_{S}) → H^{1}(S, M − N). Moreover, h^{2}(S, E(−N)) = h^{0}(S, E(−M)) = 1, and we write that χ(S, E(−M)) = 2 + h^{0}(S, M ⊗ N^{∨}) − h^{1}(S, M ⊗ N^{∨}) − ℓ, that is,

h^{1}(S, E(−N)) = ℓ − 1 + h^{1}(S, M ⊗ N^{∨}) − h^{2}(S, M ⊗ N^{∨}).

□

Assuming that P_{N,ℓ} ≠ ∅, we consider the Grassmann bundle G_{N,ℓ} over P_{N,ℓ} classifying pairs (E, Λ) with [E] ∈ P_{N,ℓ} and Λ ∈ G(2, H^{0}(S, E)). If d := c_{2}(E) we define the rational map h_{N,ℓ} : G_{N,ℓ} −−→ W_{d}^{1}(|L|), by setting h_{N,ℓ}(E, Λ) := (C_{Λ}, A_{Λ}), where A_{Λ} ∈ Pic^{d}(C_{Λ}) is such that the following exact sequence on C_{Λ} holds:

0 → Λ ⊗ O_{S} →^{ev_{Λ}} E → K_{C_{Λ}} ⊗ A_{Λ}^{∨} → 0.

Lemma 3.8. If P_{N,ℓ} ≠ ∅, then dim G_{N,ℓ} = g + ℓ + h^{0}(S, M ⊗ N^{∨}).

Proof. Let [E] ∈ P_{N,ℓ}. It is clear that

dim G_{N,ℓ} = 2ℓ + dim P(Ext_{S}^{1}(N ⊗ I_{ξ}, M)) + 2(g + 1 − c_{2}(E)).

Applying Lemma 3.7, as well as the fact that l = c_{2}(E) − M · N, we find that

dim G_{N,ℓ} = 2g − 3M · N + c_{2}(E) + 1 + h^{1}(S, M − N) − h^{2}(S, M − N)

= (g + c_{2}(E) − M · N) + (g − 2M · N + 1 + h^{1}(S, M − N) − h^{2}(S, M − N)).

From Riemann-Roch, we can write

χ(S, M − N) = 2 + \frac{1}{2}(M − N)^{2} = 2 + \frac{1}{2}L^{2} − 2M · N = g + 1 − 2M · N.

The conclusion follows. □

Lemma 3.9. Assume that P_{N,ℓ} contains a Lazarsfeld-Mukai vector bundle E on S with c_{2}(E) = d, and let W ⊂ W_{d}^{1}(|L|) be the closure of the image of the rational map h_{N,ℓ} : G_{N,ℓ} −−→ W_{d}^{1}(|L|). Then dim W = g + d − M · N = g + ℓ.

Proof. Clearly W is irreducible, as G_{N,ℓ} is irreducible. If (C, A) ∈ Im(h_{N,ℓ}), then the fibre h_{N,ℓ}^{−1}(C, A) is isomorphic to the projectivization of the space of morphisms from E_{C,A} to K_{C} ⊗ A^{∨}. The latter is isomorphic to H^{0}(S, E_{C,A} ⊗ F_{C,A}), and has dimension h^{0}(S, M ⊗ N^{∨}) + 1. Therefore, the general fibre of h_{N,ℓ} has dimension h^{0}(S, M ⊗ N^{∨}). We apply now Lemma 3.8. □

Lemma 3.10. Suppose that a smooth curve C ∈ |L| has Clifford dimension one and A is a globally generated line bundle on C with h^{0}(C, A) = 2 and [E_{C,A}] ∈ P_{N,ℓ}. Then M · N ≥ gon(C).

Proof. By Lemma 3.1it follows that M|_{C} contributes to the Cliff(C). From the exact sequence 0 → N^{∨} → M → M|_{C} → 0 and from the observation that h^{1}(S, N) = 0 (see Remark 3.6), we obtain by direct computation that

Cliff(M|_{C}) = M · N + M^{2} − 2h^{0}(S, M) + 2 = M · N − 2 − 2h^{1}(S, M) ≥ k − 2,

that is, M · N ≥ k + 2h^{1}(S, M) ≥ k. □

Remark 3.11. If we drop the condition on the Clifford dimension in the hypothesis of Lemma 3.10, we obtain the inequality M · N ≥ Cliff(C) + 2.

So far, we took care of indecomposable non-simple Lazarsfeld-Mukai bundles, and computed the dimensions of the corresponding parameter spaces. The decomposable case is much simpler. Let E = E_{C,A} = M ⊕ N be a decomposable Lazarsfeld-Mukai bundle. It was proved in [La89] that the differential of the natural map h_{E} : G(2, H^{0}(S, E)) −−→ |L|_{s} at a point [Λ], with Λ = H^{0}(C, A)^{∨}, coincides with the multiplication map μ_{0,A}. Hence, if the Grassmannian G(2, H^{0}(S, E)) dominates the linear system, the multiplication map is surjective at a general point and the corresponding irreducible components of the Brill-Noether loci are zerodimensional. This case can occur only if the Brill-Noether number is non-negative.

All these intermediate results amount to the following:

Theorem 3.12. Let S be a K3 surface and L a globally generated line bundle on S, such that general curves in |L| are of Clifford dimension one. Suppose that ρ(g, 1, k) ≤ 0, where L^{2} = 2g − 2 and k is the (constant) gonality of all curves in |L|_{s}. Then a general curve C ∈ |L| satisfies the linear growth condition (2), thus Green’s Conjecture is verified for any smooth curve in |L|.

In the case ρ(g, 1, k) = 1, Green’s Conjecture is also verified for smooth curves in |L|, cf. [V05], [HR98]. To sum up, Green’s Conjecture is verified for any curve of Clifford dimension one on a K3 surfaces.

Proof. It suffices to estimate the dimension of dominating irreducible components W of W_{k+n}^{1}(|L|), with n = 0, …, g − k + 2, with general point corresponding to a non-simple indecomposable Lazarsfeld-Mukai bundle. Lemmas 3.9and 3.10yield dim W ≤ g + n, which finishes the proof. □

Remark 3.13. The proof of Theorem 3.12 shows that for d > g − k + 2, every dominating component of W_{d}^{1}(|L|) corresponds to simple Lazarsfeld-Mukai bundles. In particular, for a general curve C ∈ |L|, one has dim W_{d}^{1}(C) = ρ(g, 1, d).

Remark 3.14. The problem of deciding whether Lazarsfeld-Mukai bundles appear in a given space P_{N,ℓ} is a non-trivial one, cf. Remark 2.2.

4. A criterion for the Green-Lazarsfeld Gonality Conjecture

Along with Green’s Conjecture, another statement of similar flavor was proposed by Green and Lazarsfeld, [GL86].

Conjecture 4.1 (The Gonality Conjecture). For any smooth curve C of gonality d, every non-special globally generated line bundle L on C of sufficiently high degree satisfies K_{h^{0}(L)−d,1}(C, L) = 0.

Conjecture 4.1is equivalent to the seemingly weaker statement that there exists a globally generated line bundle L ∈ Pic(C) with h^{1}(C, L) = 0 for which the Koszul vanishing holds [Ap02]. On a curve C with the lgc property (2), line bundles of type K_{C}(x + y), where x, y ∈ C are general points, verify the Gonality Conjecture [Ap05]. In particular, Theorem 3.12 implies the following:

Corollary 4.2. Let S be a K3 surface and L a globally generated line bundle on S, such that general curves in |L| are of Clifford dimension one. Then a general curve C ∈ |L| verifies Conjecture 4.1.

The main result of this short section is a refinement of the main result of [Ap05]:

Theorem 4.3. Let C be a smooth curve of Clifford dimension one and x, y ∈ C be distinct points, and denote

Z_{n} := {A ∈ W_{k+n}^{1}(C) : h^{0}(C, A(−x − y)) ≥ 1}.

Suppose that dim Z_{n} ≤ n − 1, for all 0 ≤ n ≤ g − 2k + 2. Then the bundle K_{C}(x + y) verifies the Gonality Conjecture.

The condition in the statement of Theorem 4.3 means that passing through the points x and y is a non-trivial condition on any irreducible component of maximal allowed dimension n of the Brill-Noether locus W_{k+n}^{1}(C), for all 0 ≤ n ≤ g−2k+2.

Proof. The proof is an almost verbatim copy of the proof of [Ap05] Theorem 2. Define ν := g − 2k + 2. The idea is to show that for any 0 ≤ n ≤ ν, and for (n + 1) pairs of distinct general points x_{0} + y_{0}, x_{1} + y_{1}, …, x_{n} + y_{n} ∈ C_{2}, there is no line bundle A ∈ W_{k+n}^{1}(C) with h^{0}(C, A(−x_{i} − y_{i})) ≠ 0 for all 1 ≤ i ≤ n, such that either h^{0}(C, A(−x − y)) ≠ 0 or h^{0}(C, A(−x_{0} − y_{0})) ≠ 0. To this end, consider the incidence varieties

(∏_{i=1}^{n} C_{2}) × Z_{n} ⊃ {(x_{1} + y_{1}, …, x_{n} + y_{n}, A) : h^{0}(A(−x_{i} − y_{i})) ≠ 0, ∀i},

respectively,

(∏_{i=1}^{n+1} C_{2})×W_{k+n}^{1}(C) ⊃ {(x_{0}+y_{0}, x_{1}+y_{1}, …, x_{n}+y_{n}, A) : h^{0}(A(−x_{i}−y_{i})) ≠ 0, ∀i}.

The fibres of the projection to Z_{n} are n-dimensional, hence the incidence variety is at most (2n−1)-dimensional and it cannot dominate ∏_{i=1}^{n} C_{2}. Similarly, the second variety is at most (2n + 1)-dimensional. Note that the condition to pass through a pair of general points is a non-trivial condition on every variety of complete pencils. To conclude, apply [Ap05] Proposition 8. □

5. Curves of higher Clifford dimension

We analyze the Koszul cohomology of curves of higher Clifford dimension on a K3 surface S. This case has similarities to [ApP08], where one focused on K3 surfaces with Picard number 2. Since plane curves are known to verify Green’s Conjecture, the significant cases occur when the Clifford dimension is at least 3. Note that, unlike the Clifford index, the Clifford dimension is not semi-continous. An example was given by Donagi-Morrisson [DM89]: If ϵ : S → P^{2} is a double sextic and L = ϵ^{∗}(O_{ℙ^{2}}(3)), then the general element in |L| is isomorphic to a smooth plane sextic, hence it has Clifford dimension 2, while special points correspond to bielliptic curves and are of Clifford dimension 1.

It was proved in [CP95] and [Kn09] that, except for the Donagi-Morrisson example, if a globally generated linear system |L| on S contains smooth curves of Clifford dimension at least 2, then L = O_{S}(2D + Γ), where D, Γ ⊂ S are smooth curves, D^{2} ≥ 2 (hence h^{0}(S, O_{S}(D)) ≥ 2), Γ^{2} = −2 and D · Γ = 1. If the genus of D is r ≥ 3, then the genus of a smooth curve C ∈ |L| equals 4r − 2 ≥ 10, and gon(C) = 2r, while Cliff(C) = 2r − 3; the Clifford dimension of C is r. From now on, we assume that we are in this situation.

Green’s hyperplane section theorem implies that the Koszul cohomology is constant in a linear system. As in [ApP08], we degenerate a smooth curve C ∈ |2D+Γ| to a reducible curve X +Γ with X ∈ |2D|. In order to be able to carry out this plan, we first analyze the geometry of the curves in |2D|. Notably, we shall prove:

Theorem 5.1. The hypothesis of Theorem 4.3 are verified for a general curve X ∈ |2D| and the two points of intersection X · Γ.

The proof of Theorem 5.1 proceeds in several steps. The first result describes the fundamental invariants of a quadratic complete intersection section of S:

Lemma 5.2. Any smooth curve X ∈ |2D| has genus 4r − 3, gonality 2r − 2, and Cliff(X) = 2r − 4.

Proof. Since Cliff(D|_{X}) = 2r − 4, we obtain Cliff(X) ≤ 2r − 4 < (g(X) − 1)/2, that is, Cliff(X) is computed by a line bundle B ∈ Pic(S), cf. [GL87]. Both bundles B and B′ := O_{S}(X)⊗B^{∨} are globally generated, hence B ·Γ ≥ 0 and B′ ·Γ ≥ 0. Since X · Γ = 2, we can assume that B · Γ ≤ 1. We may also assume, cf. [Ma89] Corollary 2.3, that h^{0}(S, B) = h^{0}(X, B|_{X}) and h^{0}(S, B′) = h^{0}(X, B′_{|X}). Then if C ∈ |L| is smooth as above, we obtain the estimate

Cliff(X) = B · X − 2h^{0}(S, B) + 2 ≥ B · C − 2h^{0}(C, B|_{C}) + 1 ≥ 2r − 4.

Since X has Clifford dimension 1, it follows that gon(X) = 2r − 2. □

It suffices therefore to analyze the structure of the loci W_{2r−2+n}^{1}(X) where n ≤ 3 = g(X) − 2gon(X) + 2, and more precisely those components of dimension n.

Lemma 5.3. We fix a general X ∈ |2D|, viewed as a half-canonical curve X ⟶^{|D|} P^{r}.

• W_{2r−2}^{1}(X) is finite and all minimal pencils 𝔤_{2r−2}^{1} on X are given by the rulings of quadrics of rank 4 in H^{0}(P^{r}, I_{X/P^{r}}(2)).

• X has no base point free pencils 𝔤_{2r−1}^{1}, that is, W_{2r−1}^{1}(X) = X + W_{2r−2}^{1}(X).

• For n = 2, 3, if A ∈ W_{2r−2+n}^{1}(X) is a base point free pencil, then the vector

bundle E_{X,A} is not simple. In all cases n ≤ 3, if A belongs to an n-dimensional component of W_{2r−2+n}^{1}(X), then the corresponding DM extension

0 → M → E_{X,A} → N ⊗ I_{ξ} → 0

verifies length(ξ) = n, M · N = 2r − 2 and M · Γ = N · Γ = 1. When n = 2, 3, we can take M = N = O_{X}(D).

Proof. We use Accola’s lemma, cf. [ELMS89] Lemma 3.1. If A ∈ W_{2r−2+n}^{1}(X) is base point free with n ≤ 3, then h^{0}(X, O_{X}(D) ⊗ A^{∨}) ≥ 2 − n/2. In particular, when n = 0, 1, we find that A′ := O_{X}(D) ⊗ A^{∨} is a pencil as well. When n = 1, we find that A′ ∈ W_{2r−3}^{1}(X), which is impossible, that is, X carries no base point free pencils 𝔤_{2r−1}^{1}. If n = 0, then deg(A) = deg(A′) = 2r − 2 and this corresponds to a quadric Q ∈ H^{0}(P^{r}, I_{X/P^{r}}(2)) with rk(Q) = 4 and X ∩ Sing(Q) = ∅, such that the rulings of Q cut out on X, precisely the pencils A and A′ respectively. If n = 2, 3, we find that h^{0}(X, K_{X}(−2A)) ≠ 0, thus the kernel of the Petri map Ker μ_{0,A} = H^{0}(X, K_{X}(−2A)) ≠ 0, and then the Lazarsfeld-Mukai bundle E_{X,A} cannot be simple.

The vector bundle E = E_{X,A} is thus expressible as a DM extension

0 → M → E_{X,A} → N ⊗ I_{ξ} → 0,

and we recall that N is globally generated with h^{1}(S, N) = 0. Suppose first that n ≠ 0, thus n ∈ {2, 3}. Then h^{0}(S, M ⊗ N^{∨}) ≠ 0, for otherwise ξ = ∅, the extension is split, and the split case only produces zero-dimensional components of the Brill-Noether loci, whilst we are in the higher dimensional case. From the exact sequence defining E_{X,A} coupled with Accola’s Lemma, we obtain the isomorphisms

H^{0}(S, E(−D)) ≅ H^{0}(X, K_{X}(−A) ⊗ O_{X}(−D)) = H^{0}(X, O_{X}(D − A)) ≠ 0,

therefore H^{0}(S, M(−D)) ≅ H^{0}(S, E(−D)) ≠ 0. Choose an effective divisor F ∈ |M(−D)| and then F ∈ |O_{S}(D)(−N)| as well. From Lemmas 3.9 and 3.10 and the generality assumption on X, we find that length(ξ) = n, M ·N = gon(X) = 2r −2 and h^{1}(S, M) = 0. Furthermore, one computes that F^{2} = 0. Since, by degree reasons, h^{0}(S, O_{S}(F)) = h^{0}(X, O_{X}(D − A)) = 1 one obtains that F ≡ 0, that is, M = N = O_{X}(D).

If n = 0, then in the associated DM extension, ξ = 0, and M, N ∈ Pic(S) are globally generated, M · N = 2r − 2 and h^{0}(M) = h^{0}(M|_{X}) and h^{0}(N) = h^{0}(N|_{X}). The intersection of Γ with one of the bundles M or N is ≤ 1; suppose M · Γ ≤ 1. We choose a smooth curve C ∈ |2D + Γ|, and compute

Cliff(M|_{C}) = M · C − 2h^{0}(M|_{C}) + 2 ≤ M · X − 2h^{0}(M) + 2 + 1 = Cliff(X) + 1,

hence M computes Cliff(C) and M · Γ = N · Γ = 1. □

Lemma 5.4. Let X ∈ |2D| be any smooth curve, and x, y ∈ X ·Γ. For any integer n ≥ 0, and any base point free pencil A ∈ W_{2r−2+n}^{1}(X), the following are equivalent:

(1) h^{0}(X, A(−x − y)) ≠ 0;

(2) E_{X,A}|_{Γ} ≅ O_{Γ} ⊕ O_{Γ}(2).

Proof. The non-vanishing of H^{0}(X, A(−x − y)) is equivalent to

(8)  h^{0}(X, A(−x − y)) = 1.

Twisting the defining exact sequence of E_{X,A} by O_{S}(Γ), we obtain

(9) 0 → H^{0}(X, A)^{∨} ⊗ O_{S}(Γ) → E_{X,A} ⊗ O_{S}(Γ) → K_{X} ⊗ A^{∨}(x + y) → 0.

By Riemann-Roch, H^{1}(S, O_{S}(Γ)) = 0. By taking cohomology, h^{0}(C, A(−x−y)) = 1 if and only if h^{0}(S, E_{X,A} ⊗ O_{S}(Γ)) = 2r + 3 − n. On the other hand, h^{0}(S, E_{X,A}) = 2 + h^{1}(X, A) = 2r + 2 − n. Consider the (twisted) exact sequence defining Γ:

0 → E_{X,A} → E_{X,A} ⊗ O_{S}(Γ) → E_{X,A}|_{Γ}(−2) → 0.

We find by taking cohomology that x, y ∈ C lie in the same fibre of |A| if and only if h^{0}(Γ, E_{X,A|Γ}(−2)) = 1. Expressing E_{X,A}|_{Γ} = O_{Γ}(a) ⊕ O_{Γ}(b), with a + b = 2, condition (8) becomes equivalent to E_{X,A}|_{Γ} ≅ O_{Γ} ⊕ O_{Γ}(2). □

Proof of Theorem 5.1. For a base point free A ∈ W_{2r−2+n}^{1}(X) with n = 0, 2, 3, the vector bundle E := E_{X,A} appears as an extension

0 → M → E → N ⊗ I_{ξ} → 0,

with length(ξ) = n, and M · Γ = N · Γ = 1. Recall that being a Lazarsfeld-Mukai bundle is an open condition in any flat family of bundles, Remark 2.2. Hence, LM bundles in a parameter space P_{N,n} correspond to general cycles ξ ∈ S^{[n]}. For n = 0, we see immediately that E|_{Γ} = O_{Γ}(1)^{⊕2}. For n = 2, 3 the same is true for ξ such that ξ ∩ Γ = ∅. To conclude, we apply Lemma 5.4. □

Remark 5.5. The variety W_{2r−2}^{1}(|2D|) is birationally equivalent to the parameter space of pairs (Q, Π), where Q ∈ |O_{P}^{r}(2)| is a quadric of rank 4 and Π ⊂ Q is a ruling. In particular, W_{2r−2}^{1}(|2D|) is irreducible (and of dimension g).

Theorem 5.1 and Theorem 4.3 imply the following:

Corollary 5.6. For a general curve X ∈ |2D|, we have K_{2r,1}(X, K_{X} ⊗ O_{S}(Γ)) = 0.

The main result of this section is (compare to [ApP08]):

Theorem 5.7. Smooth curves of Clifford dimension at least three on K3 surfaces satisfy Green’s Conjecture.

Proof. As in [ApP08, Section 4.1], for all p ≥ 1, we have isomorphisms

K_{p,1}(X + Γ, ω_{X+Γ}) ≅ K_{p,1}(X, K_{X}(Γ)).

Corollary 5.6 shows that K_{2r,1}(X + Γ, ω_{X+Γ}) = 0, implying the vanishing of K_{2r,1}(S, L), via Green’s hyperplane section theorem. Using the hyperplane section theorem again, we obtain K_{2r,1}(C, K_{C}) = 0, for any smooth curve C ∈ |L|, that is, the vanishing predicted by Green’s Conjecture for C. □

Theorems 3.12 and 5.7, altogether complete the proof of Theorem 1.1.

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Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, RO-014700 Bucharest – Romania, and Şcoala Normală Superioară Bucureşti, Calea Griviţei 21, Sector 1, RO-010702 Bucharest, Romania

   E-mail address: aprodu@imar.ro

Humboldt-Universität zu Berlin, Institut Für Mathematik, 10099 Berlin, Germany

   E-mail address: farkas@math.hu-berlin.de