HIGHER RANK BRILL-NOETHER THEORY ON SECTIONS OF K3 SURFACES

GAVRIL FARKAS AND ANGELA ORTEGA

Abstract. We discuss the role of K3 surfaces in the context of Mercat’s conjecture in higher rank Brill-Noether theory. Using liftings of Koszul classes, we show that Mercat’s conjecture in rank 2 fails for any number of sections and for any gonality stratum along a Noether-Lefschetz divisor inside the locus of curves lying on K3 surfaces. Then we show that Mercat’s conjecture in rank 3 fails even for curves lying on K3 surfaces with Picard number 1. Finally, we provide a detailed proof of Mercat’s conjecture in rank 2 for general curves of genus 11, and describe explicitly the action of the Fourier-Mukai involution on the moduli space of curves.

1. Introduction

The Clifford index Cliff(C) of an algebraic curve C is the second most important invariant of C after the genus, measuring the complexity of the curve in its moduli space. Its geometric significance is amply illustrated for instance in the statement

K_{p,2}(C, K_{C}) = 0 ⇔ p < Cliff(C)

of Green’s Conjecture [G] on syzygies of canonical curves. It has been a long-standing problem to find an adequate generalization of Cliff(C) for higher rank vector bundles. A definition in this sense has been proposed by Lange and Newstead [LN1]: If E ∈ U_{C}(n, d) denotes a semistable vector bundle of rank n and degree d on a curve C of genus g, one defines its Clifford index as

γ(E) := μ(E) − \frac{2}{n}h^{0}(C, E) + 2 ≥ 0,

and then the higher Clifford indices of C are defined as the quantities

Cliff_{n}(C) := min{γ(E) : E ∈ U_{C}(n, d), d ≤ n(g − 1), h^{0}(C, E) ≥ 2n}^{1}.

Note that Cliff_{1}(C) = Cliff(C) is the classical Clifford index of C. By specializing to sums of line bundles, it is easy to check that Cliff_{n}(C) ≤ Cliff(C) for all n ≥ 1. Mercat [Me] proposed the following interesting conjecture, which we state in the form of [LN1] Conjecture 9.3, linking the newly-defined invariants Cliff_{n}(C) to the classical geometry of C:

(M_{n}) : Cliff_{n}(C) = Cliff(C).

Mercat’s conjecture (M_{2}) holds for various classes of curves, in particular general kgonal curves of genus g > 4k − 4, or arbitrary smooth plane curves, see [LN1]. In [FO] Theorem 1.7, we have verified (M_{2}) for a general curve [C] ∈ M_{g} with g ≤ 16. More generally, the statement (M_{2}) is a consequence of the Maximal Rank Conjecture (see [FO] Conjecture 2.2), therefore it is expected to be true for a general curve [C] ∈ M_{g}. However, for every genus g ≥ 11 there exist curves [C] ∈ M_{g} with maximal Clifford index Cliff(C) = [\frac{g−1}{2}] carrying stable rank 2 vector bundles E with h^{0}(C, E) = 4 and γ(E) < Cliff(C), see [FO] Theorems 3.6 and 3.7 and [LN2] Theorem 1.1 for an improvement. For these curves, the inequality Cliff_{2}(C) < Cliff(C) holds.

1The invariant Cliff_{n}(C) is denoted in the paper [LN1] by γ′_{n}(C). Since the appearance of [LN1], it has become abundantly clear that Cliff_{n}(C), defined as above, is the most relevant Clifford type invariant for rank n vector bundles on C. Accordingly, the notation Cliff_{n}(C) seems appropriate.

Obvious questions emerging from this discussion are whether such results are specific to (i) rank 2 bundles with 4 sections, or to (ii) curves with maximal Clifford index [\frac{g−1}{2}]. First we prove that under general circumstances, curves on K3 surfaces carry rank 2 vector bundles E with a prescribed (and exceptionally high) number of sections invalidating Mercat’s inequality γ(E) ≥ Cliff(C):

Theorem 1.1. We fix integers p ≥ 1 and a ≥ 2p + 3. There exists a smooth curve C of genus 2a + 1 and Clifford index Cliff(C) = a, lying on a K3 surface C ⊂ S ⊂ P^{2p+2} with Pic(S) = ℤ · C ⊕ ℤ · H, where H^{2} = 4p + 2, H · C = deg(C) = 2a + 2p + 1, as well as a stable rank 2 vector bundle E ∈ SU_{C}(2, O_{C}(H)), such that h^{0}(C, E) = p + 3. In particular γ(E) = a − \frac{1}{2} < Cliff(C) and Mercat’s conjecture (M_{2}) fails for C.

It is well-known cf. [M2], [V1], that a curve [C] ∈ M_{2a+1} lying on a K3 surface S possesses a rank 2 vector bundle F ∈ SU_{C}(2, K_{C}) with h^{0}(C, F) = a + 2. In particular, γ(F) = a ≥ Cliff(C) (with equality if Pic(S) = ℤ · C), hence such bundles satisfy condition (M_{2}). Let us consider the K3 locus in the moduli space of curves

K_{g} := {[C] ∈ M_{g} : C lies on a K3 surface}.

When g = 11 or g ≥ 13, the variety K_{g} is irreducible and dim(K_{g}) = 19 + g, see [CLM] Theorem 5. For integers r, d ≥ 1 such that d^{2} > 4(r − 1)g and 2r − 2 ∤ d, we define the Noether-Lefschetz divisor inside the locus of sections of K3 surfaces

𝔑𝔏_{g,d}^{r} := \{[C] ∈ K_{g} | \begin{matrix}C lies on a K3 surface S, Pic(S) ⊃ ℤ · C ⊕ ℤ · H, \\ H ∈ Pic(S) is nef, H^{2} = 2r − 2, C · H = d, C^{2} = 2g − 2 \end{matrix} \} .

A consequence of Theorem 1.1 can be formulated as follows:

Corollary 1.2. We fix integers p ≥ 1 and a ≥ 2p + 3 and set g := 2a + 1. Then Mercat’s conjecture (M_{2}) fails generically along the Noether-Lefschetz locus 𝔑𝔏_{g,2a+2p+1}^{2p+2} inside K_{g}, that is, Cliff_{2}(C) < Cliff(C) for a general point [C] ∈ 𝔑𝔏_{g,2a+2p+1}^{2p+2}.

It is natural to wonder whether it is necessary to pass to a Noether-Lefschetz divisor in K_{g}, or perhaps, all curves [C] ∈ K_{g} give counterexamples to conjecture (M_{2}). To see that this is not always the case and all conditions in Theorem 1.1 are necessary, we study in detail the case g = 11. Mukai [M3] proved that a general curve [C] ∈ M_{11} lies on a unique K3 surface S with Pic(S) = ℤ · C, thus, M_{11} = K_{11}.

Theorem 1.3. For a general curve [C] ∈ M_{11} one has the equality Cliff_{2}(C) = Cliff(C), that is, Mercat’s conjecture holds generically on M_{11}. Furthermore, the locus

{[C] ∈ M_{11} : Cliff_{2}(C) < Cliff(C)}

can be identified with the Noether-Lefschetz divisor 𝔑𝔏_{11,13}^{4} on M_{11}.

In Section 5, we describe in detail the divisor 𝔑𝔏_{11,13}^{4} and discuss, in connection with Mercat’s conjecture, the action of the Fourier-Mukai involution F M : F_{11} → F_{11} on the moduli space of polarized K3 surfaces of genus 11. The automorphism F M acts on the set of Noether-Lefschetz divisors and in particular it (i) fixes the 6-gonal locus M_{11,6}^{1} and it maps the divisor 𝔑𝔏_{11,13}^{4} which corresponds to certain elliptic K3 surfaces, to the Noether-Lefschetz divisor corresponding to K3 surfaces carrying a rational curve of degree 3.

Next we turn our attention to the conjecture (M_{n}) for n ≥ 3. It was observed in [LMN] that Mukai’s description [M4] of a general curve of genus 9 in terms of linear sections of a certain rational homogeneous variety, and especially the connection to rank 3 Brill-Noether theory, can be used to construct, on a general curve [C] ∈ M_{9}, a stable vector bundle E ∈ SU_{C}(3, K_{C}) such that h^{0}(C, E) = 6. In particular γ(E) = \frac{10}{3} < Cliff(C), that is, Mercat’s conjecture (M_{3}) fails for a general curve [C] ∈ M_{9}. A similar construction is provided in [LMN] for a general curve of genus 11. In what follows we outline a construction illustrating that the results from [LMN] are part of a larger picture and curves on K3 surfaces carry vector bundles E of rank at least 3 with γ(E) < Cliff(C).

Let S be a K3 surface and C ⊂ S a smooth curve of genus g. We choose a linear series A ∈ W_{d}^{r}(C) of minimal degree such that the Brill-Noether number ρ(g, r, d) is non-negative, that is, d := r + [\frac{r(g+1)}{r+1}]. The Lazarsfeld bundle M_{A} on C is defined as the kernel of the evaluation map, that is,

0 ⟶ M_{A} ⟶ H^{0}(C, A) ⊗ O_{C} ⟶^{ev_{C}} A ⟶ 0.

As usual, we set Q_{A} := M_{A}^{∨}, hence rank(Q_{A}) = r and det(Q_{A}) = A. Following a procedure that already appeared in [L], [M2], [V1], we note that C carries a vector bundle of rank r+1 with canonical determinant and unexpectedly many global sections:

Theorem 1.4. For a curve C ⊂ S and A ∈ W_{d}^{r}(C) as above there exists a globally generated vector bundle E on C with rank(E) = r + 1 and det(E) = K_{C}, expressible as an extension

0 ⟶ Q_{A} ⟶ E ⟶ K_{C} ⊗ A^{∨} ⟶ 0,

satisfying the condition h^{0}(C, E) = h^{0}(C, A)+h^{0}(C, K_{C} ⊗A^{∨}) = g −d+2r +1. If moreover r ≤ 2 and Pic(S) = ℤ · C, then the above extension is non-trivial.

When r = 1 the rank 2 bundle E constructed in Theorem 1.4 is well-known and plays an essential role in [V1]. In this case γ(E) ≥ [\frac{g−1}{2}]. For r = 2 and g = 9 (in which case A ∈ W_{8}^{2}(C)), or for g = 11 (and then A ∈ W_{10}^{2}(C)), Theorem 1.4 specializes to the construction in [LMN]. When rank(E) = 3, we observe by direct calculation that γ(E) < [\frac{g−1}{2}]. In view of providing counterexamples to Mercat’s conjecture (M_{3}), it is thus important to determine whether E is stable.

Theorem 1.5. Fix C ⊂ S as above with g = 7, 9 or g ≥ 11 such that Pic(S) = ℤ · C, as well as A ∈ W_{d}^{2}(C), where d := [\frac{2g+8}{3}]. Then any globally generated rank 3 vector bundle E on C lying non-trivially in the extension

0 ⟶ Q_{A} ⟶ E ⟶ K_{C} ⊗ A^{∨} ⟶ 0,

and with h^{0}(C, E) = h^{0}(C, A) + h^{0}(C, K_{C} ⊗ A^{∨}) = g − d + 5, is stable.

As a corollary, we note that for sufficiently high genus Mercat’s statement (M_{3}) fails to hold for any smooth curve of maximal Clifford index lying on a K3 surface.

Corollary 1.6. We fix an integer g = 9 or g ≥ 11 and a curve [C] ∈ K_{g}. Then the inequality Cliff_{3}(C) < [\frac{g−1}{2}] holds. In particular, Mercat’s conjecture (M_{3}) fails generically along K_{g}.

We close the Introduction by thanking Herbert Lange and Peter Newstead for making a number of very pertinent comments on the first version of this paper.

2. Higher rank vector bundles with canonical determinant

In this section we treat Mercat’s conjecture (M_{3}) and prove Theorems 1.4 and 1.5. We begin with a curve C of genus g lying on a smooth K3 surface S such that Pic(S) = ℤ · C, and fix a linear series A ∈ W_{d}^{2}(C) of minimal degree d := [\frac{2g+8}{3}]. Under such assumptions both A and K_{C} ⊗ A^{∨} are base point free. From the onset, we point out that the existence of vector bundles of higher rank on C having exceptional Brill-Noether behaviour has been repeatedly used in [L], [M2] and [V1]. Our aim is to study these bundles from the point of view of Mercat’s conjecture and discuss their stability.

We define the Lazarsfeld-Mukai sheaf F_{A} via the following exact sequence on S:

0 ⟶ F_{A} ⟶ H^{0}(C, A) ⊗ O_{S} ⟶^{ev_{S}} A ⟶ 0.

Since A is base point free, F_{A} is locally free. We consider the vector bundle E_{A} := F_{A}^{∨} on S, which by dualizing, sits in an exact sequence

(1)  0 ⟶ H^{0}(C, A)^{∨} ⊗ O_{S} ⟶ E_{A} ⟶ K_{C} ⊗ A^{∨} ⟶ 0.

Since K_{C} ⊗ A^{∨} is assumed to be base point free, the bundle E_{A} is globally generated. It is well-known (and follows from the sequence (1), that c_{1}(E_{A}) = O_{S}(C) and c_{2}(E_{A}) = d. Proof of Theorem 1.4. We write down the following commutative diagram

0  0
↓  ↓
H^{0}(C, A) ⊗ O_{S}(−C) ⟶^{=} H^{0}(C, A) ⊗ O_{S}(−C)
↓  ↓
0 ⟶  F_{A}  ⟶  H^{0}(C, A) ⊗ O_{S}  ⟶ A ⟶ 0
↓  ↓  ↓^{=}
0 ⟶  M_{A}  ⟶  H^{0}(C, A) ⊗ O_{C}  ⟶ A ⟶ 0
↓  ↓
0  0

from which, if we set F_{A} := F_{A} ⊗ O_{C} and E_{A} := E_{A} ⊗ O_{C}, we obtain the exact sequence

0 ⟶ M_{A} ⊗ K_{C}^{∨} ⟶ H^{0}(C, A) ⊗ K_{C}^{∨} ⟶ F_{A} ⟶ M_{A} ⟶ 0

(use that Tor_{O_{S}}^{1}(M_{A}, O_{C}) = M_{A} ⊗ K_{C}^{∨}). Taking duals, we find the exact sequence

(2)  0 ⟶ Q_{A} ⟶ E_{A} ⟶ K_{C} ⊗ A^{∨} ⟶ 0.

Since S is regular, from (1) we obtain that h^{0}(S, E_{A}) = h^{0}(C, A) + h^{0}(C, K_{C} ⊗ A^{∨}) while H^{0}(S, E_{A} ⊗ O_{S}(−C)) = 0, that is,

h^{0}(S, E_{A}) ≤ h^{0}(C, E_{A}) ≤ h^{0}(C, A) + h^{0}(C, K_{C} ⊗ A^{∨}).

Thus the sequence (2) is exact on global sections.

We are left with proving that the extension (2) is non-trivial. We set r = 2 and then rank(E_{A}) = 3 and place ourselves in the situation when Pic(S) = ℤ · C (the case r = 1 works similarly). By contradiction we assume that E_{A} = Q_{A} ⊕ (K_{C} ⊗ A^{∨}) and denote by s : E_{A} → Q_{A} a retract and by s̃ : E_{A} → Q_{A} the induced map. We set M := Ker{E_{A} ⟶^{s̃} Q_{A}}, hence M can be regarded as an elementary transformation of the Lazarsfeld-Mukai bundle E_{A} along C. By direct calculation we find that

c_{1}(M) = O_{S}(−C) and c_{2}(M) = 2d − 2g + 2,

hence the discriminant of M equals Δ(M) := 6c_{2}(M) − 2c_{1}^{2}(M) = 4(3d − 4g + 4) < 0. Thus the sheaf M is O_{S}(C)-unstable. Applying [HL] Theorems 7.3.3 and 7.3.4, there exists a subsheaf M′ ⊂ M such that if ξ_{M,M′} := \frac{c_{1}(M′)}{rank(M′)} − \frac{c_{1}(M)}{rank(M)} ∈ Pic(S)_{ℝ}, then

(i) ξ_{M,M′} · C > 0 and (ii) ξ_{M,M′}^{2} ≥ −\frac{Δ(M)}{18}.

Since Pic(S) = ℤ · C, we may write c_{1}(M′) = O_{S}(aC) and also set r′ := rank(M′). The Lazarsfeld-Mukai bundle E_{A} is O_{S}(C)-stable, in particular μ_{C}(M′) ≤ μ_{C}(E_{A}), which yields a ≤ 0. Then from (i) we write that 0 ≤ \frac{a}{r′} + \frac{1}{3} ≤ \frac{1}{3}, whereas from (ii) one finds

\frac{1}{9} ≥ \frac{4(g − 1) − 3d}{9(g − 1)} ⇔ d ≥ g − 1,

which is a contradiction. It follows that the extension (2) is non-trivial. □

It is natural to ask when is the above constructed bundle E_{A} stable. We give an affirmative answer under certain generality assumptions, when r < 3.

We fix a K3 surface S such that Pic(S) = ℤ·C and as before, set d := [\frac{2g+8}{3}]. Under these assumptions, it follows from [L] that C satisfies the Brill-Noether theorem. We prove the stability of every globally generated non-split bundle E sitting in an extension of the form (2) and having a maximal number of sections. Proof of Theorem 1.5. We first discuss the possibility of a destabilizing sequence

0 ⟶ F ⟶ E ⟶ B ⟶ 0,

where F is a vector bundle of rank 2 and deg(F) ≥ \frac{4}{3}(g − 1). Since E is globally generated, it follows that B is globally generated as well, hence h^{0}(C, B) ≥ 2, in particular deg(B) ≥ (g + 2)/2 and hence deg(F) ≤ \frac{3}{2}g − 3. Since deg(B) ≤ \frac{2}{3}(g − 1) and C is Brill-Noether general, it follows that h^{0}(C, B) = 2, therefore h^{0}(C, F) ≥ g − d + 3. There are two cases to distinguish, depending on whether F possesses a subpencil or not.

Assume first that F has no subpencils. We apply [PR] Lemma 3.9 to find that h^{0}(C, det(F)) ≥ 2h^{0}(C, F) − 3 ≥ 2g − 2d + 3. Writing down the inequality

ρ(g, 2g − 2d + 2, deg(F)) ≥ 0

and using that deg(F) < \frac{3}{2}g − 3, we obtain a contradiction. If on the other hand, F has a subpencil, then as pointed out in [FO] Lemma 3.2, γ(F) ≥ Cliff(C), but again this is a contradiction. This shows that E cannot have a rank 2 destabilizing subsheaf.

We are left with the possibility of a destabilizing short exact sequence

0 ⟶ B ⟶ E ⟶ F ⟶ 0,

where B is a line bundle with deg(B) ≥ \frac{2}{3}(g − 1) and F is a rank 2 bundle. The bundle Q_{A} is well-known to be stable and based on slope considerations, B cannot be a subbundle of Q_{A}, that is, necessarily H^{0}(C, K_{C} ⊗ A^{∨} ⊗ B^{∨}) ≠ 0. Since the bundle E is not decomposable, it follows that deg(B) ≤ deg(K_{C} ⊗ A^{∨}) − 1 = 2g − 3 − d. Furthermore h^{1}(C, B) ≥ 3.

If F is not stable, we reason along the lines of [LMN] Proposition 3.5 and pullback a destabilizing line subbundle of F to obtain a rank 2 subbundle F′ ⊂ E such that

deg(F′) ≥ deg(B) + \frac{1}{2}(deg(E) − deg(B)) ≥ \frac{4}{3}(g − 1),

which is the case which we have already ruled out. So we may assume that F is stable. We write h^{0}(C, B) = a+1, hence h^{0}(C, F) ≥ g −d−a+4. Assume first that F admits no subpencils. Then from [PR] Lemma 3.9 we find the following estimate for the number of sections of the line bundle det(F) = K_{C} ⊗ B^{∨},

h^{0}(C, K_{C} ⊗ B^{∨}) ≥ 2h^{0}(C, F) − 3 ≥ 2g − 2d − 2a + 5,

which, after applying Riemann-Roch to B, leads to the inequality

3a ≥ g − 2d + 5 + deg(B).

Combining this estimate with the Brill-Noether inequality ρ(g, a, deg(B)) ≥ 0 and substituting the actual value of d, we find that 3a + 3 ≥ g. On the other hand a ≤ h^{0}(C, K_{C} ⊗ A^{∨}) − 2 = g − d < \frac{g−3}{3}, and this is a contradiction.

Finally, if F admits a subpencil, then γ(F) ≥ Cliff(C). Combining this with the classical Clifford inequality for B, we find that γ(E) ≥ Cliff(C), which again is a contradiction. We conclude that the rank 3 bundle E must be stable. □

3. Rank 2 bundles and Koszul classes

The aim of this section is to prove Theorem 1.1. We shall construct rank 2 vector bundles on curves using a connection between vector bundles on curves and Koszul cohomology of line bundles, cf. [AN] and [V2]. Let us recall that for a smooth projective variety X, a sheaf F and a globally generated line bundle L on X, the Koszul cohomology group K_{p,q}(X; F, L) is defined as the cohomology of the complex:

⋀^{p+1} H^{0}(L)⊗H^{0}(F ⊗L^{q−1}) ⟶^{d_{p+1,q−1}} ⋀^{p} H^{0}(L)⊗H^{0}(F ⊗L^{q}) ⟶^{d_{p,q}} ⋀^{p−1} H^{0}(L)⊗H^{0}(F ⊗L^{q+1}).

Most of the time F = O_{X}, and then one writes K_{p,q}(X; O_{X}, L) := K_{p,q}(X, L).

A Koszul class [ζ] ∈ K_{p,1}(X, L) is said to have rank ≤ n, if there exists a subspace W ⊂ H^{0}(X, L) with dim(W) = n and a representative ζ ∈ ∧^{p}W ⊗ H^{0}(X, L). The smallest number n with this property is the rank of the syzygy [ζ].

Next we discuss a connection due to Voisin [V2] and expanded in [AN], between rank 2 vector bundles on curves and syzygies. Let E be a rank 2 bundle on a smooth curve C with h^{0}(C, E) ≥ p + 3 ≥ 4 and set L := det(E). Let

λ : ∧^{2}H^{0}(C, E) → H^{0}(C, L)

be the determinant map, and we assume that there exists linearly independent sections e_{1} ∈ H^{0}(C, E) and e_{2}, …, e_{p+3} ∈ H^{0}(C, E), such that the map

λ(e_{1} ∧ − ) : ⟨e_{2}, …, e_{p+3}⟩ → H^{0}(C, L)

in injective onto its image. Such an assumption is automatically satisfied for instance if E admits no subpencils. We introduce the subspace

W := ⟨s_{2} := λ(e_{1} ∧ e_{2}), …, s_{p+3} := λ(e_{1} ∧ e_{p+3})⟩ ⊂ H^{0}(C, L).

By assumption, dim(W) = p + 2. Following [AN] and [V2], we define the tensor

ζ(E) := ∑_{i<j}(−1)^{i+j} s_{2} ∧ … ∧ ŝ_{i} ∧ … ∧ ŝ_{j} ∧ … ∧ s_{p+3} ⊗ λ(e_{i} ∧ e_{j}) ∈ ∧^{p}W ⊗ H^{0}(C, L).

One checks that d_{p,1}(ζ(E)) = 0, hence [ζ(E)] ∈ K_{p,1}(C, L) is a non-trivial Koszul class of rank at most p + 2. Conversely, starting with a non-trivial class [ζ] ∈ K_{p,1}(C, L) represented by an element ζ of ∧^{p}W ⊗ H^{0}(C, L) where dim(W) = p + 2, Aprodu and Nagel [AN] Theorem 3.4 constructed a rank 2 vector bundle E on C with det(E) = L, h^{0}(C, E) ≥ p + 3 and such that [ζ(E)] = [ζ]. This correspondence sets up a dictionary between the Brill-Noether loci in {E ∈ SU_{C}(2, L) : h^{0}(C, E) ≥ p+3} and Koszul classes of rank at most p + 2 in K_{p,1}(C, L).

Let us now fix integers p ≥ 1 and a ≥ 2p + 3. Using the surjectivity of the period mapping, see e.g. [K] Theorem 1.1, one can construct a smooth K3 surface S ⊂ P^{2p+2} of degree 4p + 2 containing a smooth curve C ⊂ S of degree d := 2a + 2p + 1 and genus g := 2a + 1. The surface S can be chosen with Pic(S) = ℤ · H ⊕ ℤ · C, where H^{2} = 4p + 2, H · C = d and C^{2} = 4a. The smooth curve H ⊂ C is the hyperplane section of S and has genus g(H) = 2p + 2. The following observation is trivial:

Lemma 3.1. Keeping the notation above, we have that H^{0}(S, O_{S}(H − C)) = 0.

Proof. It is enough to notice that H is nef and (H − C) · H = 2p − 2a + 1 < 0. □

We consider the decomposable rank 2 bundle K_{H} = A ⊕ (K_{H} ⊗ A^{∨}) on H, where A ∈ W_{p+2}^{1}(H). Via the Green-Lazarsfeld non-vanishing theorem [GL1] (or equivalently, applying [AN]), one obtains a non-zero Koszul class of rank p + 1

β := [ζ(A ⊕ (K_{H} ⊗ A^{∨}))] ∈ K_{p,1}(H, K_{H}).

Since S is a regular surface, there exist an exact sequence

0 ⟶ H^{0}(S, O_{S}) ⟶ H^{0}(S, O_{S}(H)) ⟶ H^{0}(H, K_{H}) ⟶ 0,

which induces an isomorphism [G] Theorem (3.b.7)

res_{H} : K_{p,1}(S, O_{S}(H)) ≅ K_{p,1}(H, K_{H}).

By construction, the non-trivial class α := res_{H}^{−1}(β) ∈ K_{p,1}(S, O_{S}(H)) has rank at most rank(β) + 1 = p + 2. Using [G] Theorem (3.b.1), we write the following exact sequence in Koszul cohomology:

⋯ → K_{p,1}(S; −C, H) → K_{p,1}(S, H) → K_{p,1}(C, H ⊗ O_{C}) → K_{p−1,2}(S; −C, H) → · · · .

Since H^{0}(S, O_{S}(H − C)) = 0, it follows that K_{p,1}(S; −C, H) = 0, in particular the nonzero class α ∈ K_{p,1}(S, H) can be viewed as a Koszul class of rank at most p + 2 inside the group K_{p,1}(C, O_{C}(H)). This class corresponds to a stable rank 2 bundle on C:

Proposition 3.2. Let C ⊂ S ⊂ P^{2p+2} as above and L := O_{C}(1) ∈ Pic^{2a+2p+1}(C). Then there exists a stable vector bundle E ∈ SU_{C}(2, L) with h^{0}(C, E) = p + 3.

Proof. From [AN] we know that there exists a rank 2 vector bundle E on C with det(E) = L such that [ζ(E)] = α ∈ K_{p,1}(C, L), in particular h^{0}(C, E) ≥ p + 3. Geometrically, E is the restriction to C of the Lazarsfeld-Mukai bundle E_{A} on S corresponding to a pencil A ∈ W_{p+2}^{1}(H). In particular, E is globally generated, being the restriction of a globally generated bundle on S. We also know that Cliff(C) = a (to be proved in Proposition 3.3). Since γ(E) ≤ a − \frac{1}{2} < Cliff(C), it follows that E admits no subpencils (If B ⊂ E is a subpencil, then h^{0}(C, L ⊗ B^{∨}) ≥ 2 because E is globally generated. It is easily verified that both B and L ⊗ B^{∨} contribute to Cliff(C), which brings about a contradiction). Assume now that

0 ⟶ B ⟶ E ⟶ L ⊗ B^{∨} ⟶ 0

is a destabilizing sequence, where B ∈ Pic(C) has degree at least a + p + 1. As already pointed out, h^{0}(C, B) ≤ 1, hence h^{0}(C, L ⊗ B^{∨}) ≥ p + 2. If h^{1}(C, L ⊗ B^{∨}) ≤ 1, then p + 2 ≤ h^{0}(C, L ⊗ B^{∨}) ≤ 1 + deg(L ⊗ B^{∨}) − 2a, which leads to a contradiction. If on the other hand h^{1}(C, L ⊗ B^{∨}) ≥ 2, then Cliff(L ⊗ B^{∨}) ≤ a − p − 2 < a, which is impossible. Thus E is a stable vector bundle. □

We are left with showing that the curve C ⊂ S constructed above has maximal Clifford index a. Note that the corresponding statement when p = 1 has been proved in [FO] Theorem 3.6.

Proposition 3.3. We fix integers p ≥ 1, a ≥ 2p + 3 and a K3 surface S with Picard lattice Pic(S) = ℤ · H ⊕ ℤ · C where C^{2} = 4a, H^{2} = 4p + 2 and C · H = 2a + 2p + 1. Then Cliff(C) = a.

Proof. First note that C has Clifford dimension 1, for curves C ⊂ S of higher Clifford dimension have even genus. Observe also that h^{0}(C, O_{C}(1)) = 2p + 3 and h^{1}(C, O_{C}(1)) = 2, hence O_{C}(1) contributes to the Clifford index of C and

Cliff(C) ≤ Cliff(C, O(1)) = C · H − 2(2p + 2) = 2a − 2p − 3 (≥ a).

Assume by contradiction that Cliff(C) < a. According to [GL2], there exists an effective divisor D ≡ mH + nC on S satisfying the conditions

(3)  h^{0}(S, O_{S}(D)) ≥ 2, h^{0}(S, O_{S}(C − D)) ≥ 2, C · D ≤ g − 1,

and with Cliff(O_{C}(D)) = Cliff(C). By [Ma] Lemma 2.2, the dimension h^{0}(C′, O_{C′}(D)) stays constant for all smooth curves C′ ∈ |C| and its value equals h^{0}(S, D). We conclude that Cliff(C) = Cliff(O_{C}(D)) = D · C − 2 dim |D|. We summarize the numerical consequences of the inequalities (3):

\begin{matrix}(i) & md + 2n(g − 1) ≤ g − 1 \\ (ii) & (2p + 1)m^{2} + mnd + n^{2}(g − 1) ≥ 0 \\ (iii) & (4p + 2)m + dn > 2, \end{matrix}

We claim that for any divisor D ⊂ S verifying (i)-(iii), the following inequality holds:

Cliff(O_{C}(D)) = D · C − D^{2} − 2 ≥ H · C − H^{2} − 2 = 2a − 2p − 3 ≥ a.

This will contradict the assumption Cliff(C) < a. The proof proceeds along the lines of Theorem 3 in [F], with the difference that we must also consider curves with D^{2} = 0, that is, elliptic pencils which we now characterize. By direct calculation, we note that there are no (−2)-curves in S. Equality holds in (ii) when m = −n or m = −un with u := 2a/(2p + 1). First, we describe the effective divisors D ⊂ S with self-intersection D^{2} = 0. Consider the case m = −un. If 2p + 1 does not divide a, then D ≡ 2aH − (2p + 1)C and D · C = 2a(2a − 2p − 1) > g − 1, that is, D does not verify condition (i). If a = k(2p + 1), for k ≥ 2, then D ≡ 2kH − C. Notice that D · C = a(4k − 4) + 2k(2p + 1) > 2a for k ≥ 2, that is, D does not satisfies (i). In the the case m = −n, the effective divisor D ≡ C − H, satisfies (i)-(iii) and

Cliff(O_{C}(C − H)) = 2a − 2p − 3 ≥ a.

Case n < 0. From (ii) we have either m < −n or m > −un. In the first case, by using inequality (iii), we obtain 2 < −(4p + 2)n + dn = n(2a − 2p − 1), which is a contradiction since n < 0 and 2a > 2p + 1. Suppose m > −un > 0. Inequality (i) implies that

(−n)\frac{2ad}{2p + 1} < −(g − 1)(2n − 1) = −2a(2n − 1),

then (−n)(d−(4p+2)) < 2p+1 and since d > 4p+2, this yields 2a+2p+1 = d < 6p+3 which contradicts the hypothesis a ≥ 2p + 3.

Case n > 0 . Again, by condition (ii), we have either that m < −un or m > −n. In the first case, using (iii) we write that

0 < (4p + 2)m + dn < n (d − (4p + 2)\frac{2a}{2p + 1}) ,

but one can get easily check that d(2p + 1) < 2a(4p + 2), which yields a contradiction. Suppose now −n < m < 0. By (i) we have 2a(2n − 1) ≤ −md < nd, so n < \frac{2a}{4a−d} = \frac{2a}{2a−2p−1} < 2, since a ≥ 2p + 1. This implies n = 1, therefore for n > 0 there are no divisors D ⊂ S with D^{2} > 0 satisfying the inequalities (i)-(iii).

Case n = 0. From (i), one writes m ≤ \frac{g−1}{d} = \frac{2a}{2a+2p+1} < 1, but this yields to a contradiction since by (iii) it follows that m > 0. The proof is thus finished.

□

4. Curves with prescribed gonality and small rank 2 Clifford index

The equality Cliff_{2}(C) = Cliff(C) is known to be valid for arbitrary k-gonal curves [C] ∈ M_{g,k}^{1} of genus g > (k − 1)(2k − 4). It is thus of some interest to study Mercat’s question for arbitrary curves in a given gonality stratum in M_{g} and decide how sharp is this quadratic bound. We shall construct curves C of unbounded genus and relatively small gonality, carrying a stable rank 2 vector bundle E with h^{0}(C, E) = 4 such that γ(E) < Cliff(C). In order to be able to determine the gonality of C, we realize it as a section of a K3 surface S in P^{4} which is special in the sense of Noether-Lefschetz theory. The pencil computing the gonality is the restriction of an elliptic pencil on the surface. The constraint of having a Picard lattice of rank 2 containing, apart from the hyperplane class, both an elliptic pencil and a curve C of prescribed genus, implies that the discriminant of Pic(S) must be a perfect square. This imposes severe restrictions on the genera for which such a construction could work.

Theorem 4.1. We fix integers a ≥ 3 and b = 4, 5, 6. There exists a smooth curve C ⊂ P^{4} with

deg(C) = 6a + b, g(C) = 3a^{2} + ab + 1 and gonalitygon(C) = ab,

such that C lies on a (2, 3) complete intersection K3 surface. In particular K_{1,1}(C, O_{C}(1)) ≠ 0 and conjecture (M_{2}) fails for C.

Before presenting the proof, we discuss the connection between Theorem 4.1 and conjecture (M_{2}). For C ⊂ S ⊂ P^{4} as above, we construct a vector bundle E with det(E) = O_{C}(1) and h^{0}(C, E) = 4, lying in an exact sequence

0 ⟶ E ⟶ W ⊗ O_{C}(1) ⟶ O_{C}(2) ⟶ 0,

where W ∈ G(3, H^{0}(C, O_{C}(1)) has the property that the quadric Q ∈ Sym^{2}H^{0}(C, O_{C}(1)) induced by S is representable by a tensor in W ⊗ H^{0}(C, L). This construction is a particular procedure of associating vector bundles to non-trivial syzygies, cf. [AN]. The proof that E is stable is standard and proceeds along the lines of e.g. [GMN] Theorem 3.2. Next we compute the Clifford invariant:

γ(E) = 3a + \frac{b}{2} < ab − 2 = Cliff(C),

since b ≥ 4, so not only Cliff_{2}(C) < Cliff(C), but the difference Cliff(C) − Cliff_{2}(C) becomes arbitrarily positive.

Proof. By means of [K] Theorem 6.1, there exist a smooth complete intersection surface S ⊂ P^{4} of type (2, 3) such that Pic(S) = ℤ · H ⊕ ℤ · C, where H^{2} = 6, H · C = d = 6a + b and C^{2} = 2(g − 1) (Note that such a surface exists when d^{2} > 12g, which is satisfied when b ≥ 4). The divisor E := C − aH verifies E^{2} = 0, E · H = b and E · C = ab. In particular E is effective. The class E is primitive, hence it follows that h^{0}(S, E) = h^{0}(C, O_{C}(E)) = 2, where the last equality follows by noting that H^{1}(S, O_{S}(E − C)) = 0 by Kodaira vanishing. Furthermore, h^{1}(C, O_{C}(E)) ≥ 3a^{2} +2, that is, O_{C}(E) contributes to Cliff(C) and then we write that

gon(C) = Cliff(C) + 2 ≤ Cliff(C, O_{C}(E)) + 2 = ab.

We shall show that O_{C}(E) computes the Clifford index of C. First, we classify the primitive effective divisors F ≡ mH + nC ⊂ S having selfintersection zero. By solving the equation (mH + nC)^{2} = 0, where m, n ∈ ℤ, we find the following primitive solutions: E_{1} ≡ (3a + b)H − 3C for b ≠ 6 (respectively E_{2} ≡ (a + 2)H − C for b = 6), and E_{3} = E ≡ C − aH. A simple computation shows that E_{i} · C > ab for i = 1, 2.

Since Cliff(C) ≤ ab − 2 < [\frac{g−1}{2}], the Clifford index of C is computed by a bundle defined on S. Following [GL2], there exists an effective divisor D ≡ mH + nC on S, satisfying the following numerical conditions:

(4) h^{0}(S, D) = h^{0}(C, O_{C}(D)) ≥ 2, h^{0}(S, C − D) ≥ 2, D^{2} ≥ 0 and D · C ≤ g − 1,

and such that

f(D) := Cliff(O_{C}(D)) + 2 = D · C − D^{2} = Cliff(C) + 2.

Furthermore, D can be chosen such that h^{1}(S, D) = 0, cf. [Ma]. To bound f(D) and show that f(D) ≥ ab, we distinguish two cases depending on whether D^{2} > 0 or D^{2} = 0.

By a complete classification of curves with self-intersection zero, we have already seen that for any elliptic pencil |D| satisfying (4), one has f(D) ≥ ab = f(E). We are left with the case D^{2} > 0 and rewrite the inequalities (4):

\begin{matrix}(i) & (6a + b)m + (2n − 1)(3a^{2} + ab) ≤ 0 \\ (ii) & (m + an)(3an + 3m + bn) > 0 \\ (iii) & 6m + (6a + b)n > 2, \end{matrix}

where (ii) comes from the assumption D^{2} > 0 and (iii) from the fact that D · H > 2. Furthermore,

(5)  f(m, n) := D · C − D^{2} = −6m^{2} + m(d − 2nd) + (n − n^{2})(2g − 2).

We prove that for any divisor D satisfying (i) -(iii), the inequality f(m, n) ≥ ab holds, from which we conclude that Cliff(C) = ab − 2. Case n < 0. From (iii) we find that m > 0. Then m < −an or 3m > −(3a + b)n. When m < −an, from (iii) we have that 2 < 6m + dn < −6an + dn = nb < 0, which is a contradiction. Suppose (3a + b)n + 3m > 0. For a fixed n the function f(m, n) reaches its maximum at m_{0} := \frac{d(1−2n)}{12}. So when 3m_{0} + (3a + b)n ≤ 0, we have f(m, n) ≥ f(\frac{(1−2n)(g−1)}{d}, n), since by condition (i), m ≤ \frac{(1−2n)(g−1)}{d}. A simple computation gives that whenever n < 0, one has the inequality:

f (\frac{(1 − 2n)(g − 1)}{d}, n) = (2n^{2} − 2n)(g − 1)\frac{b^{2}}{d^{2}} + (g − 1) (1 − \frac{6(g − 1)}{d^{2}})
≥ 4(g − 1)\frac{b^{2}}{d^{2}} + \frac{g − 1}{d^{2}}(18a^{2} + b^{2} + 6ab) ≥ \frac{3a^{2} + ab}{2} ≥ ab.

Assume now that 3m_{0} +(3a+b)n > 0. Since m ∈ (−\frac{(3a+b)n}{3}, \frac{(1−2n)(g−1)}{d}], we have

f(m, n) ≥ min \{f(−\frac{(3a + b)n}{3}, n), f(\frac{(1 − 2n)(g − 1)}{d}, n)\}.

A direct computation yields

f(−\frac{(3a + b)n}{3}, n) = −n (ab + \frac{b^{2}}{3} ) ≥ ab + \frac{b^{2}}{3} ≥ ab.

Case n > 0. If m ≥ 0 we get a contradiction to (i). Suppose m < 0, then we have either 3m + (3a + b)n < 0, or else m > −an. The first case contradicts (iii), so it does not appear. Suppose m > −an. Reasoning as before, observe that m_{0} < (1 − 2n)(g − 1)/d, where m_{0} is the maximum of f(m, n) for a fixed n, and m takes values in the interval (−an, \frac{(1−2n)(g−1)}{d}]. If −an ≥ m_{0}, then f(m, n) ≥ f(\frac{(1−2n)(g−1)}{d}, n). Since we are assuming −an < \frac{(1−2n)(g−1)}{d}, we have that n < \frac{3a}{b} + 1. We use this bound to directly show, like in the previous case, that f(\frac{(1−2n)(g−1)}{d}, n) ≥ ab. When −an < m_{0} we have that

f(m, n) ≥ min\{f(−an, n), f(\frac{(1 − 2n)(g − 1)}{d}, n)\}.

In this case it is enough to note that f(−an, n) = nab ≥ ab.

Case n = 0. From inequalities (i) and (iii) with n = 0, we have 1 ≤ m ≤ \frac{g−1}{d}. Note that f(m, 0) = −6m^{2} + md reaches its maximum at \frac{d}{12}. So, since \frac{g−1}{d} ≤ \frac{d}{12}, we conclude that f(m, 0) ≥ f(1, 0) = 6a + b − 6. Finally, we observe that 6a + b − 6 ≥ ab if and only if b ≤ 6. This finishes the proof. □

5. The Fourier-Mukai involution on F_{11}

The aim of this section is to provide a detailed proof of Mercat’s conjecture (M_{2}) in one non-trivial case, that of genus 11, and discuss the connection to Mukai’s work [M1], [M3]. We denote as usual by F_{g} the moduli space parametrizing pairs [S, ℓ], where S is a smooth K3 surface and ℓ ∈ Pic(S) is a primitive nef line bundle with ℓ^{2} = 2g − 2. Furthermore, we introduce the parameter space

P_{g} := {[S, C] : S is a smooth K3 surface, C ⊂ S is a smooth curve , [S, O_{S}(C)] ∈ F_{g}}

and denote by π : P_{g} → F_{g} the projection map [S, C] ↦ [S, O_{S}(C)]. If S is a K3 surface, following [M1], we set H̃(S, ℤ) := H^{0}(S, ℤ) ⊕ H^{2}(S, ℤ) ⊕ H^{4}(S, ℤ) and

ÑS(S) := H^{0}(S, ℤ) ⊕ NS(S) ⊕ H^{4}(S, ℤ).

We recall the definition of the Mukai pairing on H̃(S, ℤ):

(α_{0}, α_{2}, α_{4}) · (β_{0}, β_{2}, β_{4}) := α_{2} ∪ β_{2} − α_{4} ∪ β_{0} − α_{0} ∪ β_{4} ∈ H^{4}(S, ℤ) = ℤ.

Let now r, s ≥ 1 be relatively prime integers such that g = 1 + rs. For a polarized K3 surface [S, ℓ] ∈ F_{g} one defines the Fourier-Mukai dual Ŝ := M_{S}(r, ℓ, s), where

M_{S}(r, ℓ, s) = {E : E is an ℓ − stable sheaf on S, rk(E) = r, c_{1}(E) = ℓ, χ(S, E) = r + s}.

Setting v := (r, ℓ, s) ∈ H̃(S, ℤ), there is a Hodge isometry, see [M1] Theorem 1.4:

ψ : H^{2}(M_{S}(r, ℓ, s), ℤ) ⟶^{≅} v^{⊥}/ℤv.

We observe that ℓ̂:= ψ^{−1}((0, ℓ, 2s)) is a nef primitive vector with (ℓ̂)^{2} = 2g − 2, and in this way the pair (Ŝ, ℓ̂) becomes a polarized K3 surface of genus g. The Fourier-Mukai involution is the morphism F M : F_{g} → F_{g} defined by F M([S, ℓ]) := [Ŝ, ℓ̂].

We turn to the case g = 11, when we set r = 2 and s = 5. For a general curve [C] ∈ M_{11}, the Lagrangian Brill-Noether locus

SU_{C}(2, K_{C}, 7) := {E ∈ U_{C}(2, 20) : det(E) = K_{C}, h^{0}(C, E) = 7}

is a smooth K3 surface. The main result of [M3] can be summarized as saying a general [C] ∈ M_{11} lies on a unique K3 surface which moreover can be realized as { \atop SU_{C}(2, K_{C}, 7)}. Furthermore, there is a birational isomorphism

ϕ_{11} : M_{11} −−→ P_{11}, ϕ_{11}([C]) := [{ \atop SU_{C}(2, K_{C}, 7)}, C]

and we set q_{11} := π ∘ ϕ_{11} : M_{11} −−→ F_{11}. On the moduli space M_{11} there exist two distinct irreducible Brill-Noether divisors

M_{11,6}^{1} := {[C] ∈ M_{11} : W_{6}^{1}(C) ≠ ∅} and M_{11,9}^{2} := {[C] ∈ M_{11} : W_{9}^{2}(C) ≠ ∅}.

Via the residuation morphism W_{6}^{1}(C) ∋ L ↦ K_{C} ⊗ L^{∨} ∈ W_{14}^{5}(C), the Hurwitz divisor is the pull-back of a Noether-Lefschetz divisor on F_{11}, that is, M_{11,6}^{1} = q_{11}^{∗}(D_{6}^{1}) where

D_{6}^{1} := {[S, ℓ] ∈ F_{11} : ∃H ∈ Pic(S), H^{2} = 8, H · ℓ = 14}.

Similarly, via the residuation map W_{9}^{2}(C) ∋ L ↦ K_{C} ⊗ L^{∨} ∈ W_{11}^{3}(C), one has the equality of divisors M_{11,9}^{2} = q_{11}^{∗}(D_{9}^{2}), where

D_{9}^{2} := {[S, ℓ] ∈ F_{11} : ∃H ∈ Pic(S), H^{2} = 4, H · ℓ = 11}.

Next we establish Mercat’s conjecture for general curves of genus 11.

Theorem 5.1. The equality Cliff_{2}(C) = Cliff(C) holds for a general curve [C] ∈ M_{11}.

Proof. We fix a curve [C] ∈ M_{11} such that (i) W_{7}^{1}(C) is a smooth curve, (ii) W_{9}^{2}(C) = ∅ (in particular, any Petri general curve will satisfy these conditions) and (iii) the rank 2 Brill-Noether locus SU_{C}(2, K_{C}, 7) is a smooth K3 surface of Picard number 1. As discussed in both [LMN] Proposition 4.5 and [FO] Question 3.5, in order to verify (M_{2}), it suffices to show that C possesses no bundles E ∈ U_{C}(2, 13) with h^{0}(C, E) = 4. Suppose E is such a vector bundle. Then L := det(E) ∈ W_{13}^{4}(C) is a linear series such that the multiplication map ν_{2}(L) : Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2}) is not injective. For each extension class

e ∈ P_{L} := P(Coker ν_{2}(L))^{∨} ⊂ P(H^{0}(C, L^{⊗2}))^{∨} = PExt^{1}(L, K_{C} ⊗ L^{∨}),

one obtains a rank 2 vector bundle F on C sitting in an exact sequence

(6)  0 ⟶ K_{C} ⊗ L^{∨} ⟶ F ⟶ L ⟶ 0,

such that h^{0}(C, F) = h^{0}(C, L) + h^{0}(C, K_{C} ⊗ L^{∨}) = 7. We claim that any non-split vector bundle F with h^{0}(C, F) = 7 and which sits in an exact sequence (6), is semistable. Indeed, let us assume by contradiction that M ⊂ F is a destabilizing line subbundle with deg(M) ≥ 11. Since deg(M) > deg(K_{C} ⊗ L^{∨}), the composite morphism M → L is non-zero, hence we can write that M = L(−D), where D is an effective divisor of degree 1 or 2. Because W_{9}^{2}(C) = ∅, one finds that h^{0}(C, K_{C} ⊗ L^{∨}(D)) = 2 and L must be very ample, that is, h^{0}(C, L(−D)) = h^{0}(C, L) − deg(D). We obtain that

h^{0}(L)+h^{0}(K_{C} ⊗L^{∨}) = h^{0}(F) ≤ h^{0}(M)+h^{0}(K_{C} ⊗M^{∨}) = h^{0}(L)−deg(D)+h^{0}(K_{C} ⊗L^{∨}),

a contradiction. Thus one obtains an induced morphism u : P_{L} → SU_{C}(2, K_{C}, 7). Since SU_{C}(2, K_{C}, 7) is a K3 surface, this also implies that Coker ν_{2}(L) is 2-dimensional, hence P_{L} = P^{1}.

We claim that u is an embedding. Setting A := K_{C} ⊗ L^{∨} ∈ W_{7}^{1}(C), we write the exact sequence 0 → H^{0}(C, O_{C}) → H^{0}(C, F^{∨} ⊗ L) → H^{0}(C, K_{C} ⊗ A^{⊗(−2)}), and note that the last vector space is the kernel of the Petri map H^{0}(C, A) ⊗ H^{0}(C, L) → H^{0}(C, K_{C}), which is injective, hence h^{0}(C, F^{∨} ⊗ L) = 1. This implies that u is an embedding. But this contradicts the fact that Pic SU_{C}(2, K_{C}, 7) = ℤ, in particular SU_{C}(2, K_{C}, 7) contains no (−2)-curves. We conclude that ν_{2}(L) is injective for every L ∈ W_{13}^{4}(C). □

This proof also shows that the failure locus of statement (M_{2}) on M_{11} is equal to the Koszul divisor

𝔖𝔶𝔷_{11,13}^{4} := {[C] ∈ M_{11} : ∃L ∈ W_{13}^{4}(C) such that K_{1,1}(C, L) ≠ 0}.

Suppose now that [C] ∈ 𝔖𝔶𝔷_{11,13}^{4} is a general point corresponding to an embedding

C ↪^{|L|} P^{4} such that C lies on a (2, 3) complete intersection K3 surface S ⊂ P^{4}. Then S = { \atop SU_{C}(2, K_{C}, 7)} and ρ(S) = 2 and furthermore Pic(S) = ℤ · C ⊕ ℤ · H, where H^{2} = 6, C · H = 13 and C^{2} = 20. In particular we note that S contains no (−2)-curves, hence S and Ŝ are not isomorphic.

Let us define the Noether-Lefschetz divisor

D_{13}^{4} := {[S, ℓ] ∈ F_{11} : ∃H ∈ Pic(S), H^{2} = 6, H · ℓ = 13},

therefore 𝔖𝔶𝔷_{11,13}^{4} = q_{11}^{∗}(D_{13}^{4}).

Proposition 5.2. The action of the Fourier-Mukai involution F M : F_{11} → F_{11} on the three distinguished Noether-Lefschetz divisors is described as follows:

(i) F M(D_{6}^{1}) = D_{6}^{1}.

(ii) F M(D_{9}^{2}) = {[S, ℓ] ∈ F_{11} : ∃R ∈ Pic(S) such thatR^{2} = −2, R · ℓ = 1}.
(iii) F M(D_{13}^{4}) = {[S, ℓ] ∈ F_{11} : ∃R ∈ Pic(S) such thatR^{2} = −2, R · ℓ = 3}.

Proof. For [S, ℓ] ∈ F_{11}, we set v := (2, ℓ, 5) ∈ H̃(S, ℤ) and ℓ̂:= (0, ℓ, 10) ∈ H̃(S, ℤ) for the class giving the genus 11 polarization. We describe the lattice ψ(NS(Ŝ)) ⊂ ÑS(S).

In the case of a general point of D_{6}^{1} with lattice NS(S) = ℤ · ℓ ⊕ ℤ · H, by direct calculation we find that ψ(NS(Ŝ)) is generated by the vectors ℓ̂and (2, ℓ + H, 12). Furthermore, (2, ℓ + H, 12)^{2} = 8 and (2, H + ℓ, 12) · ℓ̂ = 14, that is, Pic(Ŝ) ≅ Pic(S), hence D_{6}^{1} is a fixed divisor for the automorphism F M.

A similar reasoning for a general point of the divisor D_{9}^{2} shows that the Neron-Severi groups ψ(NS(Ŝ)) is generated by ℓ̂and (−1, H −ℓ, −2), where (−1, H −ℓ, −2)^{2} = −2 and (−1, H − ℓ, −2) · ℓ̂= 1. In other words, the class (−1, H − ℓ, −2) corresponds to

a line in the embedding Ŝ ↪^{|ℓ̂|} P^{11}. Finally, for a general point of D_{13}^{4} corresponding to a lattice ℤ · ℓ ⊕ ℤ · H, the Picard lattice of the Fourier-Mukai partner is spanned by the vectors ℓ̂and (−1, H−ℓ, −1), where (−1, H−ℓ, −1)^{2} = −2 and (−1, H−ℓ, −1)·ℓ̂= 3. □

Remark 5.3. The fact that the divisor D_{6}^{1} is fixed by the automorphism F M is already observed and proved with geometric methods in [M3] Theorem 3.

Remark 5.4. It is instructive to point out the difference between a general element of D_{13}^{4} and its Fourier-Mukai partner. As a polarized K3 surface, SU_{C}(2, K_{C}, 7) is characterized by the existence of a degree 3 rational curve u(P_{L}) ⊂ SU_{C}(2, K_{C}, 7). On

the other hand, the complete intersection surface S ⊂ P^{4} containing C ↪^{|L|} P^{4}, where L ∈ W_{13}^{4}(C), carries no smooth rational curves. It contains however elliptic curves in the linear system |O_{S}(C − H)|. Thus the involution F M assigns to a K3 surface with a degree 7 elliptic pencil, a K3 surface containing a (−2)-curve. Since S = { \atop SU_{C}(2, K_{C}, 7)}, it also follows that the complete intersection S is a smooth K3 surface, which a priori is not at all obvious.

References

[AN] M. Aprodu and J. Nagel, Non-vanishing for Koszul cohomology of curves, Comment. Math. Helvetici 82 (2007), 617-628.

[CLM] C. Ciliberto, A. Lopez and R. Miranda, Projective degenerations of K3 surfaces, Gaussian maps and Fano threefolds, Inventiones Math. 114 (1993), 641-667.

[DM] R. Donagi and D. Morrison, Linear systems on K3 sections, Journal of Differential Geometry 29 (1989), 49-64.

[F] G. Farkas, Brill-Noether loci and the gonality stratification of M_{g}, J. reine angew. Mathematik 539 (2001), 185-200.

[FO] G. Farkas and A. Ortega, The maximal rank conjecture and rank two Brill-Noether theory, Pure Appl. Math. Quarterly 7 (2011), 1265-1296.

[G] M. Green, Koszul cohomology and the cohomology of projective varieties, Journal of Differential Geometry 19 (1984), 125-167.

[GL1] M. Green and R. Lazarsfeld, The non-vanishing of certain Koszul cohomology groups, Journal of Differential Geometry 19 (1984), 168-170.

[GL2] M. Green and R. Lazarsfeld, Special divisors on curves on a K3 surface, Inventiones Math. 89 (1987), 357-370.

[GMN] I. Grzegorczyk, V. Mercat and P.E. Newstead, Stable bundles of rank 2 with 4 sections, International Journal of Mathematics 22 (2011), 1743-1762.

[HL] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Second Edition, Cambridge University Press 2010.

[K] A. Knutsen, Smooth curves on projective K3 surfaces, Math. Scandinavica 90 (2002), 215-231.

[LN1] H. Lange and P.E. Newstead, Clifford indices for vector bundles on curves, arXiv:0811.4680, in: Affine Flag Manifolds and Principal Bundles (A. H. W. Schmitt editor), Trends in Mathematics, 165-202, Birkhäuser (2010).

[LN2] H. Lange and P.E. Newstead, Further examples of stable bundles of rank 2 with 4 sections, Pure Appl. Math. Quarterly 7 (2011), 1517-1528.

[LN3] H. Lange and P.E. Newstead, Vector bundles of rank 2 computing Clifford indices, arXiv:1012.0469.

[LMN] H. Lange, V. Mercat and P.E. Newstead, On an example of Mukai, arXiv:1003.4007, to appear in Glasgow Math. Journal.

[L] R. Lazarsfeld, Brill-Noether-Petri without degenerations, Journal of Differential Geometry 23 (1986), 299-307.

[Ma] G. Martens, On curves on K3 surfaces, in: Algebraic curves and projective geometry, Trento 1998, Lecture Notes in Mathematics Vol. 1389, Springer (1989), 174-182.

[Me] V. Mercat, Clifford’s theorem and higher rank vector bundles, International Journal of Mathematics 13 (2002), 785-796.

[M1] S. Mukai, On the moduli space of bundles on K3 surfaces, in: Vector bundles on algebraic varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 11 (1987), 341-413.

[M2] S. Mukai, Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proceedings of the National Academy Sciences USA, Vol. 86, 3000-3002, 1989.

[M3] S. Mukai, Curves and K3 surfaces of genus eleven, in: Moduli of vector bundles, Lecture Notes in Pure and Applied Math. Vol. 179, Dekker (1996), 189-197.

[M4] S. Mukai, Curves and symmetric spaces II, Annals of Mathematics 172 (2010), 1539-1558.

[PR] K. Paranjape and S. Ramanan, On the canonical ring of a curve , in: Algebraic Geometry and commutative Algebra (1987), 503-516.

[V1] C. Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math. 168 (1992), 249-272.

[V2] C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compositio Math. 141 (2005), 1163-1190.

Humboldt-Universität zu Berlin, Institut Für Mathematik, Unter den Linden 6 10099 Berlin, Germany

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

Humboldt-Universität zu Berlin, Institut Für Mathematik, Unter den Linden 6 10099 Berlin, Germany

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