Restricted Lazarsfeld–Mukai bundles and canonical curves
Marian Aprodu, Gavril Farkas, Angela Ortega
Development of Moduli Theory — Kyoto 2013
Dedicated to Professor Shigeru Mukai on his sixtieth birthday, with admiration For a K3 surface S, a smooth curve C ⊂ S and a globally generated linear series A ∈ W r d (C) with h 0 (C, A) = r + 1, the Lazarsfeld-Mukai vector bundle E C,A is defined via the following elementary modification on S The bundles E C,A have been introduced more or less simultaneously in the 80's by Lazarsfeld [L1] and Mukai [M1] and have acquired quite some prominence in algebraic geometry. On one hand, they have
... used to show that curves on general K3 surfaces verify the Brill-Noether theorem [L1], and this is still the only class of smooth curves known to be general in the sense of Brill-Noether theory in every genus. When ρ(g, r, d) = 0, the vector bundle E C,A is rigid and plays a key role in the classification of Fano varieties of coindex 3. For g = 7, 8, 9, the corresponding Lazarsfeld-Mukai bundle has been used to coordinatize the moduli space of curves of genus g , thus giving rise to a new and more concrete model of M g , see [M2], [M3], [M4]. Furthermore, Lazarsfeld-Mukai bundles of rank two have led to a characterization of the locus in M g of curves lying on K3 surfaces in terms of existence of linear series with unexpected syzygies [F], [V]. For a recent survey on this circle of ideas, see [A]. Recently, Lazarsfeld-Mukai bundles have proven to be effective in shedding some light on an interesting conjecture of Mercat in Brill-Noether theory, see [FO1], [FO2], [LMN]. Recall that the Clifford index of a semistable vector bundle E ∈ U C (n, d) on a smooth curve C of genus g is defined as Then the higher Clifford indices of the curve C are defined as the quantities For any line bundle L on C such that h i (C, L) ≥ 2 for i = 0, 1, that is, contributing to the Clifford index Cliff(C), by computing the invariants of the strictly semistable vector bundle E := L ⊗n , one finds Cliff n (C) ≤ Cliff(C). Mercat [Me1] predicted that for any smooth curve C of genus g, the following equality ). should hold. Counterexamples to (M 2 ) have been found on curves lying on K3 surfaces that are special in Noether-Lefschetz sense, see [FO1], [FO2] and [LN2]. However, (M 2 ) is expected to hold for a general curve of genus g, and in fact even for a curve C lying on 1 2 M. APRODU, G. FARKAS, AND A. ORTEGA a K3 surface S such that Pic(S) = Z·C. For instance, it is known that (M 2 ) holds on M 11 outside a certain Koszul divisor (which also admits a Noether-Lefschetz realization), see [FO2] Theorem 1.3. It is also known that (M 2 ) holds generically on M g for g ≤ 16, see [FO1]. It has been proved in [LMN] that rank three restricted Lazarsfeld-Mukai bundles invalidate statement (M 3 ) in genus 9 and 11 respectively, that is, Mercat's conjecture in rank three fails generically on M 9 and M 11 respectively. This was then extended in [FO2] Theorem 1.4, to show that on a K3 surface S with Pic( 3 ⌋, the restriction to C of the Lazarsfeld-Mukai bundle E C,A is stable and has Clifford index strictly less than ⌊ g−1 2 ⌋, in particular, statement (M 3 ) fails for the curve C. For further background on this problem, we also refer to [Me1], [LN1] and [GMN]. The restricted Lazarsfeld-Mukai bundle E| C := E C,A ⊗ O C sits in the following exact sequence One then shows [V], [FO2] that the sequence (2) is exact on global sections, that is, , it becomes clear that for sufficiently high g, one has γ(E| C ) < Cliff(C), that is, E| C , when semistable, provides a counterexample to Mercat's conjecture (M r+1 ). We prove the following result, extending to rank 4 a picture studied in smaller ranks in the papers [M1], [V], respectively [FO2]. Theorem 0.1. Let S be a K3 surface with Pic(S) = Z · L, where L 2 = 2g − 2 and write g = 4i − 4 + ρ and d = 3i + ρ, with ρ ≥ 0 and i ≥ 6. Then for a general curve C ∈ |L| and a globally generated linear series A ∈ W 3 d (C) with h 0 (C, A) = 4, the restriction to C of the Lazarsfeld-Mukai bundle E C,A is stable. Note that in Theorem 0.1, dim W 3 d (C) = ρ. The rank 3 version of this result was proved in [FO2] . We record the following consequence of Theorem 0.1: The curves C appearing in Corollary 0.2 are Brill-Noether general, that is, they satisfy Cliff(C) = ⌊ g−1 2 ⌋, see [L1]. We also show that under mild restrictions, on a very general K3 surface, the extension (2) is non-trivial and the restricted Lazarsfeld-Mukai bundle E| C is simple (see Theorem 1.3). We expect that the bundle E| C remains stable also for RESTRICTED LAZARSFELD-MUKAI BUNDLES AND CANONICAL CURVES 3 higher ranks r + 1 = h 0 (C, A), at least when Pic(S) = Z · C. However, our method of proof based on the Bogomolov inequality, seem not to extend easily for r ≥ 4. The second topic we discuss in this paper concerns the connection between normal bundles of canonical curves and Mercat's conjecture. For a smooth canonically embedded curve C ⊂ P g−1 of genus g, we consider the normal bundle N C := N C/P g−1 , and then we define the twist of the conormal bundle E := N ∨ C ⊗ K ⊗2 C . By direct calculation det(E) = K ⊗(g−5) C and rk(E) = g − 2.