Stability of rank-3 Lazarsfeld-Mukai bundles on K 3 surfaces

Abstract

ABSTRACT. Given an ample line bundle L on a K3 surface S, we study the slope stability with respect to L of rank-3 Lazarsfeld-Mukai bundles associated with complete, base point free nets of type g 2 d on curves C in the linear system |L|. When d is large enough and C is general, we obtain a dimensional statement for the variety W 2 d (C). If the Brill-Noether number is negative, we prove that any g 2 d on any smooth, irreducible curve in |L| is contained in a g r e which is induced from a line bundle on S, thus answering a conjecture of Donagi and Morrison. Applications towards transversality of Brill-Noether loci and higher rank Brill-Noether theory are then discussed.

Lazarsfeld-Mukai reflexive sheaves and their stability

“…The dual of F is called the Lazarsfeld-Mukai bundle on X associated to the pair (C, A). Lelli-Chiesa [17] has studied the (semi)stability of the Lazarsfeld-Mukai bundles on K3surfaces, and similar results have been obtained by us on abelian surfaces [22]. Also, [2] and references therein give a general survey of Lazarsfeld-Mukai bundles with other applications.…”

Mukai's program for curves on a K3 surface

“…A ′ is trigonal). 9 (iii) s = 6 and L 2 = 2 and L · A = 5 (i.e. A ′ is isomorphic to a nonsingular plane quintic).…”

Semistability of Lazarsfeld–Mukai bundles via parabolic structures

“…Proof. This result again follows from Lelli-Chiesa [13,Theorem 4.3], whereby such an F C,A is µ L -stable. (C, A, V ), and the associated parabolic structure on F , say ( F , F E , b 1 , b 2 ).…”

“…(b) Here X is a smooth projective K3 surface and L is an ample line bundle on X such that a general curve C ∈ |L| has genus g, Clifford dimension 1 and maximal gonality k. We have ρ(g, 1, d) > 0. For a general smooth C ∈ |L| , consider the rank 2 dual LM bundle F := F C,A associated with a general complete, base-point free g 1 d , say A on C. By [13], F is µ L -stable. We have an integer m such that N −m N < 1 g−1 .…”

“…Proof. Lelli-Chiesa [13,Theorem 4.3] proves that such an F = F C,A is µ L -stable. The proof follows from Theorem 1.1 by observing that on a K3 surface C · L = (C 2 ) = 2g − 2, which is even.…”