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

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 d 2 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 d 2 ( C ) . If the Brill–Noether number is negative, then we prove that any g d 2 on any smooth, irreducible curve in | L| is contained in a g e r 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.