Stability of rank-3 Lazarsfeld-Mukai bundles on K3 surfaces

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 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 W2d. If the Brill-Noether number is negative, then we prove that any on any smooth, irreducible curve in L is contained in a 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. © 2013 London Mathematical Society.