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 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.