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

TL;DR

This paper investigates the stability of rank-3 Lazarsfeld-Mukai bundles on K3 surfaces, providing new results on Brill-Noether theory, including conditions under which certain linear series are induced from the surface.

Contribution

It proves that for large degree and general curves, the Brill-Noether loci behave as expected, and confirms a conjecture relating special linear series to line bundles on the surface.

Findings

Stability of Lazarsfeld-Mukai bundles for large degree

Negative Brill-Noether number implies linear series are induced from the surface

Results support conjecture of Donagi and Morrison

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.