Brill-Noether via wall-crossing

Abstract. Wall-crossing in the derived categories of K3 surfaces naturally leads one to reprove (and improve) Lazarsfeld's result that curves on K3 surfaces are Brill-Noether general. I will describe joint work with Chunyi Li that solves the same question for abelian surfaces. We give a precise criterion for non-emptiness Brill-Noether locus of curves in the primitive linear system, and show that it is always of expected dimension, and usually irreducible. In particular, this gives many examples of Brill-Noether loci over a family of curves with negative Brill-Noether number; our result completes prior work by Knutsen, Lelli-Chiesa and Mongardi.