Maximal Brill--Noether loci

Classical Brill–Noether theory studies linear systems on a general curve in the moduli space $\mathcal{M}_g$ of algebraic curves of genus $g$. A refined Brill–Noether theory studies the linear systems on curves with a given Brill–Noether special linear system. As a first step, one would like to understand the stratification of $\mathcal{M}_g$ by Brill–Noether loci, which parameterize curves with a particular projective embedding. In this talk, we'll introduce the Maximal Brill–Noether loci conjecture, and discuss recent progress via the established refined Brill–Noether theory for curves of fixed gonality, limit linear series, and studying unstable Lazarsfeld–Mukai bundles on K3 surfaces. This is joint work with Asher Auel, Andrei Bud, Andreas Leopold Knutsen, and Hannah Larson.