Distinguishing 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 discuss recent progress on techniques to distinguish Brill–Noether loci and we'll introduce new methods using K3 surfaces, resolving a conjecture identifying the maximal loci of the stratification. This is based on joint work with Asher Auel, Andrei Bud, Andreas Leopold Knutsen, and Hannah Larson.