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Maximal Brill-Noether loci

Maximal Brill-Noether loci

Brill-Noether theory answers the question of whether a general curve of genus g admits a linear system of rank r and degree d. A refined Brill-Noether theory hopes to answer the question of whether a general such curve also admits a linear system of rank r’ and degree d'. In other words, we want to know about the relative position between Brill-Noether loci in the moduli space of curves of genus g. I'll explain a strategy for distinguishing Brill-Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill-Noether loci with respect to containment. Via an analysis of the stability of Lazarsfeld-Mukai bundles, we obtain new lifting results for linear systems of rank 3. Together with Brill-Noether theory on gonality strara, this suffices to prove the maximal Brill-Noether loci conjecture in genus up to 23. This is joint work with Richard Haburcak and with Hannah Larson.

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## Department of Mathematics and Statistics