Harvard–MIT Algebraic Geometry Seminar: Maximal Brill-Noether loci

SEMINARS, HARVARD-MIT ALGEBRAIC GEOMETRY

Speaker:

Asher Auel - Dartmouth

Brill-Noether theory answers the question of whether a general curve of genus $g$ admits $g^r_d$, a linear system of rank $r$ and degree $d$. A refined Brill-Noether theory hopes to answer the question of whether a "general curve with a $g^r_d$" admits a $g^{r'}_{d'}$. 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 which suffice to prove the maximal Brill-Noether loci conjecture in genus 9-19, 22, and 23. This is joint work with Richard Haburcak.