# Algebra Seminar

Date:

10/05/2022 - 2:00pm to 3:00pm

Location:

POT 745

Speaker(s) / Presenter(s):

Richard Haburcak, Dartmouth College

Type of Event (for grouping events):

**Title:**Maximal Brill--Noether Loci via K3 Surfaces

**Abstract:**Classical Brill--Noether (BN) theory concerns the geometry of genus g curves admitting a linear system of degree d and dimension r. When the BN number is negative, the locus of such curves is a proper subvariety of the moduli space of genus g curves, called a special BN locus. In this talk, we summarize a strategy for distinguishing special BN loci by producing curves on polarized K3 surfaces and studying the lifting of line bundles on the curve. This allows us to relate the problem of distinguishing BN loci to lattice theoretic conditions on the Picard group of K3 surfaces. We identify the maximal special BN loci in genus 9-19, 22, and 23. We prove new lifting results for rank 3 linear systems by studying the stability of Lazarsfeld--Mukai bundles that allow us to prove these novel results. This is joint work with Asher Auel.