Maximal Brill-Noether loci via K3 surfaces

TL;DR

This paper introduces a strategy to identify maximal Brill-Noether loci in the moduli space of curves by analyzing linear systems on curves embedded in polarized K3 surfaces, leading to new results and conjectures.

Contribution

It develops a novel approach using K3 surfaces and Lazarsfeld-Mukai bundles to prove the maximal Brill-Noether loci conjecture for specific genera.

Findings

Proved the maximal Brill-Noether loci conjecture for genus 9-19, 22, and 23.

Established new lifting results for linear systems of rank 3.

Provided a framework for distinguishing Brill-Noether loci via K3 surface techniques.

Abstract

We explain a strategy for distinguishing Brill-Noether loci in the moduli space of curves 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.