Maximal Brill--Noether loci via the gonality stratification

TL;DR

This paper investigates the structure of Brill-Noether loci within the gonality stratification of the moduli space of curves, providing new proofs and bounds related to their support and maximality, especially for rank 2 systems.

Contribution

It introduces novel techniques to analyze Brill-Noether loci via gonality stratification, proving maximality results and supporting the Maximal Brill-Noether Loci Conjecture for specific genera.

Findings

Brill-Noether loci with =-1 have distinct support.

For fixed r, lower bounds are established for non-containments in the conjecture.

Brill-Noether loci for rank 2 are maximal when g  28.

Abstract

We study the restriction of Brill-Noether loci to the gonality stratification of the moduli space of curves of fixed genus. As an application, we give new proofs that Brill-Noether loci with $\rho=-1$ have distinct support, and for fixed $r$ give lower bounds on when one direction of the non-containments of the Maximal Brill-Noether Loci Conjecture hold for Brill-Noether loci of rank $r$ linear systems. Using these techniques, we also show that Brill-Noether loci corresponding to rank $2$ linear systems are maximal as soon as $g\ge 28$ and prove the Maximal Brill-Noether Loci Conjecture for $g=20$.