Brill--Noether loci in genus $\leq 12$

TL;DR

This paper advances the refined Brill--Noether theory for genus g ≤ 12 curves by analyzing the relative positions of Brill--Noether loci using Lazarsfeld--Mukai bundles on K3 surfaces, providing a comprehensive classification.

Contribution

It introduces a novel approach using Lazarsfeld--Mukai bundles to distinguish Brill--Noether loci and determines their relative positions for all genus g ≤ 12 curves.

Findings

Classified all Brill--Noether loci positions for genus ≤ 12.

Connected loci positions with classical and explicit constructions.

Provided expectations for loci arrangements in general settings.

Abstract

A refined Brill--Noether theory seeks to determine which linear series are admitted by a ``general'' curve in a particular Brill--Noether locus. However, as Brill--Noether loci are not irreducible in general, a coarse answer is given by the relative positions of Brill--Noether loci. Via an analysis of unstable Lazarsfeld--Mukai bundles on K3 surfaces, we distinguish Brill--Noether loci and provide expectations for the relative positions of Brill--Noether loci in general. Together with classical results, the refined Brill--Noether theory for curves of fixed gonality and on Hirzebruch surfaces, and explicit constructions, we identify the relative positions of all Brill--Noether loci in genus $g\leq 12$.