Maximal Brill-Noether loci via degenerations and double covers

TL;DR

This paper employs degenerations and double covers to analyze Brill--Noether loci, establishing new non-containment results and confirming expected dimensions for certain loci, advancing understanding in algebraic geometry.

Contribution

It introduces novel degeneration techniques and applies them to prove non-containments and dimension results for Brill--Noether loci, including cases involving double covers.

Findings

Proved that closures of certain Brill--Noether loci contain products of small codimension loci.

Established that all expected non-containments hold for maximal Brill--Noether loci.

Provided new proofs for the expected dimension of Brill--Noether loci with small codimension.

Abstract

Using limit linear series on chains of curves, we show that closures of certain Brill--Noether loci contain a product of pointed Brill--Noether loci of small codimension. As a result, we obtain new non-containments of Brill--Noether loci, in particular that all dimensionally expected non-containments hold for expected maximal Brill--Noether loci. Using these degenerations, we also give a new proof that Brill--Noether loci with expected codimension $-\rho\leq \lceil g/2\rceil$ have a component of the expected dimension. Additionally, we obtain new non-containments of Brill--Noether loci by considering the locus of the source curves of unramified double covers.