Brill-Noether loci

TL;DR

This paper offers a new proof of the non-emptiness of Brill-Noether loci for certain parameters and demonstrates that these loci are distinct in many cases, using a novel approach involving elliptic curves.

Contribution

It introduces a new proof technique for Brill-Noether loci non-emptiness and establishes their distinctness in various codimensions, expanding previous results.

Findings

Proved non-emptiness of Brill-Noether loci for specific codimension ranges.

Showed that loci of the same codimension are distinct in many cases.

Developed a new method to verify non-inclusion of Brill-Noether loci.

Abstract

Brill-Noether loci ${\mathcal M}^r_{g,d}$ are those subsets of the moduli space ${\mathcal M}_g$ determined by the existence of a linear series of degree $d$ and dimension $r$. By looking at non-singular curves in a neighborhood of a special chain of elliptic curves, we provide a new proof of the non-emptiness of the Brill-Noether loci when the expected codimension satisfies $-g+r+1\le \rho(g,r,d)\le 0$ and prove that for a generic point of a component of this locus, the Petri map is onto. As an application, we show that Brill-Noether loci of the same codimension are distinct when the codimension is not too large, substantially generalizing the known result in codimensions 1 and 2. We also provide a new technique for checking that Brill-Noether loci are not included in each other.