Higher-rank Brill-Noether loci on nodal reducible curves

TL;DR

This paper extends Brill-Noether theory to higher-rank sheaves on polarized nodal reducible curves, establishing their geometric properties and relations to BGN extensions, and identifying irreducible components of expected dimension.

Contribution

It introduces a framework for studying Brill-Noether loci of stable sheaves on reducible curves, revealing their structure and connection to BGN extensions.

Findings

Brill-Noether loci are closely related to BGN extensions.

Irreducible components of the loci have the expected dimension.

The work generalizes smooth curve results to reducible nodal curves.

Abstract

In this paper we deal with Brill-Noether theory for higher-rank sheaves on a polarized nodal reducible curve $(C,\underline{w})$ following the ideas of [arXiv:alg-geom/9511003v1]. We study the Brill-Noether loci of $\underline{w}$-stable depth one sheaves on $C$ having rank $r$ on all irreducible components and having small slope. In analogy with what happens in the smooth case, we prove that these loci are closely related to BGN extensions. Moreover, we produce irreducible components of the expected dimension for these Brill-Noether loci.