Brill-Noether locus of rank 1 and degree g-1 on a nodal curve

TL;DR

This paper characterizes the structure of the Brill-Noether locus of line bundles with degree g-1 on a nodal reducible curve, revealing how multidegree variations relate to the locus's components.

Contribution

It provides an explicit description of the irreducible components of the Brill-Noether locus for semistable multidegrees on nodal curves.

Findings

Explicit description of irreducible components of $W_{ ext{d}}(C)$.

Isomorphism of loci components under multidegree twisters.

Application to curves with no rational components.

Abstract

In this paper we consider the Brill-Noether locus $W_{\underline d}(C)$ of line bundles of multidegree $\underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $g\geq2$. We give an explicit description of the irreducible components of $W_{\underline d}(C)$ for a semistable multidegre $\underline d$. As a consequence we show that, if two semistable multidegrees of total degre $g-1$ on a curve with no rational components differ by a twister, then the respective Brill-Noether loci have isomorphic components.