On the rank of general linear series on stable curves

TL;DR

This paper investigates the dimension of special line bundle loci on stable curves, providing characterizations for when these loci form Theta divisors or have expected dimensions, thus advancing understanding of linear series on stable curves.

Contribution

It offers new criteria for the dimension and structure of special loci of line bundles on stable curves, including characterizations of semistability and conditions for expected dimension.

Findings

Effective locus forms a Theta divisor for degree g-1.

Loci are empty or have expected dimension for degrees g-2 and g.

Special locus has codimension at least 2 in remaining cases.

Abstract

We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree $d = g - 1$, we characterize when the effective locus gives a Theta divisor. In case of degree $g - 2$ and $g$, we show that the locus is either empty or has the expected dimension. This leads to a new characterization of semistability in these degrees. In the remaining cases, we show that the special locus has codimension at least $2$. If the multidegree in addition is non-negative on each irreducible component of the curve, we show that the special locus contains an irrreducible component of expected dimension.