On the gonality of stable curves

TL;DR

This paper investigates the gonality of stable curves using admissible covers, establishing bounds and relations to components of Hurwitz schemes, thus advancing understanding of curve degenerations.

Contribution

It introduces new bounds for gonality of stable curves and constructs admissible covers that relate to boundary components of Hurwitz schemes.

Findings

Established bounds for gonality of stable curves.

Constructed admissible covers linking to Hurwitz scheme boundaries.

Identified components of boundary in Hurwitz schemes related to these covers.

Abstract

In this paper we use admissible covers to investigate the gonality of a stable curve $C$ over $\mathbb{C}$. If $C$ is irreducible, we compare its gonality to that of its normalization. If $C$ is reducible, we compare its gonality to that of its irreducible components. In both cases we obtain lower and upper bounds. Furthermore, we show that four admissible covers constructed give rise to generically injective maps between Hurwitz schemes. We show that the closures of the images of three of these maps are components of the boundary of the target Hurwitz schemes, and the closure of the image of the remaining map is a component of a certain codimension-1 subscheme of the boundary of the target Hurwitz scheme.