Covers of stacky curves and limits of plane quintics

TL;DR

This paper develops a new compactification method for covers of stacky curves, applies it to tetragonal curves on Hirzebruch surfaces, and describes boundary divisors of plane quintic curves in moduli space.

Contribution

It introduces a well-behaved compactification of finite covers of stacky curves and applies it to specific geometric problems involving plane quintics.

Findings

Constructed a compactification using admissible cover degenerations.

Explicitly described boundary divisors in the moduli space of genus 6.

Extended the understanding of degenerations of plane quintic curves.

Abstract

We construct a well-behaved compactification of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure in $\overline{M}_6$ of the locus of smooth plane quintic curves.