Cyclic covers of Stable curves and their moduli spaces

TL;DR

This paper investigates the deformation theory of cyclic group actions on stable curves and constructs a parameter space for such curves, extending previous work to non-prime order cyclic groups.

Contribution

It introduces a new parameter space for G-marked stable curves with cyclic groups of non-prime order, expanding the understanding of their moduli.

Findings

Constructed a parameter space for G-marked stable curves of given type

Extended Catanese's work to cyclic groups of non-prime order

Analyzed components of the locus of stable curves with cyclic automorphisms

Abstract

We study the deformation of $G$-marked stable curves in the case where $G$ is a cyclic group, and construct a parameterizing space for $G$-marked stable curves of a given numerical type. This is then used in order to study the components of the locus of stable curves admitting the action of a cyclic group of non prime order $d$, extending work of F. Catanese in the case where $d$ is prime.