The irreducible components of the moduli space of dihedral covers of algebraic curves

TL;DR

This paper introduces a new invariant to classify the irreducible components of the moduli space of dihedral covers of algebraic curves, extending existing genus stabilization results to ramified cases.

Contribution

The authors develop a novel invariant for dihedral group actions on curves, enabling classification of moduli space components and extending genus stabilization results.

Findings

Classified components of the moduli space of dihedral covers.

Extended genus stabilization results to ramified covers.

Clarified correspondence between components and loci in the moduli space.

Abstract

In this paper we introduce a new invariant for the action of a finite group $G$ on a compact complex curve of genus $g$. With the aid of this invariant we achieve the classification of the components of the moduli space of curves with an effective action by the dihedral group $D_n$. This invariant has been used in the meanwhile by the authors in order to extend the genus stabilization result of Livingston and Dunfield and Thurston to the ramified case. This new version contains an appendix clarifying the correspondence between the above components and the image loci in the moduli space M_g (classifying when two such components have the same image).