# Birational geometry of moduli of curves with an $S_3$-cover

## TL;DR

This paper proves that the moduli space of curves with an $S_3$-cover is of general type for odd genus $g \,\geq\, 13$, and develops tools applicable to $G$-covers for any finite group.

## Contribution

It establishes the general type of the moduli space of $S_3$-covers for high odd genus and introduces versatile tools for studying $G$-cover moduli spaces.

## Key findings

- The moduli space $\\mathcal R_{g,S_3}^{S_3}$ is of general type for $g \geq 13$ and odd.
- Developed new methods for analyzing the birational geometry of $G$-cover moduli spaces.
- Provided a framework applicable to arbitrary finite groups $G$.

## Abstract

We consider the space $\mathcal R_{g,S_3}^{S_3}$ of curves with a connected $S_3$-cover, proving that for any odd genus $g\geq 13$ this moduli is of general type. Furthermore we develop a set of tools that are essential in approaching the case of $G$-covers for any finite group $G$.