Explicit Moduli of Superelliptic Curves with Level Structure

TL;DR

This paper explicitly constructs the moduli space of trigonal superelliptic curves with level 3 structure, providing formulas for its components and generalizing known results for hyperelliptic curves.

Contribution

It offers an explicit construction of the moduli space for these curves using point sets on the projective line, extending previous work on hyperelliptic cases.

Findings

Closed formula for the number of connected components

Explicit construction via point sets on the projective line

Generalization of hyperelliptic moduli space results

Abstract

In this article we give an explicit construction of the moduli space of trigonal superelliptic curves with level 3 structure. The construction is given in terms of point sets on the projective line and leads to a closed formula for the number of connected (and irreducible) components of the moduli space. The results of the article generalise the description of the moduli space of hyperelliptic curves with level 2 structure, due to Dolgachev and Ortland, Runge and Tsuyumine.