Explicit moduli of superelliptic curves with level structure

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.