Classifying families of superelliptic curves

TL;DR

This paper initiates a comprehensive classification of superelliptic curves of genus up to 48 based on their automorphism groups, detailing parametric equations, signatures, and family dimensions.

Contribution

It provides the first systematic classification of superelliptic curves by automorphism group for genus up to 48, including parametric equations and family structures.

Findings

Classification of superelliptic curves by automorphism group

Parametric equations for each family

Analysis of family inclusion relations

Abstract

This paper is the first version of a project of classifying all superelliptic curves of genus $g \leq 48$ according to their automorphism group. We determine the parametric equations in each family, the corresponding signature of the group, the dimension of the family, and the inclussion among such families. At a later stage it will be determined the decomposition of the Jacobians and each locus in the moduli spaces of curves.