On automorphisms of algebraic curves

TL;DR

This paper explores methods to determine the automorphism groups of algebraic curves of genus at least 2 over algebraically closed fields, providing explicit equations for these families when feasible.

Contribution

It introduces techniques for classifying automorphism groups of algebraic curves and supplies explicit equations for the associated families when possible.

Findings

Methods for determining automorphism groups are developed.

Explicit equations for families of curves with given automorphism groups are provided.

The approach applies to curves over fields of any characteristic.

Abstract

An irreducible, algebraic curve $\mathcal X_g$ of genus $g\geq 2$ defined over an algebraically closed field $k$ of characteristic $\mbox{char } \, k = p \geq 0$, has finite automorphism group $\mbox{Aut} (\mathcal X_g)$. In this paper we describe methods of determining the list of groups $\mbox{Aut} (\mathcal X_g)$ for a fixed $g\geq 2$. Moreover, equations of the corresponding families of curves are given when possible.