The locus of curves with prescribed automorphism group

TL;DR

This paper classifies and describes the loci of genus g curves with prescribed automorphism groups, providing explicit equations for genus 3 and classifications for groups up to genus 10.

Contribution

It offers a complete classification of loci of genus 3 curves with automorphism groups and classifications for larger groups up to genus 10, including explicit equations.

Findings

Complete classification for genus 3 curves with automorphism groups.

Explicit equations for genus 3 curves with prescribed automorphism groups.

Classification of loci for large groups up to genus 10.

Abstract

Let G be a finite group, and $g \geq 2$. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g=3 (including equations for the corresponding curves), and for $g \leq 10$ we classify those loci corresponding to "large" G.