The stratification by automorphism groups of smooth plane sextic curves

TL;DR

This paper classifies the automorphism groups of smooth plane sextic curves over algebraically closed fields of characteristic zero or greater than 21, providing explicit families for each group to describe their geometric structure.

Contribution

It provides a complete list of automorphism groups for smooth plane sextic curves and constructs explicit parameterized families for each group, detailing their geometric stratification.

Findings

List of automorphism groups for smooth plane sextic curves.

Explicit parameterized families for each automorphism group.

Description of the geometric stratification of the moduli space.

Abstract

We obtain the list of automorphism groups for smooth plane sextic curves over an algebraically closed field K of characteristic p=0 or p>21. Moreover, we assign to each group a geometrically complete family over K describing its corresponding stratum, that is, a generic defining polynomial equation with parameters such that any curve in the stratum is K-isomorphic to a non-singular plane model obtained by specializing the values of those parameters over K.