# A note on the stratification by automorphisms of smooth plane curves of genus 6

## TL;DR

This paper classifies the automorphism group strata of smooth plane curves of genus 6, revealing new phenomena such as a non-zero-dimensional final stratum and providing explicit families for most cases.

## Contribution

It offers a detailed classification of automorphism group strata for genus 6 plane curves, including the discovery of a unique non-zero-dimensional final stratum and explicit descriptions of most families.

## Key findings

- Existence of a non-zero-dimensional final stratum of plane curves.
- Explicit representative families for all but one automorphism group stratum.
- Application of a dimension 2 version of Luroth's theorem to prove existence.

## Abstract

In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p > 13$. We start with a classification already obtained by the first author and we use standard techniques. Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non $0$-dimensional final stratum of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give a explanation for it. We explicitly describe representative families for all strata, except for the stratum with automorphism group $\mathbb{Z}/5\mathbb{Z}$. Here we find the second difference with the lower genus cases where the previous techniques do not fully work. Fortunately, we are still able to prove the existence of such family by applying a version of Luroth's theorem in dimension $2$.