A note on the stratification by automorphisms of smooth plane curves of genus 6
Volume 159 / 2020
Abstract
We give a so-called representative classification for the strata by automorphism groups of smooth $\overline {k}$-plane curves of genus $6$, where $\overline {k}$ is a fixed algebraic closure of a perfect field $k$ of characteristic $p=0$ or $p \gt 13$. We start with a classification already obtained by Badr and Bars (2016) and we mimic the techniques of Lercier et al. (2014) and Lorenzo García (2015).
Interestingly, on the way to get these families for the different strata, we find two remarkable phenomena that have not appeared before. One is the existence of a non-zero-dimensional final stratum of plane curves. At first sight it may sound weird, but we will see that this is a normal situation for higher degrees and we will give an explanation for it.
The second phenomenon occurs when explicitly describing representative families for all strata. We do this for all strata, except for the stratum with automorphism group $\mathbb {Z}/5\mathbb {Z}$. Here we find the second difference with the lower degree cases where the previous techniques do not fully work. Fortunately, we are still able to prove the existence of such a family by applying a version of Lüroth’s theorem in dimension $2$.
