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

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 $2$.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01618727
Contributeur : Marie-Annick Guillemer <>
Soumis le : mercredi 18 octobre 2017 - 13:51:19
Dernière modification le : lundi 25 juin 2018 - 14:18:15

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  • HAL Id : hal-01618727, version 1
  • ARXIV : 1701.06065

Citation

Eslam Badr, Elisa Lorenzo García. A note on the stratification by automorphisms of smooth plane curves of genus 6. 2017. 〈hal-01618727〉

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