Reduction type of non-hyperelliptic genus 3 curves
RITZENTHALER, Christophe
Institut de Mathématiques de Marseille [I2M]
Institut de Recherche Mathématique de Rennes [IRMAR]
< Réduire
Institut de Mathématiques de Marseille [I2M]
Institut de Recherche Mathématique de Rennes [IRMAR]
Langue
en
Article de revue
Ce document a été publié dans
Algebra & Number Theory. 2021
Mathematical Sciences Publishers
Date de soutenance
2021Résumé en anglais
Let $C/K$ be a smooth plane quartic over a discrete valuation field. We give a characterization of the type of reduction (ie smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of ...Lire la suite >
Let $C/K$ be a smooth plane quartic over a discrete valuation field. We give a characterization of the type of reduction (ie smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model and, over $\bar{K}$, in terms of the valuations of the Dixmier-Ohno invariants of $C$ (if the characteristic of the residue field is not $2,\,3,\,5$ or $7$). When the reduction is (potentially) good we also provide an equation for the special fiber of a generic quartic. On the way, we gather general results on geometric invariant theory over an arbitrary ring $R$ in the spirit of {Seshadri 1977}. For instance when $R$ is a discrete valuation ring, we show the existence of a homogeneous system of parameters over $R$ and we exhibit precise ones for ternary quartic forms under the action of $SL_{3,R}$ depending only on the characteristic of the residue field.< Réduire
Mots clés en anglais
invariants
hyperelliptic
valuation
smooth plane quartic
reduction
Project ANR
Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020
Origine
Importé de halUnités de recherche