Reduction type of non-hyperelliptic genus 3 curves
Résumé
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.