Reduction type of smooth quartics

TL;DR

This paper characterizes the reduction types of smooth quartic curves over discrete valuation fields using algebraic invariants and geometric invariant theory, providing explicit criteria and models, especially for Picard curves.

Contribution

It introduces a new characterization of reduction types of smooth quartics via algebraic invariants and models, extending geometric invariant theory results over arbitrary rings.

Findings

Characterization of reduction types in terms of algebraic invariants.

Existence of homogeneous systems of parameters over discrete valuation rings.

Explicit criteria for reduction types of Picard curves.

Abstract

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. 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 certain algebraic invariants of $C$ when the characteristic of the residue field is not $2,\,3,\,5$ or $7$. On the way, we gather several results of general interest 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 $R$. We exhibit explicit ones for ternary quartic forms under the action of $\textrm{SL}_{3,R}$ depending only on the characteristic $p$ of the residue field. We illustrate our results with the case of Picard curves for which we give simple criteria for the type of reduction.