# 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.

## Key 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.