# Stable models of plane quartics with hyperelliptic reduction

## TL;DR

This paper presents an algorithm to compute stable models of smooth plane quartics with hyperelliptic reduction over discrete valuation fields, utilizing valuations of theta constants for classification.

## Contribution

It introduces a new criterion based on theta constants for determining reduction type and provides an algorithm for approximating stable models in hyperelliptic cases.

## Key findings

- Algorithm successfully computes stable models in hyperelliptic reduction cases.
- New valuation criterion for reduction type based on theta constants.
- Examples demonstrate practical computation of models.

## Abstract

Let C/K: F = 0 be a smooth plane quartic over a complete discrete valuation field K. In a previous paper the authors togetehr with Q. Liu give various characterizations of the reduction (i.e. non-hyperelliptic genus 3 curve, hyperelliptic genus 3 curve or bad) of the stable model of C: in terms of the existence of a special plane quartic model and in terms of the valuations of the Dixmier-Ohno invariants of C. The last one gives in particular an easy computable criterion for the reduction type. However, it does not produce a stable model, even in the case of good reduction. In this paper we give an algorithm to obtain (an approximation of) the stable model when the reduction of the latter is hyperelliptic and the characteristic of the residue field is not 2. This is based on a new criterion giving the reduction type in terms of the valuations of the theta constants of C. Some examples of the computation of these models are given.