Reduction of Plane Quartics and Cayley Octads

TL;DR

This paper introduces p-adic criteria based on Cayley octads to determine the stable reduction types of plane quartics over local fields, including hyperelliptic cases, supported by explicit examples and numerical illustrations.

Contribution

It provides a conjectural framework linking Cayley octads to stable reduction types of plane quartics, with practical criteria and explicit families for all types.

Findings

Criteria efficiently classify stable reduction types

All possible stable types are realized by explicit families

Numerical examples demonstrate practical application

Abstract

We give a conjectural characterisation of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This results in p-adic criteria that efficiently give the stable reduction type amongst the 42 possible types, and whether the reduction is hyperelliptic or not. These criteria are in the vein of the machinery of "cluster pictures" for hyperelliptic curves. We also construct explicit families of quartic curves that realise all possible stable types, against which we test these criteria. We give numerical examples that illustrate how to use these criteria in practice.