Reduction of Plane Quartics and Cayley Octads

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.