Semistable reduction of smooth quartics

Abstract

We develop a method for computing stable reduction of smooth plane quartics over discretely valued fields, including residue characteristic p=2. The method uses the GIT-semistable plane models constructed in an earlier part of this project, together with an intrinsic description of hyperelliptic stable curves, to characterize when the stable model is obtained from a GIT-stable plane model by resolving its cusps. More precisely, for a smooth non-hyperelliptic curve of genus 3 with semistable reduction, we show that it admits a GIT-stable plane model if and only if its stable reduction is non-hyperelliptic. In that case, the stable model is obtained from the GIT-stable plane model by replacing each cusp by a 1-tail. Together with the companion paper on explicit local stable resolution of cusps, this gives an effective approach to computing stable reduction of smooth plane quartics. The resulting algorithms are implemented in the SageMath package "StabilityFunction".