Semistable Reduction of Plane Quartics

TL;DR

This paper links the stable reduction of non-hyperelliptic genus 3 curves, represented as plane quartics, with GIT stability, providing a geometric method to compute stable models via GIT-stable models and cusp resolution.

Contribution

It establishes a precise criterion connecting GIT stability of plane quartics with their stable reduction, and describes an explicit geometric process for obtaining stable models.

Findings

GIT-stable plane quartics correspond to non-hyperelliptic stable reductions.

The stable model is the minimal semistable model dominating the GIT-stable model.

The domination morphism contracts 1-tails to cusps and is an immersion elsewhere.

Abstract

The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus $g \geq 2$ over a discretely valued field admits a unique stable model after a finite field extension. Computing this model is a central problem in arithmetic geometry. For non-hyperelliptic genus $3$ curves, which are canonically embedded as plane quartics, methods like admissible reduction become challenging in small residue characteristics. This thesis establishes a precise connection between the abstractly defined stable model and computationally accessible GIT-stable plane models. We prove that a GIT-stable plane model of a smooth plane quartic exists if and only if its stable reduction is non-hyperelliptic. When this condition holds, we show that the stable model is the unique minimal semistable model that dominates the GIT-stable model. The corresponding domination morphism is geometrically explicit: it contracts the $1$-tails of the stable reduction to cusps on the special fiber of the GIT-stable model and is an immersion elsewhere. This result provides a geometric framework for computing the stable model by first finding a GIT-stable model and then resolving its cuspidal singularities.