2025/11/19 by Max Schwegele, Schwegele, Max
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.2511.15858
openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28
The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus g ≥ 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.