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Semistable Reduction of Plane Quartics

Max Schwegele

TL;DR

The paper addresses computing stable reductions of non hyperelliptic genus 3 curves by linking abstract stable models with GIT stable plane models for plane quartics. It proves that a GIT stable plane model exists iff the stable reduction is non hyperelliptic, and that the stable model is the unique minimal semistable model dominating the GIT stable model, with a geometrically explicit contraction of 1 tails to cusps. The approach leverages a detailed analysis of the dualizing sheaf, residue theory on singular fibers, and tail twisting to realize contractions and identify the resulting special fiber structure as either a GIT stable plane quartic or a double conic. This bridge provides a transparent geometric framework for computing stable models by first finding a GIT stable plane model and then resolving cuspidal singularities, with implications for explicit algorithms in arithmetic geometry. The work also clarifies the role of hyperellipticity in obstructing the existence of GIT stable models and delineates the core tail combinatorics that control the reduction type.

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.

Semistable Reduction of Plane Quartics

TL;DR

The paper addresses computing stable reductions of non hyperelliptic genus 3 curves by linking abstract stable models with GIT stable plane models for plane quartics. It proves that a GIT stable plane model exists iff the stable reduction is non hyperelliptic, and that the stable model is the unique minimal semistable model dominating the GIT stable model, with a geometrically explicit contraction of 1 tails to cusps. The approach leverages a detailed analysis of the dualizing sheaf, residue theory on singular fibers, and tail twisting to realize contractions and identify the resulting special fiber structure as either a GIT stable plane quartic or a double conic. This bridge provides a transparent geometric framework for computing stable models by first finding a GIT stable plane model and then resolving cuspidal singularities, with implications for explicit algorithms in arithmetic geometry. The work also clarifies the role of hyperellipticity in obstructing the existence of GIT stable models and delineates the core tail combinatorics that control the reduction type.

Abstract

The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus 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 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 -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.

Paper Structure

This paper contains 23 sections, 57 theorems, 117 equations, 3 figures.

Table of Contents

  1. A Note on the Text
  2. Conventions
  3. Statement of Main Theorem
  4. Models of Curves and Semistable Reduction
  5. Non-Hyperelliptic Genus 3 Curves as Plane Quartics
  6. Hyperelliptic Curves
  7. Stable Pointed Curves
  8. Hyperelliptic Stable Curves
  9. Stable Curves of Genus g = 3
  10. Quasi-Hyperelliptic Curves
  11. Proof of Main Theorem
  12. The Dualizing Sheaf
  13. General Definition and Properties
  14. Regular Differentials on Singular Curves
  15. The Dualizing Sheaf of a Normal Model
  16. ...and 8 more sections

Key Result

Theorem 1

Let $X$ be a smooth plane quartic over $K$ whose stable model $\mathcal{X}$ exists. The following conditions are equivalent: Furthermore, when these equivalent conditions hold, the stable model $\mathcal{X}$ is the unique minimal semistable model dominating the GIT-stable model $\mathcal{X}_0$. The corresponding domination morphism induces a morphism on the special fiber that contracts the $1$-t

Figures (3)

  • Figure 1: The $7$ combinatorial types of stable curves of genus $2$.
  • Figure 2: Examples of the properties in Definition \ref{['def:comb_term']}.
  • Figure 3: The decomposition process of a curve of type Z=0e. Separating nodes are marked in blue and separating pairs are marked in red.

Theorems & Definitions (178)

  • Theorem : Main Theorem
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5: Semistable Reduction Theorem
  • Proposition 1.6: liu
  • Example 1.7
  • Remark 1.8
  • Definition 1.9: liu
  • Remark 1.10
  • ...and 168 more