Semistable Reduction of Plane Quartics

Technical Summary: Semistable Reduction of Plane Quartics

Problem Statement
The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus $g \ge 2$ over a discretely valued field $K$ admits a unique stable model after a finite field extension. While the existence of this model is guaranteed, computing it explicitly remains a central challenge in arithmetic geometry. For non-hyperelliptic curves of genus 3, which are canonically embedded as smooth plane quartics, standard methods such as admissible reduction (relying on the curve being a cover of $\mathbb{P}^1$) face significant difficulties in small residue characteristics, particularly when the characteristic $p=2$. In these cases, the monodromy group order may be divisible by $p$, rendering admissible reduction techniques inapplicable or inefficient. The thesis addresses the specific problem of computing the stable reduction of non-hyperelliptic genus 3 curves in a manner that is uniform across all characteristics.

Methodology
The author establishes a bridge between the abstract theory of stable reduction (Deligne-Mumford stability) and the algebraic framework of Geometric Invariant Theory (GIT). The core methodology involves analyzing the relationship between the abstract stable model $\mathcal{X}$ and concrete plane models $\mathcal{X}_0 \subset \mathbb{P}^2_{\mathcal{O}_K}$.

1. Classification of Stable Curves: The work begins by rigorously defining and classifying hyperelliptic stable curves of genus 3. This involves extending the notion of the hyperelliptic involution to singular curves, distinguishing between "combinatorially hyperelliptic" types (determined solely by the dual graph) and those dependent on geometric moduli or field characteristic. A key distinction is made between "hyperelliptic" curves (admitting a degree-2 map to a rational tree) and "quasi-hyperelliptic" curves (where the canonical map is not birational), utilizing results from Catanese and Rosenlicht.
2. Dualizing Sheaf and Contraction: The proof relies heavily on the properties of the dualizing sheaf $\omega_{\mathcal{X}/\mathcal{O}_K}$. The author constructs a contraction morphism from the stable model to a plane model by "twisting" the dualizing sheaf. Specifically, the dualizing sheaf is modified by adding divisors corresponding to the "1-tails" (components of arithmetic genus 1 attached at a single node) of the stable reduction.
3. GIT-Stability Analysis: The paper analyzes the special fibers of plane quartic models. A model is defined as GIT-stable if its special fiber has only ordinary nodes ($A_1$) and ordinary cusps ($A_2$) as singularities. The author investigates the conditions under which a GIT-stable model exists and how its geometry relates to the underlying stable reduction.

Key Contributions and Results
The central result is the Main Theorem, which establishes a precise equivalence between the geometric nature of the stable reduction and the existence of a specific type of plane model:

• Equivalence: A smooth plane quartic $X$ admits a GIT-stable plane model if and only if its stable reduction $\mathcal{X}$ is non-hyperelliptic.
• Domination and Morphism: When this condition holds, the stable model $\mathcal{X}$ is the unique minimal semistable model that dominates the GIT-stable model $\mathcal{X}_0$. The domination morphism $\phi: \mathcal{X} \to \mathcal{X}_0$ induces a morphism on the special fiber that:
  • Contracts the 1-tails of the stable reduction to the cusps on the special fiber of the GIT-stable model.
  • Acts as an immersion elsewhere (preserving the nodes and the core of the curve).
• Hyperelliptic Case: Conversely, if the stable reduction is hyperelliptic, no GIT-stable plane model exists. In this case, any GIT-semistable model will have a special fiber that is not stable (e.g., a double conic or other non-stable configurations), preventing the direct application of the contraction method described above.

The thesis provides a complete classification of the 42 combinatorial types of stable genus 3 curves, determining exactly which are hyperelliptic based on their core structure (2-inseparable vs. 2-separable), the number of 1-tails, and the field characteristic (notably the dependence on $p=2$ for curves with a rational core and three tails).

Significance and Claims
The paper claims to provide a "geometric framework" for computing the stable model of non-hyperelliptic genus 3 curves. By proving that the existence of a GIT-stable model is equivalent to the non-hyperelliptic nature of the reduction, the work offers a computational criterion: one can compute a GIT-semistable model (using existing algorithms like those in [SW25]) and test its stability.

• If the model is GIT-stable, the stable reduction is non-hyperelliptic, and the stable model can be recovered by resolving the cusps of the GIT-stable model (specifically, by performing weighted blow-ups at the cusps to recover the 1-tails).
• If the model is not GIT-stable, the stable reduction is hyperelliptic, and the paper notes that the method of contracting 1-tails to cusps does not apply directly, suggesting that different techniques (such as toggle models) are required for the hyperelliptic case.

The author explicitly states that this result removes the need for case-by-case verification of the canonical map's behavior for 2-inseparable stable curves, generalizing previous findings to all genera $g \ge 2$ via the theory of quasi-hyperelliptic curves. The work serves as a foundational step toward a complete algorithm for computing stable reductions, with the specific construction of the field extension and the explicit resolution of singularities detailed in companion articles.