arXiv subjects

Max Schwegele

Publications and source records attributed to Max Schwegele.

3 recordsLinked to original sources

arXiv2026

Hyperelliptic Stable Curves

We provide an intrinsic characterization of hyperelliptic stable curves of genus $g \geq 2$, independent of admissible covers or auxiliary moduli data. A stable curve is hyperelliptic if it admits an involution yielding a rational tree quotient, subject to a characteristic-dependent condition. By analyzing the action of this involution on the nodes and decomposing the curve based on its connectivity, we obtain an explicit structural description of hyperellipticity and prove that the hyperelliptic involution is unique. Furthermore, we explain the connection to the very ampleness of the dualizing sheaf. This framework applies in arbitrary characteristic, explicitly capturing the divergent geometric and combinatorial behavior in characteristic 2. We verify that this formulation precisely captures the geometric points of the moduli stack of hyperelliptic stable curves $\overline{\mathcal{H}}_g$, defined as the scheme-theoretic closure of the smooth hyperelliptic locus $\mathcal{H}_g$ within the moduli stack of stable curves $\overline{\mathcal{M}}_g$. Extending this definition to flat families yields an explicit modular description of $\overline{\mathcal{H}}_g$ over $\operatorname{Spec} \mathbb{Z}[1/2]$.

math.AG↗

arXiv2026

Semistable reduction of smooth quartics

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".

math.AG↗

arXiv2025

Semistable Reduction of Plane Quartics

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.

math.AG↗