Kletus Stern | Searcharxiv

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

Models of hypersurfaces and Bruhat-Tits buildings

We propose a new approach to constructing semistable integral models of hypersurfaces over a discretely valued complete field K. For each stable hypersurface X over K we define a continuous stability function on the Bruhat-Tits building of PGL_{n+1}(K); its global minima control semistable hypersurface models after finite extensions of K. In particular, in residue characteristic zero the problem reduces to minimizing this function on the original building and then passing to a finite extension that turns a rational minimizer into a vertex. This extends work of Kollar and of Elsenhans-Stoll on minimal hypersurface models. We implement the resulting strategy for plane curves over p-adic number fields. In a follow-up article we use our results to compute the semistable reduction of smooth plane quartics.

math.AG↗