Models of hypersurfaces and Bruhat-Tits buildings

TL;DR

This paper introduces a new method for constructing semistable models of hypersurfaces over discretely valued fields using stability functions on Bruhat-Tits buildings, extending previous work and applying it to plane curves.

Contribution

The paper defines a stability function on Bruhat-Tits buildings to control semistable hypersurface models, extending prior minimal model approaches and implementing the method for plane curves.

Findings

Successfully constructs semistable models for hypersurfaces over local fields.

Reduces the problem to minimizing a stability function on the Bruhat-Tits building.

Provides a computational approach for plane curves over p-adic fields.

Abstract

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.