Mustafin models of projective varieties and vector bundles

TL;DR

This paper explores degenerations of projective varieties and vector bundles using Mustafin models, Groebner basis techniques, and applications to p-adic geometry, advancing understanding of their arithmetic and geometric properties.

Contribution

It introduces a Groebner basis approach to study degenerations of projective varieties and extends previous work on Mustafin varieties and vector bundles.

Findings

Developed methods for Groebner bases under substitution over UFDs.

Analyzed behavior of degenerations of projective varieties.

Outlined applications to p-adic Simpson correspondence.

Abstract

Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat--Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the $p-$adic Simpson correspondence, when the respective projective variety is a curve.