Mustafin Varieties

TL;DR

Mustafin varieties are degenerations of projective space linked to point configurations in Bruhat-Tits buildings, revealing rich combinatorial and geometric structures with connections to tropical and toric geometry.

Contribution

This paper provides a comprehensive study of Mustafin varieties, including their combinatorial patterns, geometric properties, and classifications in specific cases like PGL(2) and PGL(3).

Findings

Special fiber is reduced and Cohen-Macaulay.

Configurations in one apartment relate to regular mixed subdivisions.

Classification of Mustafin triangles into 38 types.

Abstract

A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of PGL(2) and in the two-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types.