Local models, Mustafin varieties and semi-stable resolutions

TL;DR

This paper investigates the singularities of integral models of Shimura varieties by exploring local models, using Mustafin varieties, and constructing semi-stable resolutions, extending previous approaches and providing new candidates under specific conditions.

Contribution

It introduces a new semi-stable resolution candidate for local models of Shimura varieties using Mustafin varieties, generalizing Genestier's approach and aligning with Görtz's resolutions in certain cases.

Findings

Explicit calculations show Genestier's approach does not always work.

Mustafin varieties describe local models as flat degenerations.

The new candidate generalizes existing semi-stable resolutions under certain assumptions.

Abstract

Our goal is to analyse singularities of integral models of Shimura varieties. One approach is to construct local models, which model the singularities of the corresponding integral model using linear algebra dada and find resolutions with mild singularities thereof. More precisely we will attack the question of existence of semi-stable resolutions. We will discuss an approach developed by Genestier. In this approach a candidate for a semi-stable resolution was given as the blow-up of a Grassmannian variety in Schubert varieties of its special fiber. Explicit calculations show that this approach does not work in general. Using the flatness of the local models, we describe these local models as Mustafin varieties for Grassmannian varieties. We combine several results on the structure of Mustafin varieties for projective spaces with the Pl\"ucker embedding to construct a candidate for a semi-stable resolution of local models. Under some additional assumptions this candidate generalises the approach suggested by Genestier. Furthermore under the same assumptions the new candidate agrees with the semi-stable resolution constructed by G\"ortz for small dimensions.