Local models, Mustafin varieties and semi-stable resolutions

Abstract

In this thesis we will analyse singularities of local models. More precisely we will attack the question of existence of semi-stable resolutions. We will discuss an approach mentioned in [Gen00]. 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 with Sage described in Appendix D show that this approach is not working in general. Starting from the proof of flatness of the local models in [Gor01], we describe these local models as Mustafinvarieties over Grassmannian varieties. We are combining several results on the structure of Mustafin varieties over projective spaces (cf. [CHSW11],[AL17]) with the Plucker embedding to be able to construct a candidate for a semi-stable resolution of local models. Under some additional assumptions this candidate is generalising the approach suggested by Genestier. Furthermore under the same assumptions the new candidate agrees with the semi-stable resolution constructed in [Gor04] for small dimensions.