# PhD project: Semi-stable resolutions of local models

### Ulrich Görtz (Universität Duisburg-Essen)

The goal of this project is the investigation of a topic in arithmetic
algebraic geometry by algorithmic and experimental methods. Local models
describe the étale-local structure of integral models of certain Shimura
varieties, and therefore, as well as for other reasons, are of great interest
in arithmetic geometry. However, in general their singularities are so
complicated that it would be desirable to pass to a model with less severe
singularities, in the best case to a semistable model. In general it is not
known whether such a model exists. This is what we will investigate by explicit
computations.

In cases of "small rank" computations (by the principal investigator, among
others) have shown that a semistable resolution exists. In the general case
there are candidates for semistable resolutions, for example by Genestier and
Faltings, but so far (without using computers) their semistability could not be
proved.

In addition, this and similar questions can also be investigated for other
classes of schemes, for instance for certain degenerations of quiver
Grassmannians.

For this project, I am looking for a PhD student. A corresponding position (75%
E13 for up to 3 years) is available. Please distribute! Candidates should send
a short application to me at ulrich.goertz@uni-due.de.