Semistable resolutions of local models

本计画的目标是以演算法与实验法探讨算术代数几何的一个主题。局部模型描述了某些志村簇的积分模型的étale局部结构，因此，以及其他原因，在算术几何中有很大的兴趣。然而，一般来说，它们的奇异性是如此复杂，以至于希望传递到具有不太严重奇异性的模型，在最好的情况下传递到半稳定模型。一般来说，不知道是否存在这样的模式。这就是我们将通过显式计算来研究的问题。在“小秩”的情况下，计算（由主要研究者进行）表明存在半稳定的分辨率。在一般情况下，有候选人的半稳定决议，例如由Genestier和Faltings，但到目前为止（不使用计算机），他们的半稳定性无法证明。此外，这个问题和类似的问题也可以研究其他类型的方案，例如某些退化的Grassmannian。

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.