A survey on the singularities of spherical varieties

TL;DR

This survey compiles known combinatorial criteria for singularities in spherical varieties, relevant to the Minimal Model Program and Fano varieties, providing clear proofs and additional results for a comprehensive overview.

Contribution

It consolidates and clarifies existing criteria for singularities in spherical varieties, enhancing understanding and accessibility for researchers.

Findings

Collection of known singularity criteria

Clarification and proof of existing results

Addition of new related results

Abstract

We list combinatorial criteria of some singularities, which appear in the Minimal Model Program or in the study of (singular) Fano varieties, for spherical varieties. Most of the results of this paper are already known or are quite easy corollary of known results. We collect these results, we precise some proofs and add few results to get a coherent and complete survey.