Normal singularities with torus actions

TL;DR

The paper introduces a combinatorial method to desingularize normal affine varieties with torus actions, enabling analysis of their singularities and classification of factorial varieties, especially in complexity one cases.

Contribution

It provides a new combinatorial approach to desingularization and classification of torus action varieties, including criteria for various singularity types.

Findings

Criteria for rational, factorial, and Gorenstein singularities.

Method to construct factorial affine varieties with torus actions.

Complete classification of factorial varieties with complexity one torus actions.

Abstract

We propose a method to compute a desingularization of a normal affine variety X endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of X. In particular, we give criteria for X to have only rational, (QQ-)factorial, or (QQ-)Gorenstein singularities. We also give partial criteria for X to be Cohen-Macaulay or log-terminal. Finally, we provide a method to construct factorial affine varieties with a torus action. This leads to a full classification of such varieties in the case where the action is of complexity one.