Polyhedral divisors and SL_2-actions on affine T-varieties

TL;DR

This paper classifies SL_2-actions on normal affine T-varieties using combinatorial methods, extending Demazure's roots, and applies this to describe special actions and classify certain threefolds.

Contribution

It introduces a combinatorial framework for classifying SL_2-actions on T-varieties, generalizing Demazure's roots and connecting to existing classifications.

Findings

Classification of SL_2-actions on T-varieties

Description of special SL_2-actions on affine varieties

Connection to classification of SL_2-threefolds

Abstract

In this paper we classify SL_2-actions on normal affine T-varieties that are normalized by the torus T. This is done in terms of a combinatorial description of T-varieties given by Altmann and Hausen. The main ingredient is a generalization of Demazure's roots of the fan of a toric variety. As an application we give a description of special SL_2-actions on normal affine varieties. We also obtain, in our terms, the classification of quasihomogeneous SL_2-threefolds due to Popov.