Affine Toric SL(2)-embeddings

TL;DR

This paper characterizes when affine SL(2)-embeddings are toric, linking the divisibility of an integer parameter to the existence of a torus action, and generalizes conditions for toric G/H-embeddings using Cox's construction.

Contribution

It provides a necessary and sufficient condition for affine SL(2)-embeddings to be toric based on the divisibility of r by q-p, and extends this to general affine G/H-embeddings via Cox's construction.

Findings

X is toric if and only if r is divisible by q-p

A normal affine G/H-embedding is toric if it admits a (G×T)-module structure with a G-equivariant isomorphism

The proof utilizes D. Cox's construction to establish the toric criterion.

Abstract

In 1973 V.L.Popov classified affine SL(2)-embeddings. He proved that a locally transitive SL(2)-action on a normal affine three-dimensional variety X is uniquely determined by a pair (p/q, r), where 0<p/q<=1 is an uncancelled fraction and r is a positive integer. Here r is the order of the stabilizer of a generic point. In this paper we show that the variety X is toric, i.e. admits a locally transitive action of an algebraic torus, if and only if r is divisible by q-p. To do this we prove the following necessary and sufficient condition for an affine G/H-embedding to be toric. Suppose X is a normal affine variety, G is a simply connected semisimple algebraic group acting regularly on X, H is a closed subgroup of G such that the character group $\mathfrak{X}(H)$ is finite and G/H -> X is a dense open equivariant embedding. Then X is toric if and only if there exist a quasitorus T and a $(G\times T)$-module V such that $X\stackrel{G}{\cong} V//T$. The key role in the proof plays D. Cox's construction.