Affine Toric SL(2)-embeddings

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.