Affine toric $\operatorname{SL}(2)$-embeddings

In the theory of affine $\operatorname{SL}(2)$-embeddings, which was constructed in 1973 by Popov, a locally transitive action of the group $\operatorname{SL}(2)$ on a normal affine three-dimensional variety $X$ is determined by a pair $(p/q,r)$, where $0<p/q\le1$ is a rational number written as an irreducible fraction and called the height of the action, while $r$ is a positive integer that is the order of the stabilizer of a generic point. In the present paper it is shown that the variety $X$ is toric, that is, it admits a locally transitive action of an algebraic torus if and only if the number $r$ is divisible by $q-p$. For that, the following criterion for an affine $G/H$-embedding to be toric is proved. Let $X$ be a normal affine variety, $G$ a simply connected semisimple group acting regularly on $X$, and $H\subset G$ a closed subgroup such that the character group $\mathfrak X(H)$ of the group $H$ is finite. If an open equivariant embedding $G/H\hookrightarrow X$ is defined, then $X$ is toric if and only if there exist a quasitorus $\widehat T$ and a $(G\times\widehat T)$-module $V$ such that $X\stackrel G\cong V/\!/\widehat T$. In the substantiation of this result a key role is played by Cox's construction in toric geometry.
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