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Article

Affine toric SL(2)-embeddings

Sbornik Mathematics. 2008. Vol. 199. No. 3. P. 319-339.

In the theory of affine SL(2)-embeddings, which was constructed in 1973 by Popov, a locally transitive action of the group SL(2) on a normal affine three-dimensional variety X is determined by a pair (p/q, r), where 0 < p/q ≤ 1 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 ⊂ G a closed subgroup such that the character group X(H) of the group H is finite. If an open equivariant embedding G/H → X is defined, then X is toric if and only if there exist a quasitorus T̂ and a (G × T̂)-module V such that X ≅G V//T̂. In the substantiation of this result a key role is played by Cox's construction in toric geometry.