Affine toric SL(2)-embeddings

Abstract

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 is determined by a pair , where is a rational number written as an irreducible fraction and called the height of the action, while 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 is toric, that is, it admits a locally transitive action of an algebraic torus if and only if the number is divisible by . For that, the following criterion for an affine -embedding to be toric is proved. Let be a normal affine variety, a simply connected semisimple group acting regularly on , and a closed subgroup such that the character group of the group is finite. If an open equivariant embedding is defined, then is toric if and only if there exist a quasitorus and a -module such that . In the substantiation of this result a key role is played by Cox's construction in toric geometry.Bibliography: 12 titles.