Affine toric SL(2)-embeddings

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 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 bT and a (G × b T )-module V such that X G ∼

Cox rings, semigroups and automorphisms of affine algebraic varieties

“…This construction has already been used in affine algebraic geometry. For example, the calculation of the Cox ring allowed to characterize affine SL(2)-embeddings that admit the structure of a toric variety [11]. In [4], it is shown that Cox realization leads to a remarkable unified description of all affine SL(2)-embeddings, which was not known before.…”

Invariant Hilbert scheme resolution of Popov's $SL(2)$-varieties II: the non-toric case

“…(q, p) = 1. Theorem 3.2 ( [Gaȋ08], see also [BH08,Corollary 2.7]). An affine normal quasihomogeneous SL(2)-variety E l,m is toric if and only if q − p divides m.…”

“…, where E − l,m and E + l,m are different GIT quotients of H q−p corresponding to some non-trivial characters, and that the varieties E l,m , E − l,m , and E + l,m are dominated by the weighted blow-up E ′ l,m = Bl ω O (E l,m ) of E l,m with a weight ω defined by the abovementioned C * -action on E l,m . The weight ω is trivial if and only if the SL(2)-variety E l,m is toric, namely if m = a(q − p) holds for some a > 0 (see [Gaȋ08,BH08]).…”

Invariant Hilbert scheme resolution of Popov's $SL(2)$-varieties I: the toric case

“…(q, p) = 1. Theorem 2.2 ( [Gaȋ08], see also [BH08,Corollary 2.7]). An affine normal quasihomogeneous SL(2)-variety E l,m is toric if and only if q − p divides m.…”

“…The smoothness of the invariant Hilbert scheme H is independent of the pair of numbers (l, m), but the behavior of the resolution γ does depend on it: it depends on whether E l,m is toric or not. Here we remark that a necessary and sufficient condition for E l,m being a toric variety is given in [Gaȋ08] (see also [BH08, Corollary 2.7]) in terms of the numbers l = p/q and m: an affine variety E l,m is toric if and only if q − p divides m. In both toric and non-toric case, we see that the restriction of γ to the main component H main = γ −1 (U) of the invariant Hilbert scheme H factors equivariantly through the weighted blow-up E ′ l,m :…”