Polyhedral divisors and SL2-actions on affine T-varieties

Abstract

In this paper we classify SL2-actions on normal affine T-varieties that are normalized by the torus T. This is done in terms of a combinatorial description of T-varieties given by Altmann and Hausen. The main ingredient is a generalization of Demazure's roots of the fan of a toric variety. As an application we give a description of special SL2-actions on normal affine varieties. We also obtain, in our terms, the classification of quasihomogeneous SL2-threefolds due to Popov. IntroductionLet k be an algebraically closed field of characteristic zero, M be a lattice of rank n, N = Hom(M, Z) be the dual lattice of M , and T be the algebraic torus Spec k[M ], so that M is the character lattice of T and N is the one-parameter subgroup lattice of T.A T-variety X is a normal algebraic variety endowed with an effective regular action of T. The complexity of a T-action is the codimension of a general orbit, and since the T-action on X is effective, the complexity of X equals dim X − rank M . For an affine variety X, to introduce a T-action on X is the same as to endow k[X] with an M -grading. There are well known combinatorial descriptions of T-varieties. We send the reader to [Dem70] and [Ful93] for the case of toric varieties, to [KKMS73, Ch. 2 and 4] and [Tim08] for the complexity one case, and to [AH06, AHS08] for the general case. In this paper we use the approach in [AH06].Any affine toric variety is completely determined by a polyhedral cone σ ⊆ N Q . Similarly, the description of a normal affine T-varieties X due to Altmann and Hausen [AH06] involves the data (Y, σ, D) where Y is a normal semiprojective variety, σ ⊆ N Q := N ⊗ Q is a polyhedral cone, and D is a divisor on Y whose coefficients are polyhedra in N Q with tail cone σ. The divisor D is called a σ-polyhedral divisor on Y (see Section 1.1 for details).Let X be a T-variety endowed with a regular G-action, where G is any linear algebraic group. We say that the G-action on X is compatible if the image of G in Aut(X) is normalized but not centralized by T. Furthermore, we say that the G-action is of fiber type if the general orbits are contained in the T-orbit closures, and of horizontal type otherwise [FZ05, Lie10a].Let now G a = G a (k) be the additive group of k. It is well known that a G a -action on an affine variety X is equivalent to a locally nilpotent derivation (LND) of k[X]. A description of compatible G a -actions on an affine T-variety, or equivalently of homogeneous LNDs on k[X], is available in the case where X is of complexity at most one [Lie10a] or the G a -action is of fiber type [Lie10b] in terms of a generalization of Demazure's roots of a fan [Dem70] (see Sections 1.3 and 1.4).A regular SL 2 -action on an affine variety X is uniquely defined by an sl 2 -triple {δ, ∂ + , ∂ − } of derivations of the algebra k[X], where ∂ ± are locally nilpotent, δ = [∂ + , ∂ − ] is semisimple and [δ, ∂ ± ] = ±2∂ ± (see Proposition 2.1). Assume now that X is an affine T-variety. If the

On rigidity of factorial trinomial hypersurfaces

“…A description of homogeneous locally nilpotent derivations of horizontal type for an affine T -variety X of complexity one in terms of the combinatorial data (Y, D) is given in [21], see also [5,Section 1.4]. To represent such a derivation, as a first step one needs to fix a vertex v z for every coefficient ∆ z of the polyhedral divisor D in such a way that all but two of these vertices are in N, see [5, Definition 1.8(iii)].…”

“…Further we assume that n i l i1 > 1 for i = 0, 1, 2 or, equivalently, f does not contain a linear term; otherwise the hypersurface X is isomorphic to the affine space K n−1 . Under this assumption, the trinomial hypersurface X is factorial if and only if any two of d 0 , If X is a toric (not necessary affine) variety with the acting torus T , then the actions G a × X → X normalized by T can be described combinatorially in terms of the so-called Demazure roots; see [8], [22,Section 3.4] for the original approach and [21,5,3] for generalizations. It is easy to deduce from this description that every affine toric variety different from a torus admits a non-trivial G a -action.…”

Commuting homogeneous locally nilpotent derivations

“…It is known that G a -actions on an affine variety X correspond to locally nilpotent derivations of the algebra K[X] and G a -actions normalized by a torus correspond to locally nilpotent derivations on K[X] that are homogeneous with respect to the grading induced by the action of the torus T. The description of homogeneous locally nilpotent derivations in terms of proper polyhedral divisors and the so called Demazure roots is obtained by A. Liendo for actions of vertical type of arbitrary complexity (see [7]) and for actions of horizontal type of complexity one (see [8], [2]). In addition, [2] describes pairs of G a -subgroups corresponding to the root subgroups of the SL 2 (K) and PSL 2 (K) actions. An objective of this paper is to describe G 2 a -actions on the affine T-variety X in the same terms.…”

“…(1) both derivations are of vertical type; (2) one derivation is of the vertical type, the another one is of horizontal type; (3) both derivations are of horizontal type.…”

On homogeneous locally nilpotent derivations of trinomial algebras

“…Let X be a toric variety. Then G a -actions on X normalized by T (or, equivalently, locally nilpotent derivations of K[X] that are homogeneous with respect to the corresponding grading) can be described in terms of Demazure roots of the fan corresponding to X; see [4], [16,Section 3.4] for the original approach and [13,3,2] for generalizations.…”