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Polyhedral divisors and SL_2-actions on affine T-varieties

In this paper we classify SL_2-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 SL_2-actions on normal affine varieties. We also obtain, in our terms, the classification of quasihomogeneous SL_2-threefolds due to Popov.

SL 2 -action is compatible, then ∂ ± are homogeneous with respect to the M -grading on k[X] and the grading given by δ is a downgrading of the M -grading.
The main result of this paper, contained in Section 2, is a classification of compatible SL 2actions on an affine T-variety X in the case where this action is of fiber type or X is of complexity one (See Theorems 2.12 and 2.18, respectively). Our idea is to classify compatible SL 2 -actions by calculating the commutator of two homogeneous LNDs. The existence of a compatible SL 2action on X puts strong restrictions on the combinatorial data (Y, σ, D) and endows D with an additional structure. It should be noted that if the T-variety X is of complexity one and the SL 2 -action is of horizontal type, then X is spherical with respect to a bigger reductive group, namely, an extension of SL 2 by a torus. We do not use the theory of spherical varieties in this paper.
The rest of the paper is devoted to two applications of our main result: special SL 2 -actions and SL 2 -actions with an open orbit. A G-action on X is called special (or horospherical) if there exists a dense open W ⊆ X such that the isotropy group of any point x ∈ W contains a maximal unipotent subgroup of G. Special actions play an important role in Invariant Theory.
Any special action of a connected reductive group G on an affine variety X may be reconstructed from the action of a maximal torus T ⊆ G on the algebra k[X] U of invariants of a maximal unipotent subgroup U [ Pop86,Theorem 5]. This reduces the study of special actions to torus actions. In Section 3 we illustrate this phenomenon for SL 2 -actions in our terms (see Theorem 3.11 and 3.12). In particular, we show that for every special SL 2 -action on an affine variety X there is a canonical 2-torus action and the SL 2 -action is compatible and of fiber type with respect to this torus. Since the reconstruction of the G-variety X from the T -variety Spec k[X] U is an algebraic procedure, it is useful to have a geometric description of X. In Proposition 3.10 we describe a normal affine variety X with a special SL 2 -action as a T 2 -variety with respect to the canonical torus T 2 . It is worthwhile to remark that any G-action on an affine variety may be contracted to a special one [Pop86,Proposition 8]. It will be interesting to interpret contraction of SL 2 -actions in terms of polyhedral divisors.
As a corollary of our classification of special actions, we prove that if an affine T-variety X of complexity one admits a compatible special SL 2 -action of horizontal type, then X is toric with respect to a bigger torus and the SL 2 -action is compatible with respect to the big torus as well. Furthermore, using a linearization result due to Berchtold and Hausen [BH03], we show that, up to conjugation in Aut(X), any special SL 2 -action on an affine toric 3-fold X is compatible with the big torus, and thus it is given by an SL 2 -root (see Definition 2.6).
It is natural to generalize Altmann and Hausen's approach [AH06] to arbitrary reductive groups. Special actions form the most accessible class for such a generalization. Our work in this line may be regarded as a first step towards this aim. It must be noted that Timashev [Tim97] already gave a combinatorial description for G-actions of complexity one in the framework of Luna-Vust theory.
Finally, our method allows to reprove, in Section 4, Popov's classification of generically transitive SL 2 -actions on normal affine threefolds. The only fact that we use is the existence of a one dimensional torus R commuting with SL 2 . Together with the maximal torus in SL 2 , this allows us to consider a quasi-homogeneous threefold as a T 2 -variety of complexity one, where T 2 is a two dimensional torus. We also obtain, as a direct consequence of our results, the characterization of toric quasi-homogeneous SL 2 -threefolds given in [Gaȋ08] and [BH08] (see Corollaries 4.9 and 4.13). Recall that a G-variety is quasi-homogeneous if it has an open G-orbit.
In the entire paper the term variety means a normal integral scheme of finite type over an algebraically closed field k of characteristic zero. The term point always refer to a closed point.

Preliminaries
In this section we recall the results about G a -actions on affine T-varieties needed in this paper. In [AH06], a combinatorial description of normal affine T-varieties is given. In what follows we recall the main features of this description. Let σ be a pointed polyhedral cone in N Q . We define Pol σ (N Q ) to be the set of all σ-polyhedra i.e., the set of all polyhedra in N Q that can be decomposed as the Minkowski sum of a bounded polyhedron and the cone σ.
Recall that σ ∨ stands for the cone in M Q dual to σ. To a σ-polyhedron ∆ ∈ Pol σ (N Q ) we associate its support function h ∆ : σ ∨ → Q defined by  (1) Hence, h ∆ is piecewise linear, concave, and positively homogeneous.
Definition 1.1. A normal variety Y is called semiprojective if it is projective over an affine variety. A σ-polyhedral divisor on Y is a formal sum D = Z ∆ Z · Z, where Z runs over all prime divisors on Y , ∆ Z ∈ Pol σ (N Q ), and ∆ Z = σ for all but finitely many Z. For m ∈ σ ∨ we can evaluate D in m by letting D(m) be the Q-divisor where h Z is the support function of ∆ Z . A σ-polyhedral divisor D is called proper if the following hold: (i) D(m) is semiample and Q-Cartier for all m ∈ σ ∨ , and (ii) D(m) is big for all m ∈ rel. int(σ ∨ ).
Here rel. int(σ ∨ ) denotes the relative interior of the cone σ ∨ . Furthermore, a Q-Cartier divisor D on Y is called semiample if there exists r > 0 such that the linear system |rD| is base point free, and big if there exists a divisor D 0 ∈ |rD|, for some r > 0, such that the complement Y \ Supp D 0 is affine.
The following theorem gives a combinatorial description of T-varieties analogous to the classical combinatorial description of toric varieties. In the sequel, χ m denotes the character of T corresponding to the lattice vector m, and σ ∨ M denotes the semigroup σ ∨ ∩ M . Furthermore, for Conversely, any normal affine T-variety is isomorphic to X[Y, D] for some semiprojective variety Y and some proper σ-polyhedral divisor D on Y .
We call Y the base variety and the pair (Y, D) the combinatorial description of X. We also define the support of a proper σ-polyhedral divisor as Supp This combinatorial description is not unique, but can be made unique by adding some minimality conditions on the pair (Y, D), see [AH06, Section 8]. Here we only need a particular case of [AH06, Corollary 8.12]. Corollary 1.3. Let D and D ′ be two proper σ-polyhedral divisors on a normal semiprojective variety Y . If for every prime divisor Z in Y there exists a vector v Z ∈ N such that Most of this paper deals with the case where the base is a curve C isomorphic to A 1 or P 1 . Any σ-polyhedral divisor on A 1 is proper. If D = z∈C ∆ z · z is a σ-polyhedral divisor on C = P 1 , then D is proper if and only if deg D := z∈C ∆ z σ. We also need the following result from [AH06, Section 11].
Corollary 1.4. Let D be a proper σ-polyhedral divisor on a smooth curve C. Then X[C, D] is toric if and only if C = A 1 and D can be chosen (via Corollary 1.3) supported in at most one point, or C = P 1 and D can be chosen (via Corollary 1.3) supported in at most two points.
1.2. Locally nilpotent derivations and G a -actions. Let X = Spec A be an affine variety. A derivation ∂ on A is called locally nilpotent (LND for short) if for every a ∈ A there exists n ∈ Z ≥0 such that ∂ n (a) = 0. We denote by G a the additive group of the base field k. Given an LND ∂ on A, the map φ ∂ : G a × A → A, φ ∂ (t, f ) = exp(t∂)(f ) defines a G a -action on X, and any G a -action on X arises in this way [Fre06].
Let now D be a proper σ-polyhedral divisor on a semiprojective variety Y , and let A = A[Y, D] be the corresponding M -graded domain. A G a -action on X = Spec A is said compatible with the T-action on X if the image of G a in Aut(X) is normalized by the torus T. A G a -action is compatible if and only if the corresponding LND ∂ on A is homogeneous i.e., if ∂ sends homogeneous elements to homogeneous elements. Any homogeneous LND ∂ has a well defined degree given as deg ∂ = deg ∂(f ) − deg f for any homogeneous f ∈ A \ ker ∂.
A homogeneous LND ∂ on A extends to a derivation on Frac A = k(Y )(M ) by the Leibniz rule, where k(Y )(M ) is the field of fractions of k(Y ) [M ]. The LND ∂ is said to be of fiber type if ∂(k(Y )) = 0 and of horizontal type otherwise. Geometrically speaking, ∂ is of fiber type if and only if the general orbits of the corresponding G a -action on X = Spec A are contained in the orbit closures of the T-action given by the M -grading.
1.3. Locally nilpotent derivations on affine toric varieties. In this section we recall the classification of homogeneous LNDs on toric varieties given in [Lie10a]. A similar description is implicit in [Dem70, Section 4.5]. As usual, for a cone σ, we denote by σ(1) the set of all rays of σ and we identify a ray with its primitive vector.
Definition 1.5. Let σ be a pointed cone in N Q . We say that e ∈ M is a root of the cone σ if the following hold: (i) there exists ρ e ∈ σ(1) such that e, ρ e = −1; The ray ρ e is called the distinguished ray of the root e. We denote by R(σ) the set of all roots of σ.
One easily checks that any ray ρ ∈ σ(1) is the distinguished ray for infinitely many roots e ∈ R(σ). For every root e ∈ R(σ) we define a homogeneous derivation ∂ e of degree e of the algebra k[σ ∨ M ] by the formula ∂ e (χ m ) = m, ρ e · χ m+e , for all m ∈ σ ∨ M . The following theorem gives a classification of the homogeneous LNDs on k[σ ∨ M ]. Theorem 1.6. For every root e ∈ R(σ), the homogeneous derivation ∂ e on k[σ ∨ M ] is an LND of degree e with kernel ker ∂ e = k[τ e ∩ M ], where τ e is the facet of σ ∨ dual to the distinguished ray ρ e . Conversely, if ∂ = 0 is a homogeneous LND on k[σ ∨ M ], then ∂ = λ∂ e for some root e ∈ R(σ), and some λ ∈ k * .
1.4. Locally nilpotent derivations on affine T-varieties. We give first a classification of homogeneous LNDs of fiber type on T-varieties of arbitrary complexity given in [Lie10b].
Letting D = Z ∆ Z · Z be a proper σ-polyhedral divisor on a semiprojective variety Y , we let A = A[Y, D]. For every prime divisor Z ⊆ Y , we let {v i,Z | i = 1, · · · , r Z } be the set of all vertices of ∆ Z . Letting e be a root of the cone σ, we define the divisor Remark that the evaluation divisor D(m) is only defined for m ∈ σ ∨ and e / ∈ σ ∨ . The reason for the above notation is that taking (1) as the definition of support function, we obtain the above formula for the evaluation divisor, which can be evaluated at any m ∈ M Q .
For every ϕ ∈ Φ * e we let ∂ e,ϕ (f χ m ) = m, ρ e · ϕ · f χ m+e , for all m ∈ σ ∨ M , and f ∈ k(Y ) . The following theorem gives a classification of the homogeneous LNDs of fiber type on A[Y, D].
Theorem 1.7. For every root e ∈ R(σ) and ϕ ∈ Φ * e , the derivation ∂ e,ϕ is a homogeneous LND of fiber type on A = A[Y, D] of degree e with kernel where τ e ⊆ σ ∨ is the facet dual to the distinguished ray ρ e . Conversely, if ∂ = 0 is a homogeneous LND of fiber type on A, then ∂ = ∂ e,ϕ for some root e ∈ R(σ) and some ϕ ∈ Φ * e .
The classification of LNDs of horizontal type is more involved and is only available in the case of complexity one. Here, we give an improved presentation of the classification given in [Lie10a,Theorem 3.28].
Since the complexity is one the base variety Y is a smooth curve C. Let D = z∈C ∆ z · z be a proper σ-polyhedral divisor on C, and let X = X[C, D]. If A = A[C, D] admits a homogeneous LND of horizontal type, then C is isomorphic either to A 1 or to P 1 . In the following we assume that C = A 1 or C = P 1 .
We also let z 0 ∈ C ′ be such that v z ∈ N for all z ∈ C ′ \ {z 0 }. We say that D is a coloring of D and we call z 0 the marked point, z ∞ the point at infinity if C = P 1 , and v z the colored vertex of the polyhedron ∆ z .
Remark that the above notion of coloring is independent from the notion of coloring in the theory of spherical varieties.
Let D be a colored σ-polyhedral divisor on C. Letting ω ⊆ N Q be the cone generated by deg D| C ′ − v deg we let ω ⊆ (N ⊕ Z) Q be the cone generated by (ω, 0) and (v z 0 , 1) if C = A 1 , and by (ω, 0), (v z 0 , 1) and (∆ z∞ + v deg − v z 0 + ω, −1) if C = P 1 . Denote by d the minimal positive integer such that d · v z 0 ∈ N . We call ω the associated cone of the colored σ-polyhedral divisor D.
Definition 1.9. A pair ( D, e), where D is a colored σ-polyhedral divisor on C and e ∈ M , is said to be coherent if (1) There exists s ∈ Z such that e = (e, s) ∈ M ⊕ Z is a root of the associated cone ω with distinguished ray ρ = (d · v z 0 , d). In this case s = − 1 /d − v z 0 (e).
(2) v(e) ≥ 1 + v z (e), for every z ∈ C ′ \ {z 0 } and every vertex v = v z of the polyhedron ∆ z .  Let us give an explicit formula for the homogeneous LND ∂ associated to the coherent pair ( D, e). Without loss of generality we may assume z 0 = 0 and z ∞ = ∞ if C = P 1 . By Corollary 1.3 we may assume v z =0 ∈ N for all z ∈ C ′ \ {z 0 }. Letting k[C ′ ] = k[t], the homogeneous LND of horizontal type ∂ corresponding to the coherent pair ( D, e) is given by ∂(χ m · t r ) = d(v 0 (m) + r) · χ m+e · t r+s , for all (m, r) ∈ M ⊕ Z .
(2) Furthermore, if we let m = (m, r) ∈ M ⊕ Z and χ m = χ m · t r , then (2) can be written as in the toric case We also need the following two technical lemmas. They follow from Theorem 1.10.  (1) If C = A 1 then e ∈ ω ∨ ⊆ σ ∨ .

Compatible SL 2 -actions on normal affine T-varieties
In this section we give a classification of compatible SL 2 -actions on T-varieties in two cases: in the case where the T-action is of complexity one; and in arbitrary complexity provided that the general SL 2 -orbits are contained in the T-orbit closures.
2.1. SL 2 -actions on affine varieties. Let SL 2 be the algebraic group of 2 × 2 matrices of determinant 1. Every algebraic subgroup of SL 2 of positive dimension is conjugate to one of the following subgroups: −1 0 · T, and B = T · U (1) . Here T is a maximal torus, N is the normalizer of a maximal torus, B is a Borel subgroup, and U (s) is a cyclic extension of a maximal unipotent subgroup. We also define the following maximal unipotent subgroups: −1 0 . As a group SL 2 is generated by the unipotent subgroups U + and U − isomorphic to G a .
Let now X = Spec A be an affine variety endowed with an SL 2 -action. The two U ± -actions on X are equivalent to two LNDs ∂ ± on the algebra A, and the T -action on X is equivalent to a Zgrading on A. Furthermore, this Z-grading on A is also uniquely determined by its infinitesimal generator i.e., by the semisimple derivation δ given by δ(a) = deg(a) · a for every homogeneous a ∈ A.
The following well known proposition gives a criterion for the existence of an SL 2 -action on an affine variety. In the lack of a reference we provide a short proof, cf. [FZ05,4.15].
Proposition 2.1. A non-trivial SL 2 -action on an affine variety X = Spec A is equivalent to a (not necessarily effective) Z-grading on A with infinitesimal generator δ and a couple of Furthermore, the Z-grading is effective if and only if SL 2 acts effectively on X. If the Zgrading is not effective, then the kernel of SL 2 → Aut(X) is {± Id} and so PSL 2 acts effectively on X.
Proof. Assume first that SL 2 acts non-trivially on X. Let {h, e + , e − } be the sl 2 -triple in the Lie algebra sl 2 . Since e ± is tangent to the 1-parameter unipotent subgroup U ± in SL 2 , it acts on A as an LND ∂ ± . The vector h is tangent to the torus T , thus h acts on A as the infinitesimal generator δ of a Z-grading. Since [h, e ± ] = ±2e ± , the LND ∂ ± is homogeneous of degree ±2, and the relation [e + , e − ] = h implies [∂ + , ∂ − ] = δ.
Conversely, assume that we have δ, ∂ + , ∂ − as in the proposition. Then s = δ, ∂ + , ∂ − is a Lie subalgebra in Der(A) isomorphic to sl 2 . Furthermore, every element of A is contained in a finite-dimensional s-submodule. Recall that any finite-dimensional sl 2 -module has a canonical structure of SL 2 -module whose tangent representation coincides with the given one. This gives A the structure of a rational SL 2 -module. Since SL 2 is generated as a group by the subgroups U ± and the LNDs ∂ ± define the action of U ± via automorphisms, the group SL 2 acts on A via automorphisms. This proves that A is a rational SL 2 -algebra, or, equivalently, SL 2 acts regularly on X.
We restrict now to the case of affine T-varieties. The following definition determines the class of SL 2 -actions that we study in the sequel.
Definition 2.2. An SL 2 -action on a T-variety X is compatible if the image of SL 2 in Aut(X) is normalized but not centralized by the torus T.
Assume now that X is a T-variety endowed with a compatible SL 2 -action. Denote by SL 2 the image of SL 2 in Aut(X). There is a homomorphism ψ : T → Aut( SL 2 ). Since any automorphism from Aut( SL 2 ) is inner, we have Aut( SL 2 ) ≃ PSL 2 . Thus the image of T is either trivial or is a maximal torus T ⊆ PSL 2 . In the first case T centralizes SL 2 so this case is excluded by the definition of a compatible SL 2 -action. Hence T contains T and so T = T · S, where S = ker ψ is a complementary subtorus that centralizes the SL 2 -action. Let U ± be unipotent root subgroups in SL 2 with respect to the torus T . Then the SL 2 -action on X is defined by the infinitesimal generator corresponding to a Z-grading on k[X] defined by T (this is a downgrading of the M -grading) and two M -homogeneous LNDs ∂ ± corresponding to the U ± -actions. This gives the following corollary.
Corollary 2.3. (i) Let X be a normal affine T-variety endowed with a compatible SL 2 -action.
In this case, in Proposition 2.1, we may assume that δ is the infinitesimal generator corresponding to a downgrading of M and that ∂ ± are M -homogeneous LNDs. Furthermore, T = T · S, where T is the maximal torus in SL 2 and S is a complementary subtorus that centralizes the SL 2 -action. (ii) Let X be a normal affine T-variety endowed with an SL 2 -action that is centralized by T.
Then we may extend T by T so that the SL 2 -action is compatible with this bigger torus action.
The following is a generalization of a definition in [Lie10b].
Definition 2.4. We say that a compatible SL 2 -action on a T-variety is of fiber type if the general orbits are contained in the T-orbit closures and of horizontal type otherwise.
Clearly, a compatible SL 2 -action is of fiber type if and only if both derivations ∂ ± are of fiber type. The following lemma shows that a compatible SL 2 -action is of horizontal type if and only if both derivations ∂ ± are of horizontal type.
Lemma 2.5. Consider a compatible SL 2 -action on a T-variety X and assume that the LND ∂ + is of fiber type. Then the SL 2 -action is of fiber type.
Proof. Set B = T · U + ⊆ SL 2 . Then the B-action on X is of fiber type i.e., the general B-orbits are contained in the orbit closures of the T-action. We consider two cases. Case 1: The general SL 2 -orbits on X are 2-dimensional. Then for general x ∈ X one has B · x = SL 2 ·x, and the SL 2 -action is of fiber type. Case 2: The general SL 2 -orbits on X are 3-dimensional. Consider a general point x ∈ X and the stabilizer T 2 x ⊆ T of the subvariety B · x. Since any automorphism of the group B is inner and the torus T normalizes B, we may find a 1-dimensional subtorus S x ⊆ T 2 x which commutes with the B-action on B · x. But the image of the homomorphism ψ : T → Aut( SL 2 ) ≃ PSL 2 is a maximal torus, and so the subtorus S x is in the kernel of ψ. Hence, S x commutes with the SL 2 -action. In particular, S x preserves B · x and SL 2 ·x, and its action on SL 2 ·x may be lifted to the action of a maximal torus S ⊆ SL 2 by right multiplication on SL 2 via a finite covering S → S x . But it is easy to check that the (B × S)-action on SL 2 has an open orbit, so S x permutes the general B-orbits on SL 2 ·x. This provides a contradiction.
2.2. SL 2 -actions on toric varieties. In this section we give a complete classification of compatible SL 2 -actions on affine toric varieties. Since a toric variety has an open T-orbit, every SL 2 -action on a toric variety is of fiber type.
If e is an SL 2 -root, then e, ρ e = −1, e, ρ −e = 1, and e, ρ = 0 ∀ρ ∈ σ(1) \ {ρ ±e }. Thus the number of SL 2 -roots of a cone σ with r rays does not exceed r(r − 1), and this bound is attained for a regular cone of dimension r.
Theorem 2.7. The compatible SL 2 -actions on an affine toric variety X σ are in bijection with the SL 2 -roots of σ. Furthermore, for every SL 2 -root e ∈ R(σ), the corresponding SL 2 -action is effective if and only if the lattice vector ρ −e − ρ e is primitive. If ρ −e − ρ e is not primitive, then Hence, the infinitesimal generator of the corresponding G m -action is given by δ(χ m ) = m, p χ m , for all m ∈ σ ∨ M . A routine computation shows that δ, ∂ e and ∂ −e satisfy the conditions of Proposition 2.1. Furthermore, since e, p = 2 then p is primitive or p /2 is primitive. This proves the "only if" part of the theorem.
To prove the converse, let δ, ∂ − , ∂ + be three homogeneous derivations satisfying the conditions of Proposition 2.1. Since ∂ ± are LNDs, then ∂ ± = λ ± ∂ e ± for some λ ± ∈ k * and some roots e ± ∈ R(σ). Furthermore, since the derivation δ comes from a downgrading of the M -grading on A, there is a lattice element p such that δ(χ m ) = m, p χ m , for all m ∈ σ ∨ M . Since the commutator [∂ + , ∂ − ] is a homogeneous operator of degree e + + e − , we have e := e + = −e − . One checks that the commutator is given by − , and the result follows. Remark 2.8. If e is an SL 2 -root of σ, then −e is also an SL 2 -root. The corresponding SL 2 -actions are conjugate.
Example 2.9. Let X σ be an affine toric variety of dimension 2. Up to automorphism of the lattice N we may assume that σ ⊆ N Q is the cone spanned by the vectors ρ 1 = (1, 0) and ρ 2 = (a, b), where a ≥ 0, a < b, and gcd(a, b) = 1. By Theorem 2.7 X σ admits a compatible SL 2 -action if and only if there exists e ∈ M such that e, ρ 1 = 1 and e, ρ 2 = −1. The only solution is e = (1, −1) and b = a + 1. Furthermore, the action is effective if and only if b is odd.
It is well known that the toric variety X σ corresponds to the affine cone over the rational normal curve C of degree a + 1 (also known as Veronese cone). The curve C is the image of P 1 under the morphism ψ : P 1 ֒→ P a+1 , [x : y] → x a+1 : x a y : x a−1 y 2 : . . . : y a+1 .
The SL 2 -action on X σ is induced by the canonical SL 2 -action on the simple SL 2 -module V (a+1) of binary forms of degree a + 1.
Example 2.10. Let now X σ be an affine toric variety of dimension 3. Letting e be an SL 2 -root of σ, we let ρ e and ρ −e be the corresponding distinguished rays and we consider a ray ρ = ρ ±e . Since e, ρ = 0 it follows that there are at most 2 non-distinguished rays. Thus the cone σ has at most 4 rays.
Remark 2.11. In dimension 4 or greater, a cone admitting an SL 2 -root can have an arbitrary number of rays.
2.3. SL 2 -actions of fiber type on T-varieties. In the following theorem we give a classification of SL 2 -actions of fiber type on normal affine T-varieties of arbitrary complexity. Furthermore, the corresponding SL 2 -action is effective if and only if the lattice vector ρ −e − ρ e is primitive. If ρ −e − ρ e is not primitive, then 1 2 (ρ −e − ρ e ) is primitive and PSL 2 acts effectively on X.
, we let e be an SL 2 -root of σ satisfying the conditions of the theorem. As in the toric case, we let ρ ±e ∈ σ(1) be the distinguished rays of the roots ±e, respectively. Letting p = ρ −e − ρ e , we define a Z-grading on A via M . So the infinitesimal generator of the corresponding G m -action is given by Letting ϕ be any rational function on Y such that div(ϕ) + D(e) = 0, we let ∂ + = ϕ∂ e and ∂ − = ϕ −1 ∂ −e . The derivations ∂ ± are LNDs on A by Theorem 1.7. Now a routine computation shows that δ, ∂ + and ∂ − satisfy the conditions of Proposition 2.1. Furthermore, since e, p = 2 then p is primitive or p /2 is primitive. This proves the "only if" part of the theorem.
2.4. SL 2 -actions of horizontal type on T-varieties. In this section we give the more involved classification of SL 2 -actions of horizontal type in the case of T-varieties of complexity one. Here, we use the notation of Section 1.4.
Letting D be a proper σ-polyhedral divisor on the curve C = A 1 or C = P 1 , we let X = X[C, D] and we assume that X admits a compatible SL 2 -action of horizontal type. By Proposition 2.1, an SL 2 -action on X is completely determined by two homogeneous LNDs ∂ ± with deg ∂ + = − deg ∂ − = e. By Theorem 1.10, the LNDs of horizontal type are in bijection with coherent pairs. Let ∂ ± be the LND given by the coherent pair ( D ± , ±e), respectively, where In the sequel, we will assume z + 0 = 0, and z + ∞ = ∞ if C = P 1 . We also let q(t) be a coordinate around z − 0 having point at infinity z − ∞ if C = P 1 i.e., q is a Möbious transformation In the case where C = A 1 , we have c = 0 and so we may choose a = d = 1 and b = −z − 0 . By Corollary 1.3 we may and will assume v − z = 0, for all z ∈ C\{z − 0 , z − ∞ }. We also let d ± be the smallest positive integer such that d ± · v ± z ± 0 is contained in the lattice N , and In this setting (2) yields To obtain a similar expression for ∂ + , we let is a rational function whose divisor is D ′ (m) − D(m) for all m ∈ σ ∨ M , and ϕ m · ϕ m ′ = ϕ m+m ′ . In this setting (2) yields Recall that the LNDs ∂ ± on k[X] correspond to the U ± -actions on X of a compatible SL 2action on X and so by Corollary 2.3 (i), the commutator δ = [∂ + , ∂ − ] is a downgrading of the M -grading on k[X] i.e., there exits p ∈ N such that To lighten the notation, we use the "prime notation" to denote the partial derivative with respect to t i.e., d dt (f ) = f ′ .
Proposition 2.15. If X[C, D] admits an SL 2 -action of horizontal type, then the marked points and the infinity points (if C = P 1 ) of D + and D − can be chosen to be equal i.e., in the notation above, without loss of generality, we may assume z + 0 = z − 0 = 0 and z + ∞ = z − ∞ = ∞. Moreover, we have d + = d − := d.
Since compatible SL 2 -actions on toric varieties are described in Theorem 2.7, in the sequel we restrict to the case where the SL 2 -action on X is not compatible with a bigger torus. In the next lemma we show that if X[C, D] admits an SL 2 -action of horizontal type, then D has a very special form.
For a subset S ⊆ N Q we denote the convex hull of S by Conv(S). For a vector e ∈ M Q we let e ⊥ = {p ∈ N Q | e, p = 0} be the subspace of N Q orthogonal to e.
Lemma 2.17. Let X = X[C, D] be a normal affine T-variety of complexity one endowed with a compatible SL 2 -action of horizontal type. Assume that the SL 2 -action is not compatible with a bigger torus and let e ∈ M be the degree of the homogeneous LND ∂ + on k[X] corresponding to the U + -action on X. Then C = A 1 or C = P 1 , D = z∈C ∆ z · z, and the σ-polyhedra ∆ z can be chosen (via Corollary 1.3) as in one of the following cases: , v + 1 (e) = −1, and Π ⊆ e ⊥ is a bounded polyhedron; or , v + 1 (e) = −1, and Π ⊆ e ⊥ is a bounded polyhedron. For the proof of this lemma we need the following notation: With this definition α m+m ′ = α m + α m ′ . More explicitly, For a rational function R(t) = P (t) /Q(t), we define the degree deg R = deg P − deg Q so that deg(R 1 · R 2 ) = deg(R 1 ) + deg(R 2 ). We also let the principal part of R be the result of the polynomial division between P and Q. Then deg(R) = 0 if and only if the principal part of R is a non-zero constant.
A direct computation shows that the principal part of νv 0 (m) + α e v 0 (m) + να m + tα ′ m + α e α m is given by This is a contradiction since Γ(e) has to be independent of t. In the following, we assume L(e) = 0. This yields deg(ϕ e t ν− 1 /d ) = 0.
Case III : The condition v 0 = 0 implies v − 0 = v + 0 and ν = − 1 /d. Hence d = 1 or d = 2 since ν − 1 /d = − 2 /d appears as the exponent of t in Γ. If d = 1, then by Definition 1.8, we can change the marked points of D ± so that v − z 0 = v + z 0 . Hence, this case reduces to Case I or Case II. Assume now that d = 2 so that v − 0 = v + 0 ∈ 1 2 N \ N . The condition deg(ϕ e t ν− 1 /d ) = 0 implies deg(ϕ e ) = 1 and so we can assume v + z (e) = 0 for all z = 1 and v + 1 (e) = −1. This yields v + z = 0 for all z = 1. The commutator is now given by By (11) we have δ(χ m t r ) = m, −2v + 1 · χ m t r , for all (m, r) ∈ M ⊕ Z . By the same argument as in Case II we obtain ∆ z = σ for all z = 0, 1, ∞ and the only vertices in ∆ 0 and ∆ 1 are {v − 0 } and {0, v + 1 }, respectively. Finally, if C = P 1 , let v be a vertex of ∆ ∞ . By Corollary 1.3, we can assume that 2v − 0 (e) = 1, and Definition 1.9 (4) shows that for every vertex v of ∆ ∞ , we have v(e) = 0. This corresponds to the second case in the lemma. The proof is now completed.
To obtain a full classification of compatible SL 2 -actions of horizontal type on X we only need conditions for existence of the homogeneous LNDs ∂ ± of horizontal type on k[X] defined in (6) and (7).
The following theorem provides the announced classification of compatible SL 2 -actions of horizontal type on T-varieties of complexity one. For the theorem, we need the following notation. Let D be as in Lemma 2.17. Then D admits two different colorings as in (5). We let ω ± be the associated cone of D ± (see before Definition 1.9). Furthermore, for every e ∈ M , we let e ± = (±e, Theorem 2.18. Let X = X[C, D] be a normal affine T-variety of complexity one. Then X admits a compatible SL 2 -action of horizontal type that is not compatible with a bigger torus if and only if the following conditions hold. (i) The base curve C is either A 1 or P 1 .
(ii) There exists a lattice vector e ∈ M such that the σ-polyhedral divisor D may be shifted via Corollary 1.3 to one of the following two forms , v + 1 (e) = −1, and Π ⊆ e ⊥ is a bounded polyhedron; or , v + 1 (e) = −1, and Π ⊆ e ⊥ is a bounded polyhedron. (iii) The lattice vectors e ± ∈ M ⊕ Z are roots of the cones ω ± , respectively.
Moreover, if (C, σ, D) is in one of the two forms above, then the compatible SL 2 -action of horizontal type on X is given by the sl 2 -triple {δ, ∂ + , ∂ − } of derivations, where δ = [∂ + , ∂ − ], the homogeneous LNDs ∂ ± are given by the coherent pairs ( D ± , ±e), and D ± are the following colorings of D in the first case; or in the second case.
Proof. By Lemma 2.17, if X admits a compatible SL 2 -action of horizontal type that is not compatible with a bigger torus, then (i) and (ii) hold. Moreover, by the proof of Lemma 2.17, such an X admits a compatible SL 2 -action of horizontal type if and only if the derivations ∂ ± given by (6) and (7) define LNDs on k[X]. By Theorem 1.10, the derivations ∂ ± define LNDs on k[X] if and only if there exists e ∈ M such that ( D ± , ±e) are coherent pairs. Furthermore, ( D ± , ±e) are coherent pairs if and only if e ± is a root of the cone ω ± and Definition 1.9 (2)-(4) hold. It is a routine verification that Definition 1.9 (2)-(4) hold for ( D ± , ±e), and so the theorem is proved.

Special SL 2 -actions
In this section we give a classification of special SL 2 -actions on normal affine varieties. This generalizes Theorem 1 in [Arz97]. Let us first state the necessary definitions and results for an arbitrary reductive group.
Let G be a connected reductive algebraic group, T ⊆ B be a maximal torus and a Borel subgroup of G, and X + (G) be the semigroup of dominant weights of G with respect to the pair (T, B). Any regular action of the group G on an affine variety X defines a structure of rational G-algebra on the algebra of regular functions k[X]. In particular, we have the isotypic decomposition where k[X] λ is the sum of all the simple G-submodules in k[X] with the highest weight λ.
Definition 3.1. A G-action on X is called special (or horospherical), if there exists a dense open W ⊆ X such that the isotropy group of any point x ∈ W contains a maximal unipotent subgroup of the group G.
Remark 3.2. If a G-action is special, then the isotropy group G x contains a maximal unipotent subgroup for all x ∈ X.

Theorem 3.3 (See [Pop86, Theorem 5]). A G-action on an affine variety X is special if and only if
Corollary 3.4. For a special action, the isotypic decomposition is a X + (G)-grading on the algebra k[X]. This defines an action of an algebraic torus S on X, and this action commutes with the G-action.
Furthermore, since S acts on every isotypic component by scalar multiplication, every Ginvariant subspace in k[X] is S-invariant. In particular, S preserves every G-invariant ideal in k[X], and thus every G-invariant closed subvariety in X. This shows that the torus S preserves all G-orbit closures on X.
We return now to the case of SL 2 -actions on T-varieties.
Proposition 3.5. Every compatible SL 2 -action of fiber type on an affine T-variety X is special.
Proof. For a general x ∈ X, let Y = SL 2 ·x. Then Y ⊆ T · x. Denote by T Y the stabilizer of the subvariety Y in T. Since the torus T normalizes the SL 2 -action, it permutes SL 2 -orbit closures, and thus T Y acts on Y with an open orbit. Since T Y also normalizes the SL 2 -action, there exists a subtorus S Y ⊆ T Y of codimension 1 that centralizes the SL 2 -action on Y . In particular, it preserves the open orbit SL 2 ·x ֒→ Y . But SL 2 ·x ≃ SL 2 /H, where H is the isotropy group of x in SL 2 . We have S Y ⊆ Aut SL 2 (SL 2 /H) and thus rank Aut SL 2 (SL 2 /H) ≥ dim(SL 2 /H) − 1 . But Aut SL 2 (SL 2 /H) ≃ N SL 2 (H)/H and rank N SL 2 (H)/H ≤ 1, so dim(SL 2 /H) ≤ 2. If H coincides either with a maximal torus or with its normalizer in SL 2 , then the group N SL 2 (H)/H is finite, a contradiction. So, H is a finite extension of a maximal unipotent subgroup of SL 2 , and the SL 2 -action is special.
Corollary 3.6. Every compatible SL 2 -action on a toric variety is special.
In the following proposition we come to a partial converse of Proposition 3.5. Namely, we realize any special action of SL 2 as a compatible action of fiber type with respect to a canonical 2-dimensional torus action.
Proposition 3.7. Every special SL 2 -variety admits the action of a 2-dimensional torus such that the SL 2 -action is compatible with the torus action and of fiber type.
Proof. Let T 2 be the 2-dimensional torus T · S, where T is a maximal torus in SL 2 and S is one constructed in Corollary 3.4. By construction, the actions of T and S on X commute, preserve every SL 2 -orbit closure and T 2 has an open orbit on every such orbit closure.
In the following proposition we determine the special SL 2 -actions among the compatible SL 2actions on a complexity one affine T-variety.
Proposition 3.8. Let X be a normal affine T-variety of complexity one endowed with a compatible SL 2 -action. Then the SL 2 -action is special if and only if it is either of fiber type; or it is of horizontal type, X is toric, and the SL 2 -action is compatible with the big torus. In particular, the T-varieties of complexity one that admit a non-special compatible SL 2 -action are given in Theorem 2.18.
Proof. If the SL 2 -action is of fiber type, then the proposition follows from Proposition 3.5.
Assume that the SL 2 -action is of horizontal type and special. Since the SL 2 -action is compatible, T is a product of a maximal torus T of SL 2 and a subtorus T ′ which commutes with the SL 2 -action. In particular, T preserves all the SL 2 -isotypic components in k[X].
On the other hand, the one-dimensional torus S constructed in Corollary 3.4 acts on any isotypic component by a scalar multiplication. Thus S commutes with T. We know that general closures of the canonical 2-torus (T · S)-orbits coincide with closures of SL 2 -orbits (see Proposition 3.7). Since the SL 2 -action is compatible, T is contained in T and since the SL 2 -action is of horizontal, the closures of SL 2 -orbits are not contained in the closures of the T-orbits. Hence S is not contained in T, so we may extent T by S and get a big torus which acts on X with an open orbit.
For the rest of this section we let T 2 be a 2-dimensional algebraic torus. In the following, we give a description of compatible SL 2 -actions of fiber type on T 2 -varieties. By Propositions 3.5 and 3.7, this gives a description of all special SL 2 -actions on normal affine varieties.
The following example gives a construction of certain T 2 -varieties admitting a compatible SL 2 -action of fiber type.
Example 3.9. Let M be a lattice of rank 2, and σ be the cone spanned in N Q by the vectors (1, 0) and (r − 1, r), for some r ∈ Z >0 . By Example 2.9 the cone σ admits the SL 2 -root e = (1, −1).
We have D(e) = D(−e) = 0 and so Theorem 2.12 yields that the T 2 -variety X = X[Y, D] admits an SL 2 -action of fiber type and the generic isotropy subgroup in SL 2 is U (r) .
Furthermore, if we let X ′ be the T 2 -variety obtained with the above construction with the data Y and H replaced by Y ′ and H ′ , then X is isomorphic to X ′ if and only if Y ≃ Y ′ and under this isomorphism H is linearly equivalent to H ′ . Indeed, since H is ample [Dem88, Proposition 3.3] implies that Y is unique up to isomorphism. Finally, Corollary 1.3 shows that H and H ′ are linearly equivalent.
Proposition 3.10. Every T 2 -variety X endowed with an SL 2 -action of fiber type is isomorphic to one in Example 3.9 above.
Proof. Let X = X[Y, D] where D = Z ∆ Z · Z is a proper σ-polyhedral divisor on a semiprojective variety Y . Since X is endowed with an SL 2 -action of fiber type the cone σ admits an SL 2 -root. By Example 2.9 we can assume that σ is the cone spanned in N Q by the vectors (1, 0) and (r − 1, r). In this case e = (1, −1).
By Remark 2.13 (2) the σ-polyhedra ∆ Z is v Z + σ, where v Z ∈ N Q . The divisor D(e) is principal by Theorem 2.12 and is given by Furthermore, by Corollary 1.3 we can assume that D(e) = 0, so that for every Z v Z = α Z (1, 1) for some α Z ∈ Q .
The following theorem is a direct consequence of Proposition 3.7 and Proposition 3.10.
Theorem 3.11. Every normal affine variety X of dimension k + 2 endowed with a special SL 2action is uniquely determined by a positive integer r, a semiprojective variety Y of dimension k, and a linear equivalence class [H] of ample Q-Cartier divisors on Y .
Remark 3.12. The variety X can be recovered from the data in Theorem 3.11 as follows. Let σ be the cone spanned in N Q ≃ Q 2 by the rays (1, 0) and (r−1, r), r > 0, and let B s = H 0 (Y, O Y (sH)). Then X is equivariantly isomorphic to Spec A, where A is the M -graded algebra The pair (Y, [H]) defines, via Demazure's construction [Dem88], a T 1 -variety W of dimension k + 1. Then the variety W with the new non-effective T 1 -action given by T 1 → T 1 , t → t r , is nothing but Spec k[X] U + endowed with the action of the maximal torus T ⊆ SL 2 . The corresponding non-effective Z-grading on k[X] U + is given by From this classification we obtain the following corollary.
Corollary 3.13. Let X be an affine toric variety endowed with a special SL 2 -action. If the canonical torus T 2 of the special action is contained in the big torus, then the SL 2 -action is normalized by the big torus.
Proof. By Proposition 3.7 and Proposition 3.10 we can assume that X regarded as T 2 -variety is given by the combinatorial data X = X[Y, D], where Y is a normal semiprojective variety Y and D is the proper σ-polyhedral divisor given by where σ is the cone spanned in N Q ≃ Q 2 by the vectors (1, 0) and (r − 1, r), ∆ Z = α Z (1, 1) + σ, and Z α Z · Z is an ample Q-Cartier divisor on Y . In this case, the SL 2 -root of the cone σ is e = (1, −1). Furthermore, the SL 2 -action of fiber type corresponding to e is unique.
Since X is toric and T 2 is a subtorus of the big torus, by [AH06, Section 11] we can assume that Y is the toric variety given by a fan Σ ⊆ N Q and that D is supported in the toric divisors of Y . Denote by Z ρ the toric divisor corresponding to a ray ρ ∈ Σ(1). In this case, the X is the toric variety given by the cone σ in N Q ⊕ N Q spanned by (σ,0) and (∆ Zρ , ρ) ∀ρ ∈ Σ(1) .
Remark 3.14. In the case of a special SL 2 -action on a 3-dimensional toric variety X, by [BH03] the canonical torus T 2 is conjugated to a subtorus of the big torus. So up to conjugation in Aut(X), every special SL 2 -action on X is normalized by the big torus. In higher dimension, it is an open problem whether T 2 is conjugated to a subtorus of the big torus.
Corollary 3.15. Consider a special SL 2 -action on the affine space A s . Assume that the action of the canonical torus T 2 on A s is linearizable. Then there exists an SL 2 -equivariant isomorphism A s ≃ k 2 ⊕ k s−2 , where k 2 is the tautological SL 2 -module and the SL 2 -action on k s−2 is identical.
Proof. By Corollary 3.13 we may assume that the SL 2 -action on A s is normalized by the torus of all diagonal matrices T s and, moreover, this action is given by the SL 2 -root (1, −1, 0, . . . , 0).

Quasi-homogeneous SL 2 -threefolds
In this section we study SL 2 -actions with an open orbit on a normal affine threefold X.
4.1. SL 2 -threefolds via polyhedral divisors. It is a byproduct of a classification due to Popov [Pop73] that most quasi-homogeneous SL 2 -threefolds admit the action of a two dimensional torus making the SL 2 -action compatible (see below). Hence, they can be classified with the methods of Section 2.4. In this section T 2 denotes an algebraic torus of dimension 2, and so rank M = 2.
Proposition 4.1. Let X be an affine normal quasi-homogeneous SL 2 -threefold. Then X admit the action of a two dimensional torus T 2 making the SL 2 -action compatible except in the case where X is equivariantly isomorphic to SL 2 /H, with H < SL 2 non-commutative and finite. Furthermore, T 2 = T × R, where T is the maximal torus in SL 2 and R is a one-dimensional torus commuting with the SL 2 -action.
In the following we restrict to the case where X ≃ SL 2 /H with H < SL 2 non-commutative and finite so that X can be regarded as a T 2 -variety of complexity one. Up to conjugation, the only finite commutative subgroups of SL 2 are the cyclic groups Popov's classification is given in terms of the order of the generic stabilizer r X and the socalled height h X . We propose to replace the height by the slope X . See Definitions 4.6 and 4.11 for a precise definition. The main result of this section is the following theorem. The invariants r X , h X and X are given in the table below but the computation of their values will be performed after the proof of the theorem. In the table we also give the number N X of SL 2 -orbits of X. This number is only given for reference as it is not proved in the text.
Furthermore, X[C, D] is an (SL 2 /µ r )-embedding, and X is an homogeneous space of SL 2 if and only if σ = {0}.
Before proving this theorem, we need a preliminary result. Let X = X[C, D] for some σpolyhedral divisor on a smooth curve. Since the SL 2 -action has an open orbit, the general isotropy group is finite, and so the SL 2 -action in non-special. Hence, by Proposition 3.8 the SL 2 -action is of horizontal type, C = A 1 or C = P 1 , and we may and will assume in the sequel that D is as in Theorem 2.18.
We will use the following general lemma to identify the σ-polyhedral divisors in Theorem 2.18 that give rise to quasihomogeneous SL 2 -actions. Here the dimension of a domain A is the dimension of the algebraic variety Spec A.
Lemma 4.3. Let X be a normal affine variety endowed with a non-special SL 2 -action. Then X has finite generic stabilizer if and only if dim k[X] SL 2 ≤ dim X − 3.
Proof. Assume first that X has finite generic stabilizer. By Rosenlicht's Theorem, the transcendence degree of k(X) SL 2 equals the codimension of the general orbit [Dol03, Corollary 6.2], so Assume now that the generic stabilizer has positive dimension. Since the SL 2 -action is nonspecial, the generic stabilizer is one-dimensional and coincides either with T or N . In both cases the subgroup has a finite index in its normalizer in SL 2 . By [Lun75, Corollaire 3], we obtain that the general SL 2 -orbits are closed in X. Thus, they are separated by regular invariants and so dim k[X] SL 2 = dim X − 2.
We now proceed to the proof of the main theorem in this section.
Proof of Theorem 4.2. We prove first the "only if" part. Let X = X[C, D] be an affine T 2 -variety admitting a compatible SL 2 -action with an open orbit. We have C = A 1 or C = P 1 and we can assume that D is as in Theorem 2.18. Let ∂ ± be the homogeneous LNDs of horizontal type corresponding to the U ± -action on X and let ±e be the degree of ∂ ± .
Since v + 1 (e) = −1, e is a primitive lattice vector, so up to automorphism of the lattice M , we may and will assume e = (1, −1). Assume for a moment that C = P 1 . In this case the cone σ is full dimensional and so σ ∨ is pointed. Hence ±e / ∈ σ ∨ . This yields e ⊥ = Q(1, 1) intersects with σ only once and so ∆ ∞ = (a, a) + σ.
We have k[X] U ± = ker ∂ ± . Since SL 2 is generated by U ± as a group, we have k[X] SL 2 = ker ∂ + ∩ ker ∂ − . Hence, the compatible SL 2 -action on X[C, D] has an open orbit if and only if ker ∂ + ∩ ker ∂ − = k .
By Lemma 1.11, if deg D| A 1 has only two vertices then ker ∂ + ∩ ker ∂ − k. Hence the second family of σ-polyhedral divisors in Theorem 2.18 does not give quasihomogeneous SL 2 -threefolds. In the following, we assume that D is as in the first family in Theorem 2.18.
Up to an automorphism of the lattice N , we can assume v − 0 = (1, 0) and v + 1 = (r − 1, r) with r ∈ Z ≥0 . If r = 0 then again deg D| A 1 has only two vertices and so ker ∂ + ∩ ker ∂ − k. Hence, r ≥ 1. This shows that ∆ 0 and ∆ 1 have the form given in the theorem.
This yields ρ 1 = cone(a + 1, a) and ρ 2 = cone(r + a − 1, r + a) with a > 0. This proves the "only if" part. Now, let X = X[C, D] be as in the theorem. By Theorem 2.18, a simple verification shows that X admits an SL 2 -action and by Lemma 1.11 we have ker ∂ + ∩ ker ∂ − = k. Hence the SL 2 -action has finite generic stabilizer and so X is a quasi-homogeneous SL 2 -threefold.
The last assertion of the theorem is shown in Lemmas 4.4 and 4.5 below.
4.2. Parameters. In the remaining of this section we define and compute the parameters r X , h X , and X given in the table in Theorem 4.2. First, we give a geometric interpretation of the parameter r X . Proof. Assume that X is a homogeneous space. The statement is equivalent to the fact that the general orbit of the acting torus is closed. Let us consider G = SL 2 ×R, where R is the one-dimensional torus commuting with SL 2 , see Proposition 4.1 so that X is a homogeneous space of G. Then the acting torus T 2 = T × R is a reductive subgroup of G. By [Lun72] the general T 2 -orbit on X is closed.
To complete the proof, we only need to show that r in the definition of ∆ 1 is the order of the generic stabilizer of the SL 2 -action on X. By [Kra84, II.3.1, Satz 3], the algebra k[SL 2 ] seen as SL 2 × SL 2 -module has the isotypic decomposition where V (d) is the simple SL 2 -module of binary forms of degree d. The one-dimensional subtorus R commuting with the (left) SL 2 -action may be identified with the maximal torus in the second (right) SL 2 . Then, the homogeneous space SL 2 /µ r is obtained as the quotient of SL 2 by the cyclic subgroup of order r in R. So simple SL 2 -submodules in k[SL 2 /µ r ] have the form V (d)⊗w, where w runs through R-weight vectors of V (d) whose weight is divisible by r. Now, the subalgebra of U + -invariants of k[SL 2 /µ r ] is spanned by the elements v ⊗ w ∈ V (d) ⊗ w, where v is highest weight vector in V (d). Let T be the maximal torus in the (left) SL 2 -action. We have shown that the order r of the generic stabilizer r is the minimal integer such that ker ∂ + contains a T -weight vector of weight r which is not R-invariant.
Hence, the minimal weight such that the ker ∂ + contains a T -weight vector which is not Rinvariant is r, and the lemma follows.
Lemma 4.5. Let X = X[C, D] be as in Theorem 4.2. Then r = r X is the order of the generic stabilizer and X is an (SL 2 /µ r )-embedding.
Proof. In the following we assume that X is not equivariantly isomorphic to a homogeneous space. Let r be the order of the generic stabilizer of X. The open embedding SL 2 /µ r ֒→ X induces an inclusion of the algebras of U + -invariants k[X] U + ֒→ k[SL 2 /µ r ] U + . Both these algebras are semigroup algebras. Moreover, the cones of these semigroup algebras share a common ray in M Q . This ray will be denoted by ρ U + .
Let ω ⊆ M Q ≃ Q 2 be a full dimensional cone and let ρ be one of its rays. It is well known that, up to automorphism of the lattice M , we can assume ρ = cone ((1, 0)) and ω = cone((1, 0), (b, c)) with 1 ≤ b ≤ c and gcd(b, c) = 1. We define the slope of ω with respect to ρ as b /c ∈ Q ∩ (0, 1].
Definition 4.6. Let X be a non-homogeneous quasi-homogeneous SL 2 -threefold. We define the slope X of X as the slope of the cone of the ring of U + -invariants with respect to the ray ρ U + .
Remark 4.7. This definition does not coincide with the height defined by Popov and used in the literature [Pop73,Kra84,Gaȋ08,BH08]. The height will be introduced below and denoted by the plain letter h X . We will also show the relation between slope and height. The main motivation to use a different definition is that the results have simpler statements.
Let X = X[C, D] be as in Theorem 4.2 and assume that X is not a homogeneous space. If C = A 1 and σ = cone ((1, 1)), then by Theorem 1.10 the cone of the ring of U + -invariants k[X] U + is given by cone((0, 1)(−1, 1)) and so the slope of X is X = 1. Assume now that C = P 1 . In this case, by Theorem 1.10, the cone of the ring k[X] U + is given by cone((0, 1), (−a, a + 1)) and the common ray of the cones of k[X] U + and k[SL 2 /µ r ] U + is spanned by the lattice vector (0, 1). Hence, the slope of X is Since the function defining X in terms of a is one to one, we have the following corollary.
Corollary 4.8. Two non-homogeneous quasi-homogeneous SL 2 -threefolds X and X ′ are equivariantly isomorphic if and only if r X = r X ′ and X = X ′ .
In the following corollary we give a criterion for a quasihomogeneous SL 2 -threefold to be toric. This result is also given in [Gaȋ08] and [BH08] in terms of the height of X.
Corollary 4.9. Let X be a quasi-homogeneous SL 2 -threefold. Then X is a toric variety if and only if X is non-homogeneous and X = p /p+1 for some p ∈ Z >0 .
Proof. Let X ≃ X[C, D] with C = A 1 or C = P 1 and D as in Theorem 4.2. By Corollaries 1.3 and 1.4, we obtain that X is toric if and only if C = P 1 and a is an integer. Let now = p /q with gcd(p, q) = 1 and p, q ≥ 0. By (12) we have a = p /q−p and so the result follows.
Remark 4.10. In Corollary 4.9, the SL 2 -action is not compatible with the big torus, since otherwise the SL 2 -action would be special. 4.3. Relation between slope and height. Let X be a non-homogeneous quasi-homogeneous SL 2 -threefold.
Definition 4.11. The height h X of X is defined as follows. If r X = 1 then the height of X is the same as the slope of X i.e., h X = X . If r X > 1 then there is a unique non-homogeneous quasi-homogeneous SL 2 -threefold X ′ with r X ′ = 1 such that X = X ′ /µ r (see [Pop73] or [Kra84, III.4.9, Satz 1]). The height of X is defined as the slope of X ′ i.e., h X = X ′ .
In this section we compute the height of X in terms of the slope and the order of the generic stabilizer. We also state Corollary 4.9 in terms of the height.
Corollary 4.12. Let X be a non-homogeneous quasi-homogeneous SL 2 -threefold. Then Finally, a direct computation shows that in terms of the height Corollary 4.9 takes the form as in [Gaȋ08,BH08].
Corollary 4.13. Let X be a non-homogeneous quasi-homogeneous SL 2 -threefold and let h X = p /q, where gcd(p, q) = 1 and p, q > 0. Then X is a toric variety if and only if q − p divides r.

4.4.
Generically transitive SL 2 ×T s -action. Consider now the reductive group G = SL 2 ×T s for some s ∈ Z ≥0 . Any action of this group on a normal affine variety is compatible with the action of the torus T = T × T s , where T ⊆ SL 2 is a maximal torus. The results of Section 2.4 may be regarded as a classification of generically transitive G-actions under the assumption that the complexity of the corresponding T-action does not exceed one. In Section 4 we deal with the case s = 1 and because of Proposition 4.1 this yields a classification of generically transitive Gactions with s = 0. The following example shows that our techniques does not allow to describe all generically transitive G-actions with s = 1.
Example 4.14. Let G = SL 2 ×k * and X = V 3 = x 3 , x 2 y, xy 2 , y 3 be a simple SL 2 -module of binary forms of degree 3, where k * acts by scalar multiplication. The module V 3 contains a 1-parameter family of general SL 2 -orbits, namely SL 2 ·(α(x 3 + y 3 )), α ∈ k \ {0}. Thus, G acts on V 3 with an open orbit isomorphic to G/µ 2 . The torus T = T × k * acts on X with complexity two. Assume that T may be extended by a torus R commuting with G. Then R commutes with k * and its action descends to the projectivization P(V 3 ). Then R maps to a subtorus of PGL 4 . Since V 3 is simple, by Schur's Lemma there are no non-identity elements in PGL 4 commuting with the image of SL 2 .

Appendix: The commutator formulas
In this appendix we prove the commutator formulas (9) and (10) used in Section 2.4. The computations are routine, but cumbersome, so we put them in an appendix instead of the main body of the paper for easy reading.
We keep the notation as in Section 2.4. The main idea is that from (6) and (7) we can obtain formulas for ∂ ± (χ m ) and for ∂ ± (t) by applying the Leibniz rule. Then we use this formulas to compute the commutator δ = [∂ + , ∂ − ].