Equivariant Embeddings of Commutative

Linear Algebraic Groups of Corank One

Ivan Arzhantsev, Polina Kotenkova

Received: January 14, 2015 Revised: June 20, 2015

Communicated by Thomas Peternell

Abstract. Let 𝕂 be an algebraically closed field of characteristic zero, 𝔾_{m} = (𝕂 \ {0}, ×) be its multiplicative group, and 𝔾_{a} = (𝕂, +) be its additive group. Consider a commutative linear algebraic group 𝔾 = (𝔾_{m})^{r} × 𝔾_{a}. We study equivariant 𝔾-embeddings, i.e. normal 𝔾-varieties X containing 𝔾 as an open orbit. We prove that X is a toric variety and all such actions of 𝔾 on X correspond to Demazure roots of the fan of X. In these terms, the orbit structure of a 𝔾-variety X is described.

2010 Mathematics Subject Classification: Primary 14M17, 14M25, 14M27; Secondary 13N15, 14J50 Keywords and Phrases: Toric variety, Cox ring, locally nilpotent derivation, Demazure root

1. Introduction

Let 𝕂 be an algebraically closed field of characteristic zero, 𝔾_{m} = (𝕂 \ {0}, ×) be its multiplicative group, and 𝔾_{a} = (𝕂, +) be its additive group. It is well known that any connected commutative linear algebraic group 𝔾 over 𝕂 is isomorphic to (𝔾_{m})^{r} × (𝔾_{a})^{s} with some non-negative integers r and s, see [20, Theorem 15.5]. We say that r is the rank of the group 𝔾 and s is the corank of 𝔾. The aim of this paper is to study equivariant embeddings of commutative linear algebraic groups. Let us recall that an equivariant embedding of an algebraic group G is a pair (X, x), where X is an algebraic variety equipped with a regular action G × X → X and x ∈ X is a point with the trivial stabilizer such that the orbit Gx is open and dense in X. We assume that the variety X is normal. If X is supposed to be complete, we speak about equivariant compactifications of G. For the study of compactifications of reductive groups, see e.g. [26]. More generally, equivariant embeddings of homogeneous spaces of reductive groups is a popular object starting from early 1970th. Recent survey of results in this field may be found in [27]. Let us return to the case 𝔾 = (𝔾_{m})^{r} × (𝔾_{a})^{s}. If s = 0 then 𝔾 is a torus and we come to the famous theory of toric varieties, see [13], [23], [18], [12]. Another extreme r = 0 corresponds to embeddings of a commutative unipotent (=vector) group. This case is also studied actively during last decades, see [19], [8], [3], [16], [14]. The next natural step is to study the mixed case r > 0 and s > 0 and to combine advantages of both torus and additive group actions. The present paper deals with the case s = 1, i.e. from now on 𝔾 is a connected commutative linear algebraic group of corank one. In other words, 𝔾 = (𝔾_{m})^{n−1} × 𝔾_{a}, where n = dim X. Let X be a toric variety with the acting torus 𝕋. Consider an action 𝔾_{a} ×X → X normalized by 𝕋. Then 𝕋 acts on 𝔾_{a} by conjugation with some character e. Such a character is called a Demazure root of X. If T = Ker(e), then the group 𝔾 := T ×𝔾_{a} acts on X with an open orbit, and X is a 𝔾-embedding, see Proposition 6. Our main result (Theorem 2) states that all 𝔾-embeddings can be realized this way. To this end we prove that for any 𝔾-embedding X the (𝔾_{m})^{n−1}-action on X can be extended to an action of a bigger torus 𝕋 which normalizes the 𝔾_{a}-action and X is toric with respect to 𝕋. This result can not be generalized to groups of corank two; examples of nontoric surfaces which are equivariant compactifications of 𝔾_{a}^{2} can be found in [14]. Similar examples are constructed in [14], [15] for semidirect products 𝔾_{m} ⋌𝔾_{a}. Such groups can be considered as non-commutative groups of corank one. If two toric varieties are isomorphic as abstract varieties, then they are isomorphic as toric varieties [10, Theorem 4.1]. This shows that the structure of a torus embedding on a toric variety is unique up to isomorphism. A structure of a 𝔾-embedding on a given variety may be non-unique, see Examples 2, 4. Such structures are given by Demazure roots and thus the number of structures is finite if X is complete, and it is at most countable for arbitrary X. At the same time, 𝔾_{a}^{6}-embeddings into ℙ^{6} admit a non-trivial moduli space [19, Example 3.6]. The paper is organized as follows. Section 2 contains preliminaries on torus actions on affine varieties. We recall basic facts on affine toric varieties and introduce a description of affine T -varieties in terms of proper polyhedral divisors due to Altmann and Hausen [1]. A correspondence between 𝔾_{a}-actions on X normalized by T and homogeneous locally nilpotent derivations (LNDs) of the algebra 𝕂[X] is explained. We define Demazure roots of a cone and use them to describe homogeneous LNDs on 𝕂[X], where X is toric. Also we give a description of homogeneous LNDs of horizontal type on algebras with grading of complexity one obtained by Liendo [22]. In Section 3 we show that if X is a normal affine T -variety of complexity one and the algebra 𝕂[X] admits a homogeneous LND of degree zero, then X is toric with an acting torus 𝕋, T is a subtorus of 𝕋, and 𝕋 normalizes the corresponding 𝔾_{a}-action. This gives the result for affine 𝔾-embeddings. Moreover, Proposition 3 provides an explicit description of affine 𝔾-embeddings.

Section 4 deals with compactifications of 𝔾. Here we use the Cox construction and a lifting of the action of 𝔾 to the total coordinate space \overline{X} of X to deduce the result from the affine case. In Section 5 we recall basic facts on toric varieties and introduce the notion of a Demazure root of a fan following Demazure [13]. The action of the corresponding one-parameter subgroup on the toric variety is also described there. Let Σ be a fan and e be a Demazure root of Σ. In Section 6 we define a 𝔾embedding associated to the pair (Σ, e) and study the 𝔾-orbit structure of X. It turns out that the number of 𝔾-orbits on X is finite. Finally, in Section 7 we prove that any 𝔾-embedding is associated with some pair (Σ, e). The idea is to reduce the general case to the complete one via equivariant compactification. At the end several explicit examples of 𝔾-embeddings are given. Some results of this paper appeared in preprint [5]. They form a part of the Ph.D. thesis of the second author [21]. The authors are grateful to the referee for careful reading of the paper and valuable suggestions.

2. 𝔾_{a}-actions on affine T -varieties

Let X be an irreducible affine variety with an effective action of an algebraic torus T, M be the character lattice of T, N be the lattice of one-parameter subgroups of T, and A = 𝕂[X] be the algebra of regular functions on X. It is well known that there is a bijective correspondence between effective T -actions on X and effective M-gradings on A. In fact, the algebra A is graded by a semigroup of lattice points in some convex polyhedral cone ω ⊆ M_{ℚ} = M ⊗_{ℤ} ℚ. So we have

A = ⨁_{m∈ω_{M}} A_{m}χ^{m},

where ω_{M} = ω ∩ M and χ^{m} is the character corresponding to m. A derivation ∂ on an algebra A is said to be locally nilpotent (LND) if for each f ∈ A there exists n ∈ ℕ such that ∂^{n}(f) = 0. For any LND ∂ on A the map φ_{∂} : 𝔾_{a} × A → A, φ_{∂}(s, f) = exp(s∂)(f), defines a structure of a rational 𝔾_{a}- algebra on A. This induces a regular action 𝔾_{a} × X → X, where X = Spec A. In fact, any regular 𝔾_{a}-action on X arises this way, see [17, Section 1.5]. A derivation ∂ on A is said to be homogeneous if it respects the M-grading. If f, h ∈ A\ ker ∂ are homogeneous, then ∂(fh) = f∂(h) + ∂(f)h is homogeneous too and deg ∂(f) − deg f = deg ∂(h) − deg h. So any homogeneous derivation ∂ has a well defined degree given as deg ∂ = deg ∂(f)−deg f for any homogeneous f ∈ A\ ker ∂. It is easy to see that an LND on A is homogeneous if and only if the corresponding 𝔾_{a}-action is normalized by the torus T in the automorphism group Aut(X), cf. [17, Section 3.7]. Any derivation on 𝕂[X] extends to a derivation on the field of fractions 𝕂(X) by the Leibniz rule. A homogeneous LND ∂ on 𝕂[X] is said to be of fiber type if ∂(𝕂(X)^{T}) = 0 and of horizontal type otherwise. In other words, ∂ is of fiber type if and only if the general orbits of corresponding 𝔾_{a}-action on X are contained in the closures of T -orbits. Let X be an affine toric variety, i. e. a normal affine variety with a generically transitive action of a torus T. In this case

A = ⨁_{m∈ω_{M}} 𝕂χ^{m} = 𝕂[ω_{M}]

is the semigroup algebra. Recall that for given cone ω ⊂ M_{ℚ}, its dual cone is defined by

σ = {n ∈ N_{ℚ} | ⟨n, p⟩ ⩾ 0 ∀p ∈ ω},

where ⟨, ⟩ is the pairing between dual lattices N and M. Let σ(1) be the set of rays of a cone σ and n_{ρ} be the primitive lattice vector on the ray ρ. For ρ ∈ σ(1) we set

S_{ρ} := {e ∈ M | ⟨n_{ρ}, e⟩ = −1 and ⟨n_{ρ′}, e⟩ ⩾ 0 ∀ ρ′ ∈ σ(1), ρ′ ≠ ρ}.

One easily checks that the set S_{ρ} is infinite for each ρ ∈ σ(1). The elements of the set ℜ := ⨆_{ρ} S_{ρ} are called the Demazure roots of σ. Let e ∈ S_{ρ}. Then ρ

is called the distinguished ray of the root e. One can define the homogeneous LND on the algebra A by the rule

∂_{e}(χ^{m}) = ⟨n_{ρ}, m⟩χ^{m+e}.

In fact, every homogeneous LND on A has a form α∂_{e} for some α ∈ 𝕂, e ∈ ℜ, see [22, Theorem 2.7]. In other words, 𝔾_{a}-actions on X normalized by the acting torus are in bijection with Demazure roots of the cone σ. Clearly, all homogeneous LNDs on a toric variety are of fiber type.

Example 1. Consider X = 𝔸^{k} with the standard action of the torus (𝕂^{×})^{k}. It is a toric variety with the cone σ = ℚ_{⩾0}^{k} having rays ρ_{1} = ⟨(1, 0, …, 0)⟩_{ℚ_{⩾0}}, …, ρ_{k} = ⟨(0, 0, …, 0, 1)⟩_{ℚ_{⩾0}}. The dual cone ω is ℚ_{⩾0}^{k} as well. In this case

S_{ρ_{i}} = {(c_{1}, …, c_{i−1}, −1, c_{i+1}, …, c_{k}) | c_{j} ∈ ℤ_{⩾0}}.

[figure: S_{ρ_{1}}; S_{ρ_{2}}; M_{ℚ} = ℚ^{2}]

Denote x_{1} = χ^{(1,0,...,0)}, …, x_{k} = χ^{(0,...,0,1)}. Then 𝕂[X] = 𝕂[x_{1}, …, x_{k}]. It is easy to see that the homogeneous LND corresponding to the root e = (c_{1}, …, c_{k}) ∈ S_{ρ_{i}} is

∂_{e} = x_{1}^{c_{1}} … x_{i−1}^{c_{i−1}}x_{i+1}^{c_{i+1}} … x_{k}^{c_{k}} \frac{∂}{∂x_{i}} .

This LND gives rise to the 𝔾_{a}-action

x_{i} ↦ x_{i} + sx_{1}^{c_{1}} … x_{i−1}^{c_{i−1}}x_{i+1}^{c_{i+1}} … x_{k}^{c_{k}}, x_{j} ↦ x_{j}, j ≠ i, s ∈ 𝔾_{a}.

Let us recall that the complexity of an action of a torus T on an irreducible variety X is the codimension of a general T -orbit on X, or, equivalently, the transcendence degree of the field of rational invariants 𝕂(X)^{T} over 𝕂. In particular, actions of complexity zero are precisely actions with an open T - orbit. Now we recall a description of normal affine T -varieties of complexity one in terms of proper polyhedral divisors. Let N and M be two mutually dual lattices with the pairing denoted by ⟨, ⟩, σ be a strongly convex cone in N_{ℚ}, and ω ⊆ M_{ℚ} be the dual cone. A polyhedron Δ ⊆ N_{ℚ}, which can be decomposed as Minkowski sum of a bounded polyhedron and the cone σ, is called σ-tailed. Let C be a smooth curve. A σ-polyhedral divisor on C is a formal sum

𝔇 = ∑_{z∈C} Δ_{z} · z,

where Δ_{z} are the σ-tailed polyhedra and only finite number of them are not equal to σ. The divisor 𝔇 is trivial, if Δ_{z} = σ for all z ∈ C. The finite set Supp 𝔇 := {z ∈ C | Δ_{z} ≠ σ} is called the support of 𝔇. For every m ∈ ω_{M} we can obtain the ℚ-divisor 𝔇(m) = ∑_{z∈C} h_{z}(m) · z, where h_{z}(m) :=

min_{p∈Δ_{z}}⟨p, m⟩. So a σ-polyhedral divisor is just a piecewise-linear function from

ω_{M} to the group of ℚ-divisors on C. One can define the M-graded algebra

A[C, 𝔇] = ⨁_{m∈ω_{M}} A_{m}χ^{m},

where

A_{m} = H^{0}(C, 𝔇(m)) := {f ∈ 𝕂(X) | div f + 𝔇(m) ⩾ 0},

where the multiplication of homogeneous elements is given as in 𝕂(X). A σ-polyhedral divisor on smooth curve C is called proper if either C is affine, or C is projective and the polyhedron deg 𝔇 := ∑_{z∈C} Δ_{z} is a proper subset of σ.

The next theorem expresses the main results of [1] specialized to the case of torus actions of complexity one.

Theorem 1. (1) Let C be a smooth curve and 𝔇 a proper σ-polyhedral divisor on C. Then the M-graded algebra A[C, 𝔇] is a normal finitely generated effectively graded (rk M + 1)-dimensional domain. Conversely, for each normal finitely generated domain A with a grading of complexity one there exist a smooth curve C and a proper σ-polyhedral divisor 𝔇 on C such that A is isomorphic to A[C, 𝔇].

(2) The M-graded domains Spec A[C, 𝔇] and Spec A[C, 𝔇′] are isomorphic if and only if for every z ∈ C there exists a lattice vector v_{z} ∈ N such that

𝔇 = 𝔇′ + ∑_{z}(v_{z} + σ) · z,

and for all m ∈ ω_{M} the divisor ∑_{z}⟨v_{z}, m⟩ · z is principal.

The following result is obtained in [1, Section 11].

Proposition 1. Let 𝔇 be a proper σ-polyhedral divisor on a smooth curve C, X = Spec A[C, 𝔇], and T × X → X be the corresponding torus action. Then this action can be realized as a subtorus action an a toric variety if and only if either C = 𝔸^{1} and 𝔇 can be chosen supported in at most one point, or C = ℙ^{1} and 𝔇 can be chosen supported in at most two points.

Also we need a description of homogeneous LNDs of horizontal type for a T - variety X of complexity one from [22]. Below we follow the approach given in [7]. We have 𝕂[X] = A[C, 𝔇] for some C and 𝔇. It turns out that C is isomorphic to 𝔸^{1} or ℙ^{1} whenever there exists a homogeneous LND of horizontal type on A[C, 𝔇], see [22, Lemma 3.15]. Let C be 𝔸^{1} or ℙ^{1}, 𝔇 = ∑_{z∈C} Δ_{z} · z a σ-polyhedral divisor on C, z_{0} ∈ C,

z_{∞} ∈ C\{z_{0}}, and v_{z} a vertex of Δ_{z} for every z ∈ C. Put C′ = C if C = 𝔸^{1} and C′ = C\{z_{∞}} if C = ℙ^{1}. A collection 𝔇̃ = {𝔇, z_{0}; v_{z}, ∀z ∈ C} if C = 𝔸^{1} and 𝔇̃ = {𝔇, z_{0}, z_{∞}; v_{z}, ∀z ∈ C′} if C = ℙ^{1} is called a colored σ-polyhedral divisor on C if the following conditions hold: (∗) v_{deg} := ∑_{z∈C′} v_{z} is a vertex of deg 𝔇|_{C′}:= ∑_{z∈C′} Δ_{z};

(∗∗) v_{z} ∈ N for all z ∈ C′, z ≠ z_{0}. Let 𝔇̃ be a colored σ-polyhedral divisor on C and δ ⊆ N_{ℚ} be the cone generated by deg 𝔇|_{C′} −v_{deg}. Denote by σ̃ ⊆ (N ⊕ ℤ)_{ℚ} the cone generated by (δ, 0) and (v_{z_{0}}, 1) if C = 𝔸^{1}, and by (δ, 0), (v_{z_{0}}, 1) and (Δ_{z_{∞}} + v_{deg} − v_{z_{0}} + δ, −1) if C = ℙ^{1}. By definition, put d the minimal positive integer such that d·v_{z_{0}} ∈ N. A pair (𝔇̃, e), where e ∈ M, is said to be coherent if

(i) there exists s ∈ ℤ such that ẽ = (e, s) ∈ M ⊕ ℤ is a Demazure root of the cone σ̃ with distinguished ray ρ̃ = (d · v_{z_{0}}, d);

(ii) ⟨v, e⟩ ⩾ 1 + ⟨v_{z}, e⟩ for all z ∈ C′\{z_{0}} and all vertices v ≠ v_{z} of the

polyhedron Δ_{z}; (iii) d · ⟨v, e⟩ ⩾ 1 + ⟨v_{z_{0}}, e⟩ for all vertices v ≠ v_{z_{0}} of the polyhedron Δ_{z_{0}}; (iv) if Y = ℙ^{1}, then d · ⟨v, e⟩ ⩾ −1 − d · ∑_{z∈Y′}⟨v_{z}, e⟩ for all vertices v of the

polyhedron Δ_{z_{∞}}.

It follows from [7, Theorem 1.10] that homogeneous LNDs of horizontal type on A[C, 𝔇] are in bijection with the coherent pairs (𝔇̃, e). Namely, let (𝔇̃, e) be a coherent pair. Without loss of generality we may assume that z_{0} = 0, z_{∞} = ∞ if C = ℙ^{1}, and v_{z} = 0 ∈ N for all z ∈ C′\{z_{0}}. Let 𝕂(C) = 𝕂(t). Then the homogeneous LND of horizontal type corresponding to (𝔇̃, e) is given by

(1)  ∂(χ^{m} · t^{r}) = d(⟨v_{0}, m⟩ + r)χ^{m+e} · t^{r+s} for all m ∈ M, r ∈ ℤ.

In particular, the vector e is the degree of the derivation ∂.

3. The affine case

Let (X, x) be an equivariant embedding of the group 𝔾 = (𝔾_{m})^{n−1} × 𝔾_{a}, where n = dim X. In this section we assume that X is normal and affine. Let us denote the subgroup (𝔾_{m})^{n−1} of 𝔾 by T. Since the action of T on X is effective, it has complexity one and defines an effective grading of the algebra 𝕂[X] by the lattice M. In particular, the graded algebra 𝕂[X] has the form A[C, 𝔇] for some smooth curve C and some proper σ-polyhedral divisor on C, where σ is a cone in N_{ℚ}. Since the action of the subgroup 𝔾_{a} commutes with T -action on X, the corresponding homogeneous LND on 𝕂[X] has degree zero. Moreover, the group 𝔾 acts on X with an open orbit. It implies that the 𝔾_{a}-action on X is of horizontal type, and hence either C = 𝔸^{1} or C = ℙ^{1}.

Proposition 2. Let X = Spec A[C, 𝔇] be a T -variety of complexity one. Suppose that there exists a homogeneous LND of horizontal type and of degree zero on A[C, 𝔇]. Then

(1) if C = 𝔸^{1}, then one can assume (via Theorem 1) that 𝔇 is a trivial σ-polyhedral divisor;

(2) if C = ℙ^{1}, then one can choose 𝔇 = Δ_{∞} · [∞], where Δ_{∞} ⊊ σ is some σ-tailed polyhedron.

Proof. Let (𝔇̃, 0) be the coherent pair corresponding to the homogeneous LND of horizontal type. Without loss of generality we may assume that z_{0} = 0 and z_{∞} = ∞ if C = ℙ^{1}. By definition of a coherent pair, there exists s ∈ ℤ such that (0, s) is a Demazure root of the cone σ̃ with distinguished ray (dv_{0}, d). It implies that s = −1, d = 1, and hence v_{0} ∈ N. Further, the inequality ⟨v, 0⟩ ⩾ 1 + ⟨v_{z}, 0⟩ should be satisfied for every z ∈ C′ and every vertex v ≠ v_{z} of Δ_{z}. It means that each polyhedron Δ_{z}, where z ∈ C′, has only one vertex v_{z}. Replacing σ-polyhedral divisor 𝔇 with 𝔇′ = 𝔇 + ∑_{z∈C′}(−v_{z} + σ) · z and using Theorem 1, we obtain the assertion. The condition Δ_{∞} ⊊ σ follows from the fact that 𝔇 is a proper σ-polyhedral divisor. □

Corollary 1. Under the conditions of Proposition 2 the variety X is toric with T being a subtorus of the acting torus 𝕋.

Proof. It follows immediately from Propositions 1 and 2. □

The next proposition is a specification of Corollary 1. In particular, it shows that the 𝔾_{a}-action on X is normalized by the acting torus 𝕋.

Proposition 3. Under the conditions of Proposition 2,

(1) if C = 𝔸^{1}, then X ≅ Y ×𝔸^{1}, where Y is the toric variety corresponding to the cone σ and 𝔾_{a} acts on 𝔸^{1} by translations;

(2) if C = ℙ^{1}, then X is the toric variety with the cone σ̃ ⊂ N ⊕ ℤ generated by (σ, 0), (Δ_{∞}, −1) and (0, 1). The 𝔾_{a}-action on X is given by Demazure root ẽ = (0, −1) ∈ M ⊕ ℤ of the cone σ̃.

Proof. Let 𝕂(C) = 𝕂(t). If C = 𝔸^{1} then 𝔇 is trivial and

A[C, 𝔇] = ⨁_{m∈ω_{M}} 𝕂[t] · χ^{m} = 𝕂[ω_{M}] ⊗ 𝕂[t] = 𝕂[Y] ⊗ 𝕂[t].

Hence X ≅ Y × 𝔸^{1}. Applying formula (1), we obtain that the homogeneous LND is given by

(2)  ∂(χ^{m} · t^{r}) = rχ^{m} · t^{r−1}

for all m ∈ ω_{M} and r ∈ ℤ_{⩾0}. Thus 𝔾_{a} acts on Y × 𝔸^{1} as (y, t) ↦ (y, t + s). If C = ℙ^{1} then 𝔇 = Δ_{∞} · [∞] and we obtain

A[C, 𝔇] = ⨁_{m∈ω_{M}} ⨁_{r=0}^{h_{∞}(m)} 𝕂χ^{m} · t^{r} = ⨁_{(m,r)∈ω̃M̃} 𝕂χ^{m} · t^{r} = 𝕂[ω̃M̃],

where M̃ = M ⊕ ℤ and ω̃ ⊂ M̃_{ℚ} is the cone dual to σ̃. So we see that A[C, 𝔇] is a semigroup algebra and X is a toric variety with the cone σ̃. In this case formula (2) gives the LND corresponding to the Demazure root ẽ = (0, −1). □

4. The complete case

In this section we study equivariant compactifications of the group 𝔾. First we briefly recall the main ingredients of the Cox construction, see [4, Chapter 1] for more details. Let X be a normal variety with finitely generated divisor class group Cl(X) and only constant invertible regular functions. Suppose that Cl(X) is free. Denote by WDiv(X) the group of Weil divisors on X and fix a subgroup K ⊆ WDiv(X) which maps onto Cl(X) isomorphically. The Cox ring of the variety X is defined as

R(X) = ⨁_{D∈K} H^{0}(X, D),

where H^{0}(X, D) = {f ∈ 𝕂(X) | div f + D ⩾ 0} and multiplication on homogeneous components coincides with multiplication in 𝕂(X) and extends to R(X) by linearity. If Cl(X) has torsion, we choose a finitely generated subgroup K ⊆ WDiv(X) that projects to Cl(X) surjectively. Denote by K_{0} ⊂ K the kernel of this projection. Take compatible bases D_{1}, …, D_{s} and D_{1}^{0} = d_{1}D_{1}, …, D_{r}^{0} = d_{r}D_{r} in K and K_{0} respectively. Let us choose the set of rational functions ℱ = {F_{D} ∈ 𝕂(X)^{×} : D ∈ K_{0}} such that div(F_{D}) = D and F_{D+D′} = F_{D}F_{D′}. Suppose that D, D′ ∈ K and D − D′ ∈ K_{0}. The map f ↦ F_{D−D′}f is an isomorphism of the vector spaces H^{0}(X, D) and H^{0}(X, D′). The linear span of the elements f − F_{D−D′}f over all D, D′ with D − D′ ∈ K_{0} and all f ∈ H^{0}(X, D) is an ideal I(K, ℱ) of the graded ring T_{K}(X) := ⨁_{D∈K} H^{0}(X, D). The Cox ring of the variety X is given by

R(X) = T_{K}(X)/I(K, ℱ).

This construction does not depend on the choice of K and ℱ, see [2, Lemma 3.1 and Proposition 3.2] and [4, Proposition 1.4.2.2]. Suppose that the Cox ring R(X) is finitely generated. Then \overline{X} := Spec R(X) is a normal affine variety with an action of the quasitorus H_{X} := Spec 𝕂[Cl(X)]. There is an open H_{X}-invariant subset X̂ ⊆ \overline{X} such that the complement \overline{X}\X̂ is of codimension at least two in \overline{X}, there exists a good quotient p_{X}: X̂ → X̂//H_{X}, and the quotient space X̂//H_{X} is isomorphic to X, see [4, Construction 1.6.3.1]. So we have the following diagram

X̂ ⟶^{i} \overline{X} = Spec R(X)
↓^{//H_{X}}
X

Let us return to equivariant compactifications of 𝔾.

Proposition 4. Let 𝔾 = T × 𝔾_{a} and X be a normal compactification of 𝔾. Then the T -action on X can be extended to an action of a bigger torus 𝕋 such that 𝕋 normalizes 𝔾_{a} and X is a toric variety with the acting torus 𝕋.

Proof. The variety X is rational with torus action of complexity one. By [4, Theorem 4.3.1.5], the divisor class group Cl(X) and the Cox ring R(X) are finitely generated. There exists a finite epimorphism ϵ: 𝔾′ → 𝔾 of connected linear algebraic groups and an action 𝔾′ × X̂ → X̂ which commutes with the quasitorus H_{X} and p_{X}(g′ ·x̂) = ϵ(g′)·p_{X}(x̂) for all g′ ∈ 𝔾′ and x̂ ∈ X̂, see [4, Theorem 4.2.3.1]. The group 𝔾′ has a form T′ × 𝔾_{a}, where ϵ defines a finite epimorphism of tori T′ → T and is identical on 𝔾_{a}. Since \overline{X} = Spec 𝕂[X̂], the action of 𝔾′ extends to the affine variety \overline{X}. This variety is an embedding of the group (T′H_{X}^{0})×𝔾_{a}. By Proposition 3, it is toric with an acting torus \overline{𝕋} normalizing the 𝔾_{a}-action and T′H_{X}^{0} is a subtorus of \overline{𝕋}. Since X is complete, [25, Corollary 2.5] implies that the subset X̂ is invariant under the torus \overline{𝕋}. By [4, Lemma 4.2.1.3], the action of \overline{𝕋} descends to an action of the torus 𝕋 := \overline{𝕋}/H_{X}^{0} on X. Here 𝕋 normalizes 𝔾_{a}, its action extends the action of T on X, and X is toric with respect to 𝕋. □

5. Toric varieties and Demazure roots

We keep notations of Section 2. Let X be a toric variety of dimension n with an acting torus 𝕋 and Σ be the corresponding fan of convex polyhedral cones in the space N_{ℚ}, see [18] or [12] for details. As before, let Σ(1) be the set of rays of the fan Σ and n_{ρ} be the primitive lattice vector on the ray ρ. For ρ ∈ Σ(1) we consider the set S_{ρ} of all vectors e ∈ M such that

(1) ⟨n_{ρ}, e⟩ = −1 and ⟨n_{ρ′}, e⟩ ⩾ 0 ∀ ρ′ ∈ σ(1), ρ′ ≠ ρ;

(2) if σ is a cone of Σ and ⟨v, e⟩ = 0 for all v ∈ σ, then the cone generated by σ and ρ is in Σ as well.

Note that condition (1) implies condition (2) if Σ is a maximal fan with support |Σ|. This is the case if X is affine or complete. The elements of the set ℜ := ⨆_{ρ} S_{ρ} are called the Demazure roots of the fan Σ,

cf. [13, Définition 4] and [23, Section 3.4]. Again elements e ∈ ℜ are in bijection with 𝔾_{a}-actions on X normalized by the acting torus, see [13, Théoreme 3] and [23, Proposition 3.14]. If X is affine, the 𝔾_{a}-action given by a Demazure root e coincides with the action given by the locally nilpotent derivation ∂_{e} of the algebra 𝕂[X] as defined in Section 2. Let us denote the image under this action of the group 𝔾_{a} in Aut(X) by H_{e}. Thus H_{e} is a one-parameter unipotent subgroup normalized by 𝕋 in Aut(X). We recall basic facts from toric geometry. There is a bijection between cones σ ∈ Σ and 𝕋-orbits O_{σ} on X such that σ_{1} ⊆ σ_{2} if and only if O_{σ_{2}} ⊆ \overline{O_{σ_{1}}}. Here dim O_{σ} = n − dim⟨σ⟩. Moreover, each cone σ ∈ Σ defines an open affine 𝕋-invariant subset U_{σ} on X such that O_{σ} is a unique closed 𝕋-orbit on U_{σ} and σ_{1} ⊆ σ_{2} if and only if U_{σ_{1}} ⊆ U_{σ_{2}}. Let ρ_{e} be the distinguished ray corresponding to a root e, n_{e} be the primitive lattice vector on ρ_{e}, and R_{e} be the one-parameter subgroup of 𝕋 corresponding to n_{e}. Our aim is to describe the action of H_{e} on X.

Proposition 5. For every point x ∈ X \ X^{H_{e}} the orbit H_{e}x meets exactly two 𝕋-orbits O_{1} and O_{2} on X, where dim O_{1} = dim O_{2} + 1. The intersection O_{2} ∩ H_{e}x consists of a single point, while

O_{1} ∩ H_{e}x = R_{e}y for any y ∈ O_{1} ∩ H_{e}x.

Proof. It follows from the proof of [23, Proposition 3.14] that the affine charts U_{σ}, where σ ∈ Σ is a cone containing ρ_{e}, are H_{e}-invariant, and the complement of their union is contained in X^{H_{e}}, cf. [9, Lemma 2.4]. This reduces the proof to the case X is affine. Then the assertion is proved in [6, Proposition 2.1]. □

A pair of 𝕋-orbits (O_{1}, O_{2}) on X is said to be H_{e}-connected if H_{e}x ⊆ O_{1} ∪ O_{2} for some x ∈ X \ X^{H_{e}}. By Proposition 5, O_{2} ⊆ \overline{O_{1}} for such a pair (up to permutation) and dim O_{1} = dim O_{2} + 1. Since the torus normalizes the subgroup H_{e}, any point of O_{1} ∪ O_{2} can actually serve as a point x.

Lemma 1. A pair of 𝕋-orbits (O_{σ_{1}}, O_{σ_{2}}) is H_{e}-connected if and only if e|_{σ_{2}} ≤ 0 and σ_{1} is a facet of σ_{2} given by the equation ⟨v, e⟩ = 0.

Proof. The proof again reduces to the affine case, where the assertion is [6, Lemma 2.2]. □

6. The orbit structure

We keep notations of the previous section. Let us begin with a construction mentioned in the Introduction. Let X be a toric variety with the acting torus 𝕋. Consider a non-trivial action 𝔾_{a} × X → X normalized by 𝕋 and thus represented by a Demazure root e of the fan Σ of X. Then 𝕋 acts on 𝔾_{a} by conjugation with the character e and the semidirect product 𝕋 ⋌ 𝔾_{a} acts on X as well. Let T = Ker(e) ⊆ 𝕋 and consider the group 𝔾 := T × 𝔾_{a}.

Proposition 6. The variety X is an embedding of 𝔾.

Proof. Take a point x ∈ X whose stabilizers in 𝕋 and 𝔾_{a} are trivial. It suffices to show that the stabilizer of x in 𝔾 is trivial. To this end, note that by the Jordan decomposition [20, Theorem 15.3] any subgroup of T × 𝔾_{a} is a product of subgroups in T and 𝔾_{a} respectively. □

Remark 1. The 𝔾-embedding of Proposition 6 is defined by the pair (Σ, e).

Since ⟨n_{e}, e⟩ = −1, we have 𝕋 = T × R_{e}.

Lemma 2. Any (T × 𝔾_{a})-invariant subset in X is also 𝕋-invariant.

Proof. Note that an orbit 𝕋x does not coincide with the orbit T x if and only if the stabilizer of x in 𝕋 is contained in T. For x ∈ O_{σ} this condition is equivalent to e|_{σ} = 0. It shows that for every x ∈ X^{𝔾_{a}} we have 𝕋x = T x. If x ∈ X \ X^{𝔾_{a}}, then by Proposition 5 the orbit 𝔾_{a}x is invariant under R_{e}. This proves that any orbit of (T × 𝔾_{a}) is R_{e}- and T -invariant, thus the assertion. □

Proposition 7. Let X be a 𝔾-embedding given by a pair (Σ, e). Then any 𝔾orbit on X is either a union O_{1} ∪ O_{2} of two 𝕋-orbits on X or a unique 𝕋-orbit; the first possibility occurs if and only if the pair (O_{1}, O_{2}) is H_{e}-connected. In particular, the number of 𝔾-orbits on X is finite.

Proof. The assertion follows directly from Lemma 2 and Proposition 5. □

Proposition 8. Let X be a 𝔾-embedding given by a pair (Σ, e). Then the stabilizer of any point x ∈ X in 𝔾 is connected and the closure of any 𝔾-orbit on X is a (normal) toric variety. If X is smooth, then the closure of any 𝔾-orbit is smooth.

Proof. The stabilizer of a point x in 𝔾 is the direct product of stabilizers in T and in 𝔾_{a}. An algebraic subgroup of 𝔾_{a} is either {0} or 𝔾_{a} itself, while the stabilizer in T is the kernel of the (primitive) character e restricted to the (connected) stabilizer of x in 𝕋. Thus the stabilizer of x in 𝔾 is connected. Proposition 7 shows that any 𝔾-orbit on X contains an open 𝕋-orbit, and thus the closure of a 𝔾-orbit coincides with the closure of some 𝕋-orbit. Now the last two assertions follow from [18, Section 3.1]. □

Remark 2. If X contains l torus invariant prime divisors, then the number of 𝔾-invariant prime divisors on X is l − 1. On a toric variety, the closure of any torus orbit is an intersection of torus invariant prime divisors. In contrast, not every 𝔾-orbit closure on X is an intersection of 𝔾-invariant prime divisors, see Example 3.

7. The general case

We are going to show that every 𝔾-embedding can be realized as in Proposition 6.

Theorem 2. Let 𝔾 = T × 𝔾_{a} and X be a normal equivariant 𝔾-embedding. Then the T -action on X can be extended to an action of a bigger torus 𝕋 such that 𝕋 normalizes 𝔾_{a} and X is a toric variety with the acting torus 𝕋. In particular, every 𝔾-embedding comes from a pair (Σ, e), where Σ is a fan and e is a Demazure root of Σ.

Proof. We begin with a classical result of Sumihiro. Let X be a normal variety with a regular action G × X → X of a linear algebraic group G. By [24, Theorem 3], there exists a normal complete G-variety X such that X can be embedded equivariantly as an open subset of X. In other words, X is an equivariant compactification of X. Let X be a normal embedding of 𝔾 and X be an equivariant compactification of X. By Proposition 4, the T -action on X can be extended to an action of a bigger torus 𝕋 such that 𝕋 normalizes 𝔾_{a} and X is a toric variety with the acting torus 𝕋. Since the subset X ⊆ X is (T × 𝔾_{a})-invariant, it is invariant under 𝕋, see Lemma 2. This provides the desired structure of a toric variety on X. □

Proposition 9. A complete toric variety X admits a structure of a 𝔾embedding if and only if Aut(X)^{0} ≠ 𝕋.

Proof. The variety X admits a structure of a 𝔾-embedding if and only if Aut(X)^{0} contains at least one root subgroup. It is well known that the group Aut(X)^{0} is generated by 𝕋 and root subgroups [13, Proposition 11], [23, Section 3.4], [11, Corollary 4.7]. □

Consider two structures of a 𝔾-embedding on a variety X. We say that such structures are equivalent, if there is an automorphism of X sending one structure to the other. Since the structure of a toric variety on X is unique up to automorphism, we may assume that our two structures share the same acting torus 𝕋 and the same fan Σ, and are given by two roots e, e′ of Σ. Then the structures are equivalent if and only if e can be sent to e′ by an automorphism of the torus 𝕋. This leads to the following result.

Proposition 10. Two structures of a 𝔾-embedding given by pairs (Σ, e) and (Σ, e′) are equivalent if and only if there is an automorphism ϕ of the lattice N which preserves the fan Σ and such that the induced automorphism ϕ^{∗} of the dual lattice M sends e to e′.

Let us finish with explicit examples of 𝔾-embeddings into a given variety.

Example 2. We find all structures of 𝔾-embeddings on 𝔸^{2}. The cone of 𝔸^{2} as a toric variety is ℚ_{⩾0}^{2}. The set of Demazure roots of ℚ_{⩾0}^{2} is

ℜ = {(−1, k) | k ∈ ℤ_{⩾0}} ⊔ {(k, −1) | k ∈ ℤ_{⩾0}}, see Example 1. The 𝔾-action on 𝔸^{2} corresponding to the root (−1, k) is given by

(3)  (t, s) ∘ (x_{1}, x_{2}) = (t^{k}x_{1} + st^{k}x_{2}^{k}, tx_{2}),

where (x_{1}, x_{2}) ∈ 𝔸^{2}, s ∈ 𝔾_{a}, and t ∈ 𝕂^{×}. If k ≠ 0, then there is a line of 𝔾_{a}-fixed points and the stabilizer of a non-zero point on this line is a cyclic group of order k. If k = 0, then there is no 𝔾_{a}-fixed point. So formula (3) gives non-equivalent 𝔾-actions for different k. With k ≠ 0 we have three 𝔾-orbits on 𝔸^{2}, while for k = 0 there are two 𝔾-orbits. Note that 𝔾-actions defined by the roots (k, −1) and (−1, k) are equivalent via the automorphism x_{1} ↔ x_{2} of 𝔸^{2}.

Example 3. Let X = ℙ^{2}. It is a complete toric variety with a fan Σ generated by the vectors (1, 0), (0, 1) and (−1, −1):

[figure: N_{ℚ}; M_{ℚ}; e_{4}; e_{1}; e_{5}; e_{2}; e_{6}; e_{3}]

The set of Demazure roots is

ℜ = {e_{1} = (1, 0), e_{2} = (1, −1), e_{3} = (0, −1),
e_{4} = (−1, 0), e_{5} = (−1, 1), e_{6} = (0, 1)}.

We see that for any i and j there exists isomorphism of the fan Σ sending e_{i} to e_{j}. So any 𝔾-embedding into ℙ^{2} is equivalent to

(t, s) ∘ [z_{0} : z_{1} : z_{2}] = [tz_{0} + stz_{1} : tz_{1} : z_{2}].

This time seven 𝕋-orbits glue to five 𝔾-orbits.

Example 4. Consider the Hirzebruch surface 𝔽_{1}. The corresponding complete fan Σ is generated by the vectors (1, 0), (0, 1), (0, −1), and (−1, 1):

[figure: N_{ℚ}; M_{ℚ}; e_{2}; e_{1}; e_{3}; e_{4}]

The set of Demazure roots is

ℜ = {e_{1} = (1, 0), e_{2} = (−1, 0), e_{3} = (0, 1), e_{4} = (1, 1)}.

By an automorphism, we can send e_{1} to e_{2} and e_{3} to e_{4}. For the first equivalence class we have six 𝔾-orbits, while in the second one the number of 𝔾-orbits is seven.

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Ivan Arzhantsev National Research University Higher School of Economics (HSE) Faculty of Computer Science Kochnovskiy Proezd 3 Moscow 125319 Russia arjantsev@hse.ru Polina Kotenkova National Research University Higher School of Economics (HSE) Faculty of Computer Science Kochnovskiy Proezd 3 Moscow 125319 Russia pkotenkova@hse.ru 1054

Documenta Mathematica 20 (2015)