I Arzhantsev
email: arjantsev@hse.ru

K Kuyumzhiyan

M Zaidenberg
email: mikhail.zaidenberg@univ-grenoble-alpes.fr

INFINITE TRANSITIVITY, FINITE GENERATION, AND DEMAZURE ROOTS

Keywords: Mathematics Subject Classification: 14R20, 32M17 affine variety, toric variety, group action, one-parameter subgroup, Demazure root, transitivity

An affine algebraic variety X of dimension ≥ 2 is called flexible if the subgroup SAut(X) ⊂ Aut(X) generated by the one-parameter unipotent subgroups acts m-transitively on reg (X) for any m ≥ 1. In a preceding paper ([5]) we proved that any nondegenerate toric affine variety X is flexible. Here we show that if such a toric variety X is smooth in codimension 2 then one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property. For X = A n with n ≥ 2, three such subgroups are enough.

Introduction

Let K be an algebraically closed field of characteristic zero, and let X be an affine variety over K. A one-parameter subgroup H of Aut(X) isomorphic as an algebraic group to the additive group G a of the base field K is called a unipotent one-parameter subgroup, or a G a -subgroup, for short. One can consider the subgroup SAut(X) ⊂ Aut(X) generated by all the one-parameter unipotent subgroups. It is known [START_REF] Arzhantsev | Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity[END_REF]Thm. 2.1]) that for a toric variety X with no toric factor (that is, a nondegenerate toric variety) the group SAut(X) acts infinitely transitively in the smooth locus reg(X), that is, m-transitively for any m ≥ 1. Varieties X with this property are called flexible ( [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF], [START_REF] Arzhantsev | Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity[END_REF]). Actually, the simple transitivity of SAut(X) in reg(X) already guarantees that X is flexible ([3, Thm. 0.1]). The same is true for quasi-affine varieties ( [START_REF] Arzhantsev | Infinite transitivity on universal torsors[END_REF], [START_REF] Flenner | The Gromov-Winkelmann theorem for flexible varieties[END_REF]). In turn, the flexibility implies several other useful properties, for instance, the unirationality (see, e.g., [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF], [START_REF] Bogomolov | Unirationality and existence of infinitely transitive models[END_REF], [START_REF] Popov | On infinite dimensional algebraic transformation groups[END_REF]). The flexibility has found important applications, e.g., to the Zariski cancellation problem ( [START_REF] Flenner | Cancellation for surfaces revisited[END_REF]). It is known [START_REF] Flenner | The Gromov-Winkelmann theorem for flexible varieties[END_REF]Thm. 1.1]) that the flexibility survives upon passing to the complement of a subvariety of codimension at least 2. Different flexibility properties are intensively studied in complex analytic geometry, see, e.g., surveys [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF], [START_REF] Forstnerič | Stein Manifolds and Holomorphic Mappings. The Homotopy Principle in Complex Analysis[END_REF], [START_REF] Kaliman | On the present state of the Andersen-Lempert theory[END_REF].

The toric affine varieties with no toric factors are flexible ([5, Thm.

2.1]

). There are several other interesting classes of flexible affine varieties (see, e.g., [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF], [START_REF] Arzhantsev | Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity[END_REF], [START_REF] Arzhantsev | Infinite transitivity on universal torsors[END_REF], [START_REF] Yu | Multitransitivity of Calogero-Moser spaces[END_REF], [START_REF] Dubouloz | Flexible bundles over rigid affine surfaces[END_REF], [START_REF] Micha Lek | Flexible affine cones and flexible coverings[END_REF], [START_REF] Park | Flexible affine cones over del Pezzo surfaces of degree 4[END_REF], [START_REF] Perepechko | Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5[END_REF], [START_REF] Popov | On infinite dimensional algebraic transformation groups[END_REF], [START_REF] Prokhorov | Fano-Mukai fourfolds of genus 10 as compactifications of C 4[END_REF], [START_REF] Shafarevich | Flexibility of affine horospherical varieties of semisimple groups[END_REF]).

In fact, for a flexible X certain proper subgroups G of SAut(X) act also infinitely transitively in reg(X), or at least in a Zariski dense open subset of reg(X). This is the case for a subgroup G generated by a sufficiently rich family of G a -subgroups of SAut(X), see [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF]Thm. 2.2]. Let us stay on this in more detail.

Let LND(X) stand for the set of all nonzero locally nilpotent derivations (LNDs, for short) of the structure algebra O X (X). Any ∂ ∈ LND(X) is a generator of a G a -subgroup H = exp(K∂) of Aut(X), and any G a -subgroup H ⊂ Aut(X) has the form H = exp(K∂) for some ∂ ∈ LND (X). If a ∈ ker ∂ then a∂ is again an LND called a replica of ∂. The subgroup H(a) = exp(Ka∂) is called a replica of H.

A family F of G a -subgroups of Aut(X) is called saturated if (i) any replica of H ∈ F belongs to F, and (ii) F is closed under conjugation by the elements of the subgroup G = G(F) generated by the members of F ([3, Def. 2.1]). Notice [START_REF] Flenner | Cancellation for surfaces revisited[END_REF]Lem. 4.6]) that for any family F which verifies (i) there exists a larger family F ′ satisfying both (i) and (ii) such that G(F) = G(F ′ ). If G(F) has an open orbit in X and F is saturated then the action of G on this orbit is infinitely transitive ([3, Thm. 2.2]). By [START_REF] Flenner | The Gromov-Winkelmann theorem for flexible varieties[END_REF]Prop. 2.15] one can find such a family F composed by the replicas of just two LNDs.

We say that X is generically flexible if SAut(X) acts on X with an open orbit. For instance ( [START_REF] Gizatullin | Quasihomogeneous affine surfaces[END_REF]), any Gizatullin surface X is generically flexible. Notice that the open orbit of Aut(X) can be smaller than reg(X), see, e.g., [START_REF] Kovalenko | Transitivity of automorphism groups of Gizatullin surfaces[END_REF] for examples of Gizatullin surfaces with this property.

The assumption of saturation could be too restrictive in applications. The aim of the present paper is to elaborate more moderate conditions on the family F which still guarantee the infinite transitivity of G(F) on the open orbit. In Section 2 we formulate such conditions, see Theorem 2.2. It occurs that for a generically flexible variety already a countable family of G a -subgroups generates a group acting infinitely transitively on its open orbit, see Corollary 2.8. A useful observation in Section 3 consists in the fact that the orbits and transitivity of an algebraically generated group G ⊂ Aut(X) are not affected upon passing to the closure G, see Proposition 3.4. The remaining part of the paper appeared as a result of our discussions on the following Conjecture 1.1. Any generically flexible affine variety X admits a finite collection {H 1 , . . . , H N } of G a -subgroups of Aut(X) such that the group G = ⟨H 1 , . . . , H N ⟩ acts infinitely transitively on its open orbit.

In Section 5 we fix this conjecture for toric affine varieties under a certain mild restriction. In Section 4 we recall some basics on toric varieties and Cox rings. We deal there with the LNDs of the structure ring of a toric affine variety X which are normalized by the acting torus T. The degree e of such an LND ∂ is a lattice vector called a Demazure root. The corresponding G a -subgroup H e ⊂ Aut(X) normalized by T is called a (Demazure) root subgroup.

Our approach exploits hardly the following phenomenon. Consider a group G acting effectively on a toric affine variety X and generated by its unipotent subgroups. The closure G of G with respect to the ind-topology could contain more Demazure root subgroups than the group G itself. However, the multiple transitivity of G on its orbit is inherited by the group G, see Proposition 3.4.

To describe some extra Demazure root subgroups contained in G we develop in Section 4.2 certain degeneration techniques. Given an LND ∂ ∈ Der(O X (X)) generating a one-parameter unipotent subgroup of G we define its Newton polytope N (∂) with respect to the acting torus. The extremal points of this polytope correspond to one-parameter unipotent root subgroups which belong to G, see Proposition 4.16. To find a convenient (non-root) LND ∂ we conjugate one Demazure root subgroup by a second one which does not centralize the first. The Newton polytope N (∂) of the resulting LND ∂ occurs to be a segment one of whose end points is a desired extra Demazure root. This segment can be found explicitly by using a version of the Baker-Campbell-Hausdorff formula, see Corollary 4.14.

The simplest toric affine varieties are the affine spaces A n = A n K . In this case both the affine group Aff n and the group SL(n, K) extended by just one root subgroup act infinitely transitively on their open orbits, see, e.g., Corollary 5.3. This is based on the results of Bodnarchuk ( [START_REF] Yu | Affine group as a subgroup of biregular transformation group of an affine space[END_REF][START_REF] Yu | On generators of the tame invertible polynomial maps group[END_REF]), Edo ([19]), and Furter ( [START_REF] Furter | Polynomial composition rigidity and plane polynomial automorphisms[END_REF]) concerning cotame automorphisms of the affine spaces, see Definition 5.9. We establish the following facts, see Theorems 5.16 and 5.17.

Theorem 1.2. For any n ≥ 2 one can find three G a -subgroups of Aut(A n ) which generate a subgroup acting infinitely transitively on A n . The same is true for some n + 2 Demazure root subgroups. For n = 2, the latter holds with three Demazure root subgroups.

Our main result in this direction for toric affine varieties (see Theorem 5.20) is the following Theorem 1.3. For any toric affine variety X of dimension at least 2, with no toric factor, and smooth in codimention 2 one can find a finite collection of Demazure root subgroups such that the group generated by these ones acts infinitely transitively in the smooth locus reg(X).

Infinite transitivity on the open orbit

We are working over an algebraically closed field K of characteristic zero. We let A n stand for the affine space of dimension n over K, and G a and G m for the additive and the multiplicative groups of K, respectively, viewed as algebraic groups.

2.1. Let X be an affine variety over K of dimension n ≥ 2. Consider a finite collection of pairwise non-collinear locally nilpotent derivations ∂ 1 , ∂ 2 , . . . , ∂ k of O X (X) which contains a subset of n linearly independent derivations. For each i = 1, . . . , k fix a finitely generated subalgebra A i ⊂ ker ∂ i such that the fraction field Frac (A i ) has finite index in Frac (ker ∂ i ). Consider the following possibilities:

(α) O X (X) is generated by A 1 , . . . , A k ; (β) [Frac (ker ∂ i ) ∶ Frac (A i )] = 1 for some value of i; (γ) [Frac (ker ∂ i ) ∶ Frac (A i )] > 1 for all i = 1, . . . , k.
In the latter case we fix an extra element b 1 ∈ ker ∂ 1 such that Frac (ker ∂ 1 ) is generated by b 1 and Frac (A 1 ). In cases (α) and (β) one might consider b 1 = 0. Let G be the subgroup of SAut(X) generated by the G a -subgroups

H 0 = exp(Kb 1 ∂ 1 ) and H i (a i ) = exp(Ka i ∂ i ) where a i ∈ A i , i = 1, . . . , k . Notice that G acts on X with an open orbit O G , see [3, Corollary 1.11a].
The following theorem is the main result of this section. This theorem is actually a refined version of Theorem 2.2 in [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF]. Its proof follows, with some modifications, the lines of the proof of Theorem 2.2 in [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF].

Lemma 2.3. Let Ω ⊂ O G be a dense open subset. Then for any finite collection of distinct points Q 1 , Q 2 , . . . , Q m ∈ O G there exists g ∈ G such that g(Q i ) ∈ Ω for every i = 1, 2, . . . , m. Proof. By [3, Prop. 1.5] there is a finite collection of G a -subgroups U 1 , U 2 , . . . , U N in G such that for any x ∈ O G we have O G = (U 1 ⋅ . . . ⋅ U N ).
x. This gives a surjective morphism

ϕ x ∶ A N → O G , (t 1 , . . . , t N ) ↦ (U 1 (t 1 ) ⋅ . . . ⋅ U N (t N )).x . Letting ϕ i = ϕ Q i , i = 1, . . . , m consider the dense open subset ω = m ⋂ i=1 ϕ -1 i (Ω) ⊂ A N .
Pick up a point (t 1 , t 2 , . . . , t N ) ∈ ω, and let g = U 1 (t 1 ) ⋅ . . . ⋅ U N (t N ) ∈ G. Then for any i = 1, . . . , m one has g(Q i ) ∈ Ω.

In the sequel we use the following notation.

Notation 2.4.

Let H i = H i (1), i = 1, . . . , k. Letting W i = Spec A i consider the morphism π i ∶ X → W i induced by the inclusion A i ↪ O X (X).
There is a Zariski open, dense subset ω i ⊂ W i such that on U i ∶= π -1 i (ω i ) there exists the geometric quotient U i H i . The inclusion A i ⊂ O U i (U i ) induces a generically finite morphism q i ∶ U i H i → W i . Due to our assumption, ω i can be chosen so that q i is a finite morphism onto its image, of degree

d i ∶= [Frac (ker ∂ i ) ∶ Frac (A i )]. One can find a dense open subset Ω ⊂ O G such that (i) in each point x ∈ Ω the vectors ∂ 1 (x), . . . , ∂ k (x)
are pairwise non-collinear and generate the tangent space T x X; (ii) for i = 1, . . . , k one has π i (Ω) ⊂ reg(W i ) and the restriction π i Ω ∶ Ω → W i is a smooth morphism; (iii) for i = 1, . . . , k the fiber of π i Ω over any point w ∈ π i (Ω) is a dense open subset of the union of d i orbits of H i ; (iv) in case (γ) for i = 1 these orbits are separated by the function b 1 ∈ ker ∂ 1 as in 2.1. For each i = 1, . . . , k there is a factorization

π i ∶ Ω p i → Ω H i q i → W i .
We have the following analogue of Lemma 2.10 in [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF]. Lemma 2.5. For any pair of distinct points Q 1 , Q 2 ∈ O G there exists g ∈ G such that for every i = 1, . . . , k the points g.Q 1 and g.Q 2 are separated by A i for i = 1, . . . , k.

Proof. By Lemma 2.3 one may suppose that Q 1 , Q 2 ∈ Ω where Ω ⊂ X is as in 2.4. Assume first that for some i ∈ {1, . . . , k} the algebra A i separates Q 1 and Q 2 . This is so in cases (α) and (β); anyway, one may consider that i = 1. Let a 1 ∈ A 1 be such that a 1 (Q 1 ) = 0 and

a 1 (Q 2 ) = 1. Then H 1 (a 1 ) fixes Q 1 and moves Q 2 along its H 1 -orbit. Let Q 2 (t) = exp(t∂ 1 )(Q 2 ), t ∈ K . Given i ≥ 2 the condition (1) Q 2 (t) ∈ Ω and π i (Q 2 (t)) ≠ π i (Q 1 )
is an open condition in t ∈ K. Since Q 2 ∈ Ω, by (i) and (ii) the image

π i (H 1 (Q 2 )) in W i is one-dimensional.
It follows that (1) holds on a dense open subset in K. Moreover, the latter is true simultaneously for all i = 2, . . . , k, as required. Now one may restrict to case (γ). Suppose that A 1 does not separate

Q 1 and Q 2 . Assume further that H 1 (Q 1 ) ≠ H 1 (Q 2 ). Then b 1 separates Q 1 and Q 2 due to (iv).
Consider the flow

φ t = exp(t(b 1 -b 1 (Q 1 ))∂ 1 ) ⊂ H 0 ⋅ H 1 ⊂ G , and let Q 2 (t) = φ t (Q 2 )
. Then φ t fixes Q 1 and moves Q 2 along its H 1 -orbit. Now the same argument as before applies and proves that the algebra A i , i = 2, . . . , k separates Q 1 and Q 2 (t) for a general t ∈ K. Since by our assumptions k ≥ n ≥ 2, one may interchange now the role of A 1 and A k and achieve as before that A 1 separates the images of Q 1 and Q 2 under the action of exp(ta k ∂ k )(Q 1 ) for a suitable a k ∈ A k and a general t ∈ K. This gives the result.

Suppose further that H 1 (Q 1 ) = H 1 (Q 2 ). We claim that for every i = 2, . . . , k and for a general t ∈ K the points Q 1 (t) and Q 2 (t) are separated by A i . Indeed, assume to the contrary that π i (Q 1 (t)) = π i (Q 2 (t)) for some i ≥ 2 and for all t ∈ K. Since the image π i (H 1 (Q 1 )) in W i is one-dimensional there exists a i ∈ A i such that the restriction a i H 1 (Q 1 ) defines a non-constant polynomial

p i ∈ K[t]. Since Q 2 = Q 1 (τ ) for some nonzero τ ∈ K one has Q 2 (t) = Q 1 (t + τ ).
It follows that p i (t) = p i (t + τ ) for any t ∈ K, a contradiction. The proof ends by the argument used in the previous case for i = 1. Lemma 2.6. For any finite collection of distinct points Q 1 , . . . , Q m ∈ O G there exists an element g ∈ G such that the points g(Q 1 ), . . . , g(Q m ) are separated by A i for i = 1, . . . , k.

Proof. By Lemma 2.3 we may assume that Q j ∈ Ω ∀i = 1, . . . , m. We proceed by induction on m. For m = 1 the assertion is evidently true. Assume that the points Q 1 , . . . , Q m-1 are already separated by A i for i = 1, . . . , k. Applying Lemma 2.5 and its proof to Q m and Q 1 one may replace the cortege (Q 1 , . . . , Q m ) by a new one (Q

1 (t), . . . , Q (1)
m (t)) so that the separation still holds for Q

1 (t), . . . , Q (1)
m-1 (t) with a generic t ∈ K, and in addition it holds for Q m (t 2 ), and so for. Finally one arrives at a cortege with the desired separation property.

Lemma 2.7. For any finite collection of distinct points

Q 1 , . . . , Q m ∈ O G the stabilizer Stab Q 1 ,...,Qm (G) acts transitively on O G ∖ {Q 1 , . . . , Q m }.
Proof. We proceed by induction on m. The assertion is evidently true for m = 0. Assuming that it holds for a given m ≥ 0 consider a collection of m+1 distinct points Q 1 , . . . , Q m , Q m+1 ∈ O G . By Lemma 2.3 one may assume that these points lie in Ω. Applying Lemma 2.6 one may suppose that for i = 1, . . . , k the images π i (Q j ) ∈ W i , j = 1, . . . , m + 1 are all distinct. Then for every i = 1, . . . , k there exists a i ∈ A i which vanishes in Q 1 , . . . , Q m and does not vanish in

Q m+1 .
Consider the G a -subgroups

H i (a i ) ⊂ Stab G (Q 1 , . . . , Q m ), i = 1, . . . , k .
The orbit of Q m+1 under the action of the stabilizer Stab G (Q 1 , . . . , Q m ) is locally closed (see, e.g., Proposition 1.3 in [START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF]) and contains the one-dimensional

H i -orbits of Q m+1 , i = 1, . . . , k.
By (i) the tangent vectors to these orbits at Q m+1 span the tangent space

T Q m+1 X. It follows that the orbit Stab G (Q 1 , . . . , Q m )(Q m+1 ) is open in X whatever is the point Q m+1 ∈ O G ∖ {Q 1 , . . . , Q m }. Since an open dense orbit is unique one has Stab G (Q 1 , . . . , Q m )(Q m+1 ) = O G ∖ {Q 1 , . . . , Q m } .
Proof of Theorem 2.2. We have to show that for any two ordered corteges

(Q 1 , . . . , Q m ) and (Q ′ 1 , . . . , Q ′ m ) in O G there is g ∈ G such that g.Q j = Q ′ j , j = 1, . . . , m. Assuming by induction that Q i = Q ′ i , i = 1, . . . , m -1, by Lemma 2.7 one can find g ∈ Stab G (Q 1 , . . . , Q m-1 ) such that g.Q m = Q ′
m , as required. Corollary 2.8. Let X be a generically flexible affine variety1 of dimension n ≥ 2. Then there exists a countable collection of G a -subgroups {H 1 , . . . , H n , . . .} such that the subgroup

G = ⟨H i i ∈ N⟩ ⊂ SAut(X)
acts on X with an open orbit O G and is infinitely transitive on O G .

Proof. The generic flexibility of X implies that there is a collection of n linearly independent LNDs ∂ 1 , . . . , ∂ n ∈ LND (X). Letting H i = exp(K∂ i ) choose for any i = 1, . . . , n a finitely generated subalgebra A i ⊂ ker ∂ i separating the general H i -orbits, and let {a i,j } j∈N be a countable Hamel basis of A i viewed as a vector space over K. Letting H i,j = exp(Ka i,j ∂ i ) consider the group

G = ⟨H i,j i = 1, . . . , n, j ∈ N⟩ ⊂ SAut(X) .
Clearly, G acts on X with an open orbit O G and satisfies condition (β) of 2.1. Therefore, Theorem 2.2 applies and gives the desired conclusion.

Example 2.9. Let X be a toric affine variety of dimension ≥ 2 with no toric factor. Then the subgroup G ⊂ SAut(X) generated by all the root subgroups H e , where e runs over the (countable) set of all the Demazure roots of X (see Section 4) acts infinitely transitively in reg(X). This follows from Corollary 2.8 or, alternatively, from the proof of Theorem 2.1 in [START_REF] Arzhantsev | Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity[END_REF].

Group closures and orbits

We gather some facts that will be used in the next section.

3.1.

Recall that Aut(X) has a structure of an affine ind-group; see, e.g., [START_REF] Kumar | Kac-Moody groups, their flag varieties and representation theory[END_REF] for generalities. In more detail, following [START_REF] Furter | On the geometry of the automorphism group of affine n-space[END_REF] or [32, Prop. 2.1] we fix an embedding X ↪ A n and introduce in O X (X) a (positive) degree function. For α ∈ Aut(X) one defines deg(α) to be the maximum of the degrees of components of α. One can write Aut(X) = lim → Σ s where

• for s ≥ 1, Σ s ∶= {α ∈ Aut(X) deg(α), deg(α -1 ) ≤ s} is a closed subvariety of the affine variety Σ s+1 ; • for any r, s ≥ 1 the composition yields a morphism Σ r × Σ s → Σ rs ;
• the inversion yields an automorphism of Σ s . The Zariski closure of a subset F ⊂ Aut(X) can be defined as

F = lim → (F ∩ Σ s )
where the overline stands for the Zariski closure in Σ s . The Zariski closure of F is a closed ind-subvariety of the ind-variety Aut(X). An algebraic subgroup of Aut(X) is a subgroup which is a closed subvariety of some Σ s .

Lemma 3.2. (a) The closure G of a subgroup G ⊂ Aut(X) is a closed ind-subgroup of Aut(X). (b) If ρ∶ A 1 → Aut(X) is a morphism such that ρ(t) ∈ G for t ≠ 0 then ρ(0) ∈ G. (c) Any G-invariant closed subset Y ⊂ X is G-invariant. (d) If G acts on X with an open orbit O G then O G coincides with the open orbit O G of G. (e) If a normal subgroup G ⊂ Aut(X) acts on X with an open orbit O G then O G = O Aut(X) . Proof. (a) Let G s = G ∩ Σ s . By definition, G s = G ∩ Σ s . Since G r ⋅ G s ⊂ G rs and Σ r × Σ s → Σ rs is a morphism then G r × G s → G rs is a morphism. Since Σ -1 s = Σ s
and the inversion is an automorphism of Σ s one has G -1 s = G s and the inversion G s → Σ s extends to a morphism G s → Σ s which is still the inversion with values in G s . Now (a) follows.

(b) One has ρ(A 1 ) ⊂ Σ s for some s ≥ 1. Hence ρ(A 1 ∖ {0}) ⊂ G s , and so, ρ(0) ∈ G s . (c) The statement follows immediately from the fact that the action map Σ s × X → X is a morphism for any s ≥ 1.

(d) Suppose to the contrary that

O G ⊊ O G . Then Y = O G ∖ O G is a nonempty proper G-invariant closed subset of O G . By (c), Y is G-invariant, a contradiction.
The statement of (e) is a simple exercise.

Z = G.x ∖ G.x. Then Z is a nonempty G-invariant closed subset which meets the orbit G.x open in Y . Hence Z ⊃ G.x ⊃ G.x. This is a contradiction. (b) Choose a cortege Q of distinct points Q 1 , . . . , Q m ∈ O G .
Consider the diagonal action of Aut(X) on X m , and let D ⊂ X m be the union of the big diagonals. Assume that G acts m-transitively on O G . Then the

G-orbit of Q in X m ∖ D coincides with (O G ) m ∖ D. In particular, it is open. The image of G in Aut(X m ) is contained in the closure of the image of G in Aut(X m ). Applying (a) one concludes that G.Q = G.Q = (O G ) m ∖ D, that is, G acts m-transitively on O G .
Finally, (c) is immediate from (b). Then X = Spec A is a toric affine variety of dimension n equipped with the n-torus action defined by the grading. In fact, any toric affine variety arises in this way. The acting algebraic torus is the torus of characters T = Hom(M, G m ). By duality, M is the character lattice of T.

Consider the dual lattice N = Hom(M, Z) and the dual cone

σ ⊂ N Q , σ = {x ∈ N Q ⟨x, y⟩ ≥ 0 ∀y ∈ σ ∨ } .
A ray generator of σ is a primitive lattice vector on an extremal ray of σ. Let Ξ be the set of ray generators of σ. Assume that X has no toric factor, that is, X cannot be decomposed into a product G m × Y where Y is another toric variety. The latter is equivalent to the fact that the cone σ ∨ is pointed, that is, contains no line, and also to the fact that σ is of full dimension, that is, Ξ contains a basis of N Q . To any vector ρ ∈ N there corresponds a

G m -subgroup R ρ ⊂ T acting via t.χ m = t ⟨ρ,m⟩ χ m , t ∈ K * , m ∈ σ ∨ ∩ M .
Definition 4.2 (Demazure roots and Demazure facets). Let X = Spec A be a toric affine variety with no toric factor associated to a lattice cone σ ∨ ⊂ M Q , and let Ξ = {ρ 1 , . . . , ρ k } be the set of primitive ray generators of σ ⊂ N Q . A Demazure root which belongs to a primitive ray generator ρ i ∈ Ξ is a vector e ∈ M such that (i) ⟨ρ i , e⟩ = -1;

(ii) ⟨ρ j , e⟩ ⩾ 0 ∀j ≠ i, see [17, §3.1], [START_REF] Liendo | Affine T -varieties of complexity one and locally nilpotent derivations[END_REF], [START_REF] Liendo | G a -actions of fiber type on affine T -varieties[END_REF]. The rational convex polyhedron S i defined in the affine hyperplane H i = {⟨ρ i , e⟩ = -1} by (ii) will be called a Demazure facet of σ ∨ . The Demazure roots belonging to the ray generator ρ i ∈ Ξ are the points in S i ∩ M . (The lattice vector e is called the degree of ∂.) The proof is straightforward. Proof. To show (c) it suffices to notice that

(b) Let Σ ∨ = σ ∨ ∪ k ⋃ i=1 S i . Then ∂ = ∂ ρ,e (A) ⊂ A if and only if e ∈ Σ ∨ ∩ M
(∂ τ ) l (χ m ) = (∂ l ) lτ (χ m ) = 0 ∀m ∈ σ ∨ ∩ M and ∀l = l(m) ≫ 1 .
For the rest see [START_REF] Liendo | G a -actions of fiber type on affine T -varieties[END_REF]Lem. 1.10].

The following lemma is immediate.

Lemma 4.8. The semigroup S i ∩M is a finitely generated (τ i ∩M )-module. For any f ∈ τ i ∩M one has ρ and ρ ′ are collinear and ⟨ρ, e⟩ = ⟨ρ, e ′ ⟩ (this holds, in particular, if e, e ′ ∈ S i for some i ∈ {1, . . . , k}); ρ and ρ ′ are non-collinear and ⟨ρ ′ , e⟩ = ⟨ρ, e ′ ⟩ = 0.

χ f ∂ ρ i ,e = ∂ ρ i ,e+f ∈ LND (A) .

Given two derivations (or vector fields

) U = ∂ 1 and V = ∂ 2 in Der(A) we let (5) ad m U (V ) = [U, [U, . . . [U, V ] . . .]] = m i=0 m i ∂ m-i 1 ∂ 2 (-∂ 1 ) i
where U is repeated m times, see [START_REF] Manetti | The Baker-Campbell-Hausdorff formula[END_REF]. Thus, ad m U ∈ End (Der(A)). Lemma 4.11 (cf. [START_REF] Manetti | The Baker-Campbell-Hausdorff formula[END_REF]). If U ∈ LND(A) then ad U ∈ End (Der(A)) is locally nilpotent, that is, for any V ∈ Der(A) there exists c(V ) > 0 such that ad m U (V ) = 0 ∀m > c(V ). Proof. Let A = K[a 1 , . . . , a k ]. There exists N > 0 such that ∂ N 1 (a j ) = 0 ∀j = 1, . . . , k. In the last sum in (5) applied to a j certain members of the sum vanish so that it remains just N first members of the sum. For m ≫ 1 the first N terms vanish as well, hence ad m U (V )(a j ) = 0 for j = 1, . . . , k, and so, ad m U (V ) = 0. 4.12. We keep U = ∂ 1 ∈ Der(A) being locally nilpotent. In the sequel we use the following version of the Baker-Campbell-Hausdorff formula, see, e.g., [START_REF] Bonfiglioli | Topics in noncommutative algebra. The theorem of Campbell, Baker, Hausdorff and Dynkin[END_REF]Thm. 4.14]: [START_REF] Arzhantsev | Infinite transitivity on universal torsors[END_REF] Ad exp(U

) (V ) = exp(ad U )(V ) = V + ∞ m=1 1 m! ad m U (V ) .
Due to Lemma 4.11, for a given V = ∂ 2 the last sum is finite, hence the formula has sense. The second equality in ( 6) is just the usual exponential formula for the endomorphism ad U of the vector space Der(A). Due to the first equality in (6), if V is locally nilpotent then exp(ad U )(V ) is as well.

To apply this formula in our setting we need the following simple lemma.

r m = d 1 ! (d 1 -m)! ρ 2 -mc 2 d 1 ! (d 1 -m + 1)! ρ 1 and f m = me 1 + e 2 .
Furthermore, for any d 1 ≥ 0 one has

(8) ad m U (V ) = 0 ∀m ≥ δ + 1 and (9) ad δ U (V ) = -c 2 δ!∂ ρ 1 ,f δ ∈ LND(A) where f δ = δe 1 + e 2 ∈ S 1 ∩ M . Proof.
The assertions follow immediately from (4) by recursion on m. Proof. Applying formula [START_REF] Arzhantsev | Infinite transitivity on universal torsors[END_REF], by Lemma 4.13 one obtains:

∂ = V + δ m=1 1 m! ∂ m
where ∂ m = ∂ rm,fm ∈ Der(A) is a homogeneous derivation of degree f m = me 1 + e 2 . Now the result follows. Any subgroup T of the torus T acts by conjugation on Der(A), and so, induces a grading of Der(A) compatible with the M -grading, see, e.g., [START_REF] Kotenkova | On restriction of roots on affine T-varieties[END_REF]. The pieces which correspond to the Demazure facets (10) Given an isomorphism T ≅ G m , an element t λ ∈ T with λ ∈ G m acts on A via

S 1 ∩ M = {(-1, j) j ∈ Z ≥0 } and S 2 ∩ M = {(i, -1) i ∈ Z ≥0 } are Der (-1,j) (K[x, y]) = ay j ∂ ∂x a ∈ K resp., Der (i,-1) (K[x, y]) = ax i ∂ ∂y a ∈ K .
t λ .χ m = λ l T (m) χ m ∀m ∈ σ ∨ . It follows that t -1 λ ○ ∂ s ○ t λ = λ -s ∂ s . Therefore, one has t -1 λ ○ exp(τ ∂ s ) ○ t λ = exp(τ λ -s ∂ s ) and, furthermore, t -1 λ ○ exp(τ ∂) ○ t λ = exp lmax s=l min τ λ -s ∂ s .
Letting τ = hλ lmax one obtains:

t -1 λ ○ exp(τ ∂) ○ t λ = exp h lmax s=l min λ (lmax-s) ∂ s → exp(h∂ T ) as λ → 0
on any monomial χ m ∈ A, m ∈ σ ∨ . This convergence guarantees the convergence with respect to the ind-group structure on Aut(X) associated to any given filtration A = ⋃ ∞ r=1 A r by finite dimensional graded subspaces of A such that A r ⊂ A r+1 , see Lemma 3.2(b). Since t -1 λ ○ exp(τ ∂) ○ t λ ∈ G for any λ ∈ K * and τ ∈ K one concludes that exp(h∂ T ) ∈ G for any h ∈ K.

e i ∈ S i ∩ M , i = 1, 2. Let δ = ⟨ρ 1 , e 2 ⟩ + 1. Suppose that δe 1 + e 2 ∈ S 1 , that is, ⟨ρ 2 , e 1 ⟩ ≥ 1. Then one has H δe 1 +e 2 ⊂ ⟨H e 1 , H e 2 ⟩ .
Proof. This follows by Corollaries 4.14 and 4.17 applied to U = ∂ ρ 1 ,e 1 , V = ∂ ρ 2 ,e 2 , and ∂ = exp(ad U )(V ) = Ad exp(U ) (V ), see (6). Let T(k) ≅ (G m ) k be the standard k-torus acting on A k , and let F Cox = Hom (Cl(X), G m ) be the dual group of the group Cl(X). This is a quasitorus, that is, the direct product of an algebraic torus and a finite Abelian group. By duality, Cl(X) is the group of characters of F Cox . The Cl(X)-grading on K[x 1 , . . . , x k ] defines an action on A k of the quasitorus

F Cox ⊂ T(k). The structure ring O X (X) is canonically isomorphic to the ring of invariants K[x 1 , . . . , x k ] F Cox . This yields (canonical) isomorphisms X ≅ A k F Cox and T ≅ T(k) F Cox .
The linear forms ρ 1 , . . . , ρ k on M Q define a monomorphism of lattices ϕ∶ M ↪ Z k which extends to the linear embedding

Φ∶ M Q ↪ A k Q , v ↦ (⟨ρ 1 , v⟩, . . . , ⟨ρ k , v⟩
) . The coordinates of the image Φ(m) will be called the total coordinates of m ∈ M .

We let ∆ ∨ ≥0 ⊂ A k Q be the positive octant, and let Ŝi = S i (∆ ∨ ≥0 ) be the ith Demazure facet of ∆ ∨ ≥0 . The image Φ(e) of a Demazure root e ∈ S i is a Demazure root, say, ê ∈ Ŝi . Any root vector ê ∈ Φ(M Q ) ∩ Z k appears in this way. The action of the root subgroup H e on X induces the action of the root subgroup H Φ(e) on A k , see [START_REF] Cox | The homogeneous coordinate ring of a toric variety[END_REF]Sect. 4]. In more detail, one has the following Lemma 4.20 (cf. [START_REF] Cox | The homogeneous coordinate ring of a toric variety[END_REF]Lem. 4.4]). Let us intruduce the necessary notation.

For an arbitrary lattice vector e = (c 1 , . . . , c k ) ∈ Z k we let

x e = x c 1 1 ⋯x c k k . For e ∈ S 1 ∩ M one has ê = (-1, c 2 , . . . , c k ) ∈ Z k where c i ∈ Z ≥0 , i = 2, . . . , k. The root subgroup H ê acts on A k via (12) (x 1 , . . . , x k ) ↦ (x 1 + tx ê+ε 1 , x 2 , . . . , x k ), t ∈ K , where (ε 1 , . . . , ε k ) is the standard basis in A k and x ê+ε 1 = x c 2 2 ⋯x c k k . Lemma 4.20. (a) A Demazure root ê ∈ Ŝi ∩ Z k belongs to the image Φ(S i ∩ M ) if and only if deg(x ê) = 0, that is, x ê ∈ Frac (K[x 1 , . . . , x k ]) F Cox . (b)
The subgroups H ê and F Cox of Aut(A k ) commute if and only if ê = Φ(e) for a root e of σ ∨ . In the latter case the action of the root subgroup H ê on A k descends to the action of the root subgroup H e on X under the quotient morphism

A k → X = A k F Cox .
Proof. (a) Recall that the lattice of T-invariant divisors on X is generated by D 1 , . . . , D k . One may assume that i = 1. One has

deg(ê) = 0 ⇔ [D 1 ] = c 2 [D 2 ] + . . . + c k [D k ] in Cl(X) .
The latter equality amounts to

(13) c 2 D 2 + . . . + c k D k -D 1 = div (χ m ) ∈ Princ(X) T for some m ∈ M where div (χ m ) = k i=1 ⟨ρ i , m⟩D i .
Thus, ( 13) admits a solution m ∈ M if and only if

⟨ρ 1 , m⟩ = -1 and ⟨ρ i , m⟩ = c i ≥ 0 ∀i = 2, . . . , k , that is, if m = e ∈ S 1 ∩ M is a
∂ ε ∨ 1 ,ê = x c 2 2 ⋯x c k k ∂ ∂x 1 has zero F Cox -degree, that is, if deg(x 1 ) = deg(x ê+ε 1 ) or, which is equiv- alent, [D 1 ] = c 2 [D 2 ] + . . . + c k [D k ] .
So, the first assertion of (b) follows by an argument used in the proof of (a). The second one is a simple consequence of the first. Indeed, the LND

∂ ε ∨ 1 ,ê ∈ LND (K[x 1 , . . . , x k ]) of F Cox -degree zero restricts to the ring of invariants K[x 1 , . . . , x k ] F Cox = O X (X) yielding ∂ ρ 1 ,e ∈ LND (O X (X)).
Hence the H ê-action on A k descends to the H e -action on X.

Remark 4.21. The connected group H ê normalizes F Cox in Aut(A k ) if and only if these groups commute. Indeed, Aut(F Cox ) is a finite extension of GL(l, Z), hence a discrete group.

Infinite transitivity: the case of toric varieties

In this section we apply Theorem 2.2 to toric affine varieties X with no toric factor. It is known ([5, Thm. 2.1]) that the action of SAut(X) on the smooth locus reg(X) is infinitely transitive. However, SAut(X) is a huge group. Under a mild additional assumption we construct in Theorem 5.20 a subgroup G ⊂ SAut(X) which still acts infinitely transitively in reg(X) and is generated by a finite number of root subgroups, as it is predicted by Conjecture 1.1. We start with the case where X is an affine space.

= Z n ⊂ A n Q and M = Z n ⊂ (A n Q ) * . The cones σ ⊂ A n Q and σ ∨ ⊂ (A n Q ) *
are the positive octants. The ray generators ρ 1 , . . . , ρ n ∈ N form the standard basis of A n Q . The dual basis (ε 1 , . . . , ε n ) is the standard base of the lattice M . The LNDs associated with the Demazure roots e i = -ε i ∈ S i are the partial derivatives

∂ i = ∂ ∂x i = ∂ ρ i ,e i ∈ LND (A), i = 1, . . . , n . For a lattice vector m = (m 1 , . . . , m n ) ∈ M we write x m = x m 1 1 x m 2
2 ⋯x mn n . Given a root vector e ∈ S i ∩ M the associated root subgroup

H e = exp(Kx e+ε i ∂ i ) ⊂ SAut(A n ) acts on A n via elementary transformations x = (x 1 , . . . , x n ) ↦ (x 1 , . . . , x i-1 , x i + tx e+ε i , x i+1 , . . . , x n ) where t ∈ K .
For instance, letting H i,j = exp(Kx 2 j ∂ i ) where j ≠ i the root subgroups H 1,2 and H 2,3 act on

A n via (14) (x 1 , . . . , x n ) ↦ (x 1 + tx 2 2 , x 2 , . . . , x n ) resp. (x 1 , . . . , x n ) ↦ (x 1 , x 2 + tx 2 3 , x 3 , . . . , x n ) where t ∈ K.
To simplify the notation we write just the coordinates of the image for such an action. The following result confirms Conjecture 1.1 in the case X = A n , n ≥ 2.

Theorem 5.2. Consider the action of the symmetric group S(n) on A n by permutations. Then for any n ≥ 3 the subgroup

G = ⟨H 1,2 , S(n)⟩ ⊂ Aut(A n ) acts infinitely transitively in O G = A n ∖ {0}.
The following corollary is straightforward.

Corollary 5.3. For n ≥ 3 the subgroup ⟨H 1,2 , SL (n, K)⟩ ⊂ Aut(A n ) acts infinitely transitively in A n ∖ {0}.
The proof of Theorem 5.2 is preceded by the following lemmas.

Lemma 5.4. Assume that H u ⊂ G where u = (-1, c 2 , . . . , c n ) ∈ S 1 ∩ M with c 2 ≥ 1. Letting v = (0, -1, 2, 0, . . . , 0) ∈ S 2 ∩ M consider the root vector e = u + v = (-1, c 2 -1, c 3 + 2, c 4 , . . . , c n ) ∈ S 1 ∩ M .
Then H e ⊂ G.

Proof. This follows immediately from Lemma 4.18. Indeed, the pair (u, v) satisfies the assumptions of this lemma with δ = 1. Lemma 5.5. For three indices s, i, j ∈ {2, . . . , n} where i ≠ j consider a root vector of the form

(15) w = (-1, 1, . . . , 1, 2) + 3k s ε s + k i,j (ε i + ε j ) ∈ S 1 ∩ M . Then H w = exp(Kx w+ε 1 ∂ 1 ) ⊂ G for any k s , k i,j ∈ Z ≥0 . Proof. Let v i = -ε i +2ε i+1 , i = 1, . . . , n-1. The Demazure root u = v 1 = (-1, 2, 0, . . . , 0) ∈ S 1 ∩M generates the root subgroup H u = H 1,2 ⊂ G.
Starting with u and adding v 2 , . . . , v n-1 one gets the root vector w 0 = (-1, 1, . . . , 1, 2) ∈ S 1 ∩ M . By Lemma 5.4 the associated root subgroup H w 0 is contained in G. The same conclusion holds if one adds to w 0 the lattice vectors

(-ε i + 2ε j ) + (2ε i -ε j ) = ε i + ε j and (ε i + ε j ) + (2ε i -ε j ) = 3ε i .
Iterating one arrives at the desired conclusion.

We need also the following elementary lemma.

Lemma 5.6. For n ≥ 4 the vectors

3ε i and ε i + ε j , i ≠ j, i, j ≥ 2 span the sublattice L = ⟨ε 2 , . . . , ε n ⟩ ⊂ M of rank n -1.
Proof. One has

ε 2 = (ε 3 + ε 4 ) + 2(ε 2 + ε 4 ) -(ε 2 + ε 3 ) -3ε 4 .
Similar expressions hold for ε i , i = 3, . . . , n.

Proof of Theorem 5.2. By our assumption one has G ⊃ H i,j , i ≠ j, i, j ∈ {1, . . . , n}. By Lemma 5.5, G ⊃ H w for any root vector w in [START_REF] Cox | The homogeneous coordinate ring of a toric variety[END_REF]. Letting

w 0 = (-1, 1, . . . , 1, 2) consider ∂1 = x w 0 ∂ ∂x 1 and A 1 = K[x 3 i , x i x j i ≠ j, i, j ≥ 2] ⊂ ker ∂1 . By Lemma 5.5, G ⊃ exp(Kf ∂1 ) for any f ∈ A 1 .
The conjugation by the S(n)-action yields a collection {( ∂i , A i )} i=1,...,n and G ⊃ exp(Kf ∂i ) for any f ∈ A i .

By Lemma 5.6 for n ≥ 4 this collection satisfies condition (β) of 2.1. 2 in ( 14) by x 1 + x k 2 with k ≥ 3. Indeed, under such a replacement any g ∈ G sends the pair of points (Q, ωQ) with Q ∈ A n ∖ {0} and ω k-1 = 1 to a pair (g(Q), ωg(Q)). Thus, the 2-transitivity of G fails.

For n = 3 it sat- isfies condition (α) of 2.1, that is, the function field Frac (K[x 1 , x 2 , x 3 ]) is generated by {Frac (A i )} i=1,
2. Theorem 5.2 does not hold in the case n = 2. More generally, fixing a, b ∈ Z ≥0 consider the root subgroups Proof. Assume first that ab = 0; let, say, a = 0. Then H 1 acts on A 2 by translations, and G acts on the first coordinate also by translations. Hence one has x(g.P )-x(g.Q) = x(P )-x(Q) for any P, Q ∈ A 2 . The latter is an obstacle to the 2-transitivity.

H 1 ∶ (x, y) ↦ (x + t 1 y a , y) and H 2 ∶ (x, y) ↦ (x, y + t 2 x b ), t 1 , t 2 ∈ K . Claim. If the group G = ⟨H 1 , H 2 ⟩
Suppose that a = b = 1. Then G = SL(2, K). However, a linear group preserves the collinearity, hence it does not act 2-transitively on A n for n ≥ 2.

Let further ab > 2. Fixing a primitive root of unity ω of degree ab -1 > 1 consider the set

S = {(P, Q) ∈ A 2 × A 2 P = (x, y), Q = (ωx, ω b y)} .
It is easily seen that S is invariant under the diagonal action of G on A 2 × A 2 . Once again, this makes an obstacle to the 2-transitivity.

It would be interesting to determine the degree of transitivity of the G-action on A 2 ∖ {0} in the remaining case ab = 2. We can show that this action is 2-transitive. However, we do not know whether a higher transitivity holds.

Infinite transitivity on

A n and cotameness. 5.8. Let Aff n stand for the group of affine transformations of the affine space A n , and let SAff n = {f ∈ Aff n Jac (f ) = 1} = ⟨Transl n , SL (n, K)⟩ be the subgroup of volume preserving affine transformations, where Jac stands for the Jacobian. Notice that the subgroup of translations Transl n has generators

H i = exp(K∂ i ), i = 1, . . . , n .

Letting

H(i, j) = exp(Kx j ∂ i ), i, j ∈ {1, . . . , n}, i ≠ j the group SL (n, K) is generated by H(n, 1) and the root subgroups {H(i, i + 1)} i=1,...,n-1 corresponding to the root system of type A n-1 . Therefore, [START_REF] Cox | Toric Varieties[END_REF] SAff n = ⟨H 1 , H(1, 2), . . . , H(n -1, n), H(n, 1)⟩ is generated by n + 1 root subgroups. (In fact, there exists a smaller generating set.) Definition 5.9 (cf. [START_REF] Edo | Co-tame polynomial automorphisms[END_REF]). Let Tame n stand for the tame subgroup of Aut(A n ). Consider the subgroup STame n = {g ∈ Tame n Jac(g i+1 ⋯x mn n , i = 1, . . . , n. It is known ( [START_REF] Yu | Affine group as a subgroup of biregular transformation group of an affine space[END_REF], [START_REF] Yu | On generators of the tame invertible polynomial maps group[END_REF]1.4 and Thm. 1.8], [START_REF] Edo | Co-tame polynomial automorphisms[END_REF]) that for n ≥ 3 any triangular h ∈ Aut(A n )∖Aff n is cotame, while there is no triangular cotame h ∈ Aut(A 2 )∖Aff 2 .

) = 1} ⊂ Tame n . One says that h ∈ Aut(A n ) ∖ Aff n is cotame if ⟨Aff n , h⟩ ⊃ Tame
Using Theorem 5.10 it is not difficult to deduce the following result on infinite transitivity (see [START_REF] Yu | On the transitivity of the action of a group of biregular automorphisms containing the affine group. (Ukrainian) Dop. NAN Ukraine[END_REF], [START_REF] Yu | Some extreme properties of the affine group as an automorphisms group of the affine space[END_REF], [START_REF] Yu | On generators of the tame invertible polynomial maps group[END_REF]Thm. 1.2]). Theorem 5.12. For any n ≥ 2 and any h ∈ Aut(A n ) ∖ Aff n the group ⟨Aff n , h⟩ acts infinitely transitively on A n .

Proof. According to Theorem 5.10, Aff n is a maximal closed subgroup in Aut(A n ). Its normalizer

N n is a closed subgroup of Aut(A n ) containing Aff n . Since Aff n is not a normal subgroup of Aut(A n ), that is, N n ≠ Aut(A n ), one has N n = Aff n . Since h ∉ Aff n it does not normalize Aff n , that is, h Aff n h -1 ≠ Aff n . Pick up g ∈ h Aff n h -1 ∖ Aff n . By Theorem 5.10 one has ⟨Aff n , g⟩ ⊃ Tame n . Letting G = ⟨Aff n , hAff n h -1 ⟩ ⊃ ⟨Aff n , g⟩
one obtains G ⊃ Tame n . Since Tame n acts infinitely transitively in A n then also G does. The group G is algebraically generated. By Proposition 3.4(c), G acts infinitely transitively in A n . Since G ⊂ ⟨Aff n , h⟩ the latter group does as well.

Remark 5.13. Let g = α 1 g 1 ⋯α l g l α l+1 ∈ Aut(A n ) where Jac(g i ) = 1 and α i ∈ Aff n . If Jac(g) = 1 then there exists another decomposition g = β 1 h 1 ⋯β l h l β l+1 with Jac(h i ) = 1 and β i ∈ SAff n , i = 1, . . . , l where β i = c i-1 α i c -1 i , i = 1, . . . , l + 1, and

h i = c i g i c -1 i , i = 1, . . . , l with c 0 = 1, c i = i j=1 d j , d i = Jac(α i ), and so, c l+1 = l+1 j=1 d j = 1 .
In the setting of Theorem 5.10 this observation yields the following corollary. Proof. Suppose first that n ≥ 3 and u = (-1, c 2 , . . . , c n ) ∈ S 1 ∩ Z n . Since H u is not affine one has c 2 + . . . + c n ≥ 2. Letting e i = -ε i , i = 1, . . . , n and assuming that c i ≥ 1 one can deduce from Lemma 4. [START_REF] Dubouloz | Flexible bundles over rigid affine surfaces[END_REF] cone σ ⊂ N Q of X is regular, that is, the pair of ray generators (ρ i , ρ j ) of τ can be included in a base of the lattice N . For instance, the cone σ ⊂ A 3 Q with the primitive ray generators

ρ i = ε i , i = 1, 2, 3, ρ 4 = ε 1 + ε 2 -ε 3
defines a toric threefold X with a single singular point; this X is smooth in codimension 2.

The following theorem is the main result of this section.

Theorem 5.20. Let X be a toric affine variety of dimension n ≥ 2 with no toric factor. Suppose that X is smooth in codimension 2. Then one can find a finite collection of root subgroups H 1 , . . . , H N such that the group G = ⟨H 1 , . . . , H N ⟩ acts infinitely transitively in the regular locus reg(X).

Proof. If n = 2 then X is smooth, hence X ≅ A 2 . In this case the result (with N = 3) follows from Theorem 5.16.

Assume in the sequel that n ≥ 3. Due to Theorem 2.1 in [START_REF] Arzhantsev | Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity[END_REF], reg(X) coincides with the open orbit of the group SAut(X). By [3, Prop. 1.5] there exists a finite collection of root subgroups H 1 , . . . , H r such that the group generated by H 1 , . . . , H r acts transitively in reg(X) too. To get infinite transitivity we need to enlarge this collection.

Recall that Ξ stands for the set of the primitive ray generators ρ 1 , . . . , ρ k of the cone σ ⊂ N Q associated with X. Given a ray generator, say, ρ 1 ∈ Ξ there exists m1 ∈ M such that the hyperplane L m1 = {v ∈ N Q ⟨v, m1 ⟩ = 0} is strictly supporting for the ray Q ≥0 ρ 1 of σ, that is, ⟨ρ 1 , m1 ⟩ = 0 and ⟨ρ j , m1 ⟩ > 0 ∀j = 2, . . . , k .

Since ρ 1 is a primitive lattice vector its coordinates are coprime. So, ⟨ρ 1 , m1 ⟩ = -1 for some m1 ∈ M . Fix r ≫ 1 and a root vector e 1 = r m1 + m1 ∈ S 1 where ⟨ρ j , e 1 ⟩ ≥ 2 ∀j = 2, . . . , k .

Up to renumbering one may suppose that the 2-cones τ 1,2 and τ 1,3 spanned by the pairs of ray generators (ρ 1 , ρ 2 ) and (ρ 1 , ρ 3 ), respectively, are two-dimensional faces of σ containing the common ray Q ≥0 ρ 1 . By our assumption, τ 1,2 is regular. Hence one can find m1,2 ∈ M such that ⟨ρ 1 , m1,2 ⟩ = 0 and ⟨ρ 2 , m1,2 ⟩ = -1 .

Choose a strictly supporting hyperplane L 1,2 = {⟨v, m1,2 ⟩ = 0} of the face τ 1,2 of σ where m1,2 ∈ M satisfies ⟨ρ 1 , m1,2 ⟩ = ⟨ρ 2 , m1,2 ⟩ = 0 and ⟨ρ j , m1,2 ⟩ > 0 ∀j ≥ 3 . where the stars are integers ≥ 2.

Let τ ∨ 1,2 be the face of σ ∨ of codimension 2 dual to τ 1,2 , that is,

τ ∨ 1,2 = m1,2 ∈ M ⟨ρ i , m1,2 ⟩ = 0, i = 1, 2 ⟨ρ i , m1,2 ⟩ ≥ 0, i = 3, . . . , k .
Choosing n -2 linearly independent primitive ray generators {η 1 , . . . , η n-2 } of τ ∨ 1,2 consider the sequence of roots ( 19)

u 1 = e 2 , u 2 = e 2 + η 1 , . . . , u n-1 = e 2 + η n-2 ∈ S 2 ∩ M
with total coordinates ûi = (0, -1, * , . . . , * ) where " * " ≥ 2. The lattice vectors ( 20) Then one has H w ⊂ G 1 for any root w ∈ e 1 + M 1 ⊂ S 1 ∩ M . Proof of the claim. The assertion is true for w = e 1 . Assume by recursion that H w ⊂ G 1 for some root w ∈ e 1 + M 1 . It suffices to show that then the same holds as well for any root w + v i ∈ e 1 + M 1 , i = 1, . . . , n -1.

v 1 = u 1 + e 3 = e 2 + e 3 , v 2 = u 2 + e 3 , . . . , v n-1 = u n-1 + e 3 ∈
Notice that ŵ = (-1, * , . . . , * ) where " * " ≥ 2. Since ⟨ρ 2 , w⟩ ≥ 1 the pair (w, u i ) satisfies the assumptions of Lemma 4.18 with ⟨ρ 1 , u i ⟩ = 0 and δ = 1 for any i = 1, . . . , n -1. Applying the recursive hypothesis and Lemma 4.18 one deduces that H w+u i ⊂ ⟨H w , H u i ⟩ ⊂ G 1 ∀i = 1, . . . , n -1 .

Likewise, since ⟨ρ 3 , w + u i ⟩ ≥ 1 the pair (w + u i , e 3 ) satisfies the assumptions of Lemma 4.18 with ⟨ρ 1 , e 3 ⟩ = 0 and δ = 1 for any i = 1, . . . , n -1. Applying Lemma 4.18 one deduces by [START_REF] Edo | Co-tame polynomial automorphisms[END_REF] that H w+v i ⊂ ⟨H w+u i , H e 3 ⟩ ⊂ G 1 ∀i = 1, . . . , n -1 . This yields the inductive step and ends the recursion. Now one can constitute the data verifying the assumptions 2.1 of Theorem 2.2. Recall that we fixed already a collection of root subgroups H 1 , . . . , H r such that the open orbit of the group ⟨H 1 , . . . , H r ⟩ coincides with reg(X).

Letting ∂ 1 = ∂ ρ 1 ,e 1 ∈ LND(O X (X)) consider the subalgebra Choose linearly independent ray generators ρ 1 , . . . , ρ n ∈ Ξ. Repeating the same construction one obtains for any i = 1, 2, . . . , n a triple (G i , ∂ i , A i ) with properties similar to the ones of (G 1 , ∂ 1 , A 1 ). Let now G = ⟨H 1 , . . . , H r+s , G 1 , . . . , G n ⟩ ⊂ SAut(X) .

A 1 = K[χ v v ∈ M 1 ] = K[χ v
The group G satisfies the assumptions 2.1(γ) of Theorem 2. We cannot exclude that this number equals 2, at least in the setup of arbitrary G asubgroups; cf. Theorem 5.17. Let us remind the question (V. L. Popov [START_REF] Popov | Roots of the affine Cremona group[END_REF]Problem 3.1]) as to when (the closure of) the subgroup G = ⟨H 1 , H 2 ⟩ generated by G a -subgroups H 1 , H 2 ⊂ Aut(A n ) is an algebraic group. The third author thanks Hanspeter Kraft for an inspiring example of two root subgroups of Aut(A 2 ) whose product is an (infinite dimensional) free product. The discussions with Hanspeter Kraft resulted in the following theorem ([33, Thm. 5.5.1]) which answers, in particular, the question above.

Theorem 5.22. Given an affine variety X the subgroup G ⊂ Aut(X) generated by a family F of connected algebraic subgroups of Aut(X) is a (closed) algebraic group if and only if the Lie algebras Lie (H) for H ∈ F generate a finite dimensional Lie algebra.

The following conjecture arises naturally (cf., e.g., Lemma 4.18). Conjecture 5.23. Let X be an affine variety, and let A = O X (X) be its structure algebra. Consider the group G = ⟨H 1 , . . . , H k ⟩ generated by a finite collection of G a -subgroups H i = exp(K∂ i ) ⊂ SAut(X) where ∂ i ∈ LND(A), i = 1, . . . , k. Then the G a -subgroup H = exp(K∂) ⊂ SAut(X) where ∂ ∈ LND(A) is contained in G if and only if ∂ ∈ Lie ⟨∂ 1 , . . . , ∂ k ⟩.

Of course, the latter holds if G is an algebraic group. One more justification is provided by Lemmas 4.13 and 4.18. Indeed, letting X = Spec(A) be a nondegenerate toric affine variety of dimension n ≥ 2, in the notation of Lemma 4.13 for two LNDs U = ∂ 1 and V = ∂ 2 of A and for m = δ the nonzero homogeneous derivation W = ad δ U (V ) is an LND. According to Lemma 4.18 the associated root subgroup H W = exp(KW ) is contained in the closure G where G = ⟨H U , H V ⟩.

Theorem 2 . 2 .

 22 The action of G on O G is infinitely transitive.

( 1 )

 1 m (t). Fixing such a value t 1 ∈ K one may apply the same procedure to obtain a new cortege (Q

( 2 )

 2 j (t 2 )) j=1,...,m preserving the former property and adding the separation of Q

3. 3 .

 3 A subgroup G ⊂ Aut(X) generated by a family of connected algebraic subgroups of Aut(X) is called algebraically generated ([3]). The orbits of G are locally closed subsets of X in the Zariski toplogy; see [3, Prop. 1.3]. Proposition 3.4. Let G ⊂ Aut(X) be an algebraically generated subgroup. Then the following hold. (a) The orbits of G and of G in X are the same. In particular, if G acts on X with an open orbit O G then G does and O G = O G . (b) If G acts m-transitively on O G then also G does. (c) If G acts infinitely transitively on O G then also G does. Proof. (a) Let x ∈ X, and let Y = G.x. By Lemma 3.2(c), Y is G-invariant. The orbits G.x and G.x ⊃ G.x are both open and dense in Y . Suppose to the contrary that G.x ≠ G.x, and let

4 .

 4 Toric varieties, Cox rings, and derivations 4.1. Toric affine varieties and Demazure roots. 4.1. Recall the combinatorial description of a toric affine variety (see, e.g., [16, Ch. 1], [25, Sec. 1.3]). Let M be a lattice of rank n, let M Q = M ⊗ Q be the associated vector space over Q, and let σ ∨ ⊂ M Q be a rational convex cone with a nonempty interior (the weight cone). Fix a basis of M . For m = (m 1 , . . . , m n ) ∈ M by χ m one means a Laurent monomial x m 1 1 . . . x mn n . Consider the graded affine algebra A = ⊕ m∈M ∩σ ∨ Kχ m .

Definition 4 . 3 (

 43 Homogeneous derivations). A derivation ∂ ∈ Der(A) is called homogeneous if ∂ respects the grading, that is, sends any graded piece to another one. The following description of homogeneous derivations on toric affine varieties completes the one in [36, Sect. 2]. Proposition 4.4. (a) Any homogeneous derivation ∂ has the form ∂ = λ∂ ρ,e for some λ ∈ K, ρ ∈ N , and e ∈ M where (2) ∂ ρ,e (χ m ) = ⟨ρ, m⟩χ m+e ∀m ∈ σ ∨ ∩ M .

  and, in the case where e ∈ S i ∩ M , ∂ = λ∂ ρ i ,e for some λ ∈ K. (c) ([36, Lem. 2.6 and Thm. 2.7]) A homogeneous derivation ∂ ∈ Der(A) is locally nilpotent if and only if ∂ = λ∂ ρ i ,e for a Demazure root e ∈ S i and for some λ ∈ K.

Remark 4 . 5 .

 45 The kernel of ∂ ρ,e is spanned by the characters χ m where m ∈ M belongs to the hyperplane section τ ρ of σ ∨ defined by ⟨ρ, m⟩ = 0. If e ∈ S i ∩ M is a Demazure root thenτ i ∶= τ ρ i = ρ ∨i is a facet of σ ∨ . The affine hyperplane H i spanned by the Demazure facet S i is parallel to τ i . Definition 4.6. Given a Demazure root e ∈ S i the associated one-parameter unipotent subgroup H e = exp(K∂ ρ i ,e ) ⊂ SAut(X) is called a root subgroup. The next lemma is mainly borrowed in [36, Lem. 1.10]. Lemma 4.7. (a) Any derivation ∂ ∈ Der(A) admits a decomposition (3) ∂ = e∈Σ ∨ ∩M ∂ e where ∂ e is a homogeneous derivation of degree e. (b) The set {e ∈ Σ ∨ ∩ M ∂ e ≠ 0} is finite. Its convex hull N (∂) is a polytope (called the Newton polytope of ∂). (c) If ∂ ∈ LND (A) then for any face τ of N (∂) one has ∂ τ ∶= e∈τ ∩M ∂ e ∈ LND (A) .In particular, for any vertex e of N (∂) one has ∂ e ∈ LND (A).

4. 1 . 1 .

 11 Iterated commutators on toric varieties. For homogeneous derivations one has the following lemma (cf. [45, Prop. 1 and Lem. 2]). Lemma 4.9. (a) For two nonzero homogeneous derivations ∂ = ∂ ρ,e and ∂ ′ = ∂ ρ ′ ,e ′ one has (4) [∂, ∂ ′ ] = ∂ ρ,ê where ρ = ⟨ρ, e ′ ⟩ρ ′ -⟨ρ ′ , e⟩ρ ∈ N and ê = e + e ′ . (b) If ρ ≠ 0 then [∂, ∂ ′ ] is a homogeneous derivation of degree e + e ′ with the linear form ρ. In particular, e + e ′ ∈ Σ ∨ ∩ M . (c) ρ = 0 (that is, ∂ and ∂ ′ commute) if and only if one of the following holds:

Lemma 4 . 13 . 1 where

 4131 Consider two nonzero homogeneous LNDs U = ∂ ρ 1 ,e 1 and V = ∂ ρ 2 ,e 2 in Der(A) where e i ∈ S i ∩ M , i = 1, 2. Letting c 2 = ⟨ρ 2 , e 1 ⟩, d 1 = ⟨ρ 1 , e 2 ⟩, and δ = d 1 + 1 and assuming that c 2 ≥ 1 one has ad m U (V ) = ∂ rm,fm ∀m = 0, . . . , d

Corollary 4 . 14 .

 414 Under the assumptions of Lemma 4.13 let ∂ = Ad exp(U ) (V ). Then the Newton polytope N (∂) is the segment [e 2 , e 2 + δe 1 ].

4. 1 . 2 .

 12 Der(A) as a graded Lie algebra. Given a lattice vector e ∈ Σ ∨ consider the linear subspace L e = span{∂ ρ,e } ρ∈N ⊂ Der(A) generated by the homogeneous derivations of A of degree e. By Lemma 4.9 one has [L e , L e ′ ] ⊂ L e+e ′ . Hence, the Lie algebra Der(A) is M -graded: Der(A) = ⊕ e∈M L e where L e ≠ {0} ⇔ e ∈ Σ ∨ ∩ M , see Proposition 4.4(b) and Lemma 4.7(a).

Example 4 . 15 .

 415 Consider the standard action of the 2-torus T on A 2 = Spec K[x, y]. This action induces an N 2 -grading on Der(K[x, y]) with graded pieces Der e (K[x, y]) where e = (i, j) ∈ Z 2 runs over the lattice vectors with i, j ≥ -1 and (i, j) ≠ (-1, -1). The piece Der e (K[x, y]) consists of all the homogeneous derivations of K[x, y] of degree e. For i, j ≥ 0 one has Der e (K[x, y]) = x i y j ax ∂ ∂x + by ∂ ∂y a, b ∈ K .

e∈{l T =m}∩Z 2 ∂ 4 . 2 .

 242 Any one-parameter subgroup T ⊂ T induces a Z-grading on both K[x, y] and Der(K[x, y]). Letting l T be the integral linear form on Z 2 associated with T and l T = m be a supporting affine line for the Newton polytope N (∂), consider the corresponding T -principal part of ∂, e ∈ Der(K[x, y]) ,where ∂ e stand for the homogeneous component of ∂ of degree e in decomposition[START_REF] Arzhantsev | Flexible varieties and automorphism groups[END_REF]. According to Lemma 4.7, if ∂ is locally nilpotent then also ∂ T is. In particular, for any vertex e of the Newton polytope N (∂) the corresponding derivation ∂ e is locally nilpotent. Consequently, all the vertices of the Newton polytope N (∂) are situated on the Demazure facets[START_REF] Yu | Affine group as a subgroup of biregular transformation group of an affine space[END_REF]. Hence the Newton polytope N (∂) is either a quadruple, a triangle, a line segment, or finally a point. Degeneration techniques. We explore the M -grading on Der(A) in the following degeneration trick. Proposition 4.16. Consider a subgroup G ⊂ Aut(X) normalized by a one-parameter subgroup T of the torus T. Let H = exp(K∂) be a G a -subgroup of G where ∂ ∈ LND (A), and let∂ T ∈ LND (A) be the T -principal part of ∂. Then H T = exp(K∂ T ) is a G a -subgroup of G.Proof. Let N (∂) be the Newton polytope of ∂, let l T be the linear form on M associated with T , and let l max = max{l T N (∂) } and l min = min{l T N (∂) } . Thus, one has ∂ T = ∂ τ for the face τ of N (∂) on which l T achieves its maximal value. The action of T on Der(A) defines a Z-grading. Any ∂ ∈ Der(A) admits a decomposition according with this grading: ∂ = lmax s=l min ∂ s where ∂ s = e∈N (∂)∩{l T =s} ∂ e ∈ Der(A) with ∂ e ∈ L e .

Corollary 4 . 17 .

 417 Under the assumptions of Proposition 4.16 suppose that G is normalized by the torus T. Then any vertex e of the Newton polytope N (∂) belongs to a Demazure facet S i and the root subgroup H e is contained in G. Proof. It suffices to apply Proposition 4.16 to a one-parameter subgroup T ⊂ T such that l T N (∂) achieves its maximum at e. Lemma 4.18. Letting n ≥ 2 consider two roots

4. 3 . 4 . 19 .

 3419 Cox ring and total coordinates. Let us recall some generalities on the Cox ring R(X) of a toric affine variety X, see, e.g., [2, Ch. 2], [15, Sect. 1], [16, Ch. 5] for detailed expositions. As before, Ξ = {ρ 1 , . . . , ρ k } stands for the set of primitive ray generators of the cone σ ⊂ N Q . To any ray ρ i ∈ Ξ there corresponds a facet ρ ∨ i of the dual cone σ ∨ and a Tinvariant prime Weil divisor D i = D(ρ i ) on X. The classes [D 1 ], . . . , [D k ] generate the class group Cl(X). The Cox ring R(X) is the polynomial ring K[x 1 , . . . , x k ] graded by the class group Cl(X) in such a way that any variable x i is a homogeneous element of degree deg(x i ) = [D i ] ∈ Cl(X), i = 1, . . . , k. This defines the grading uniquely.

  Demazure root and ê = Φ(e). (b) The action (12) on A k commutes with the F Cox -action on A k if and only if the LND

5. 1 . 5 . 1 .

 151 Infinite transitivity on the affine spaces: an example. The affine space A n = Spec K[x 1 , . . . , x n ] can be regarded as a toric variety. The mutually dual lattices N and M are the standard lattices of integer vectors N

2 , 3 .

 23 By Theorem 2.2, G acts infinitely transitively on the open orbit O G . By virtue of Proposition 3.4(c) the same is true for G and the open orbit O G = A n ∖ {0}. Remarks 5.7. 1. Theorem 5.2 does not hold any longer if one replaces x 1 + x 2

  acts 2-transitively on its open orbit then one has ab = 2.

  n and topologically cotame if ⟨Aff n , h⟩ ⊃ Tame n . The following result due to Edo ([19, Thm. 1.2]) extends and refines the earlier results of Bodnarchuk ([9, Thm. 3]) and Furter ([26, Thm. D]). Theorem 5.10. For n ≥ 2 any element h ∈ Aut(A n ) ∖ Aff n is topologically cotame. Remark 5.11. Recall that the triangular (de Jonquères) subgroup B n ⊂ Aut(A n ) is the subgroup generated by the torus T and the triangular root subgroups exp(x m ∂ i ) where x m = x m i+1

Corollary 5 . 14 .

 514 For any h ∈ SAut(A n ) ∖ SAff n one has ⟨SAff n , h⟩ ⊃ STame n .The group STame n acts infinitely transitively on A n . This leads to the following result. Corollary 5.15. For any n ≥ 2 and any h∈ SAut(A n ) ∖ SAff n the subgroup ⟨SAff n , h⟩ ⊂ SAut(A n ) acts infinitely transitively on A n .Let us provide an alternative direct proof of a similar result which does not apply the notion of cotameness.Theorem 5.16. For any n ≥ 2 and any non-affine root subgroup H u ⊂ Aut(A n ) the subgroup ⟨SAff n , H u ⟩ ⊂ STame n generated by n + 2 root subgroups of Aut(A n ) acts infinitely transitively on A n . Furthermore, for n = 2 there exists a collection of three root subgroups with the latter property.

  4(c). Since G ⊃ Transl n one has O G = A n . This gives the first assertion for n ≥ 3. Let further n = 2. Consider two affine and one non-affine roots e 2 = (0, -1) ∈ S 2 , v = e 2e 1 = (1, -1) ∈ S 2 , and u = (-1, 2) ∈ S 1 .

Fixing r ≫ 1 3 .

 13 consider the root (18) e 2 = r m1,2 + m1,2 ∈ S 2 with ⟨ρ 1 , e 2 ⟩ = 0, ⟨ρ 2 , e 2 ⟩ = -1, and ⟨ρ j , e 2 ⟩ ≥ 2 ∀j ≥ Choose a root e 3 ∈ S 3 in a similar fashion. Then in the total coordinates one has ê1 = (-1, * , . . . , * ), ê2 = (0, -1, * , . . . , * ), and ê3 = (0, * , -1, * . . . , * )

  1 , . . . , χ v n-1 ] ⊂ ker(∂ 1 ) , see Remark 4.5. According to the Claim for anyf ∈ A 1 the replica exp(Kf ∂ 1 ) of H e 1 is a subgroup of G 1 . Since rank (M 1 ) = n -1 one has [Frac(ker(∂ 1 )) ∶ Frac(A 1 )] < +∞ .Hence there exists b 1 ∈ ker ∂ 1 such that Frac (ker ∂ 1 ) is generated by b 1 and Frac (A 1 ). According to Remark 4.5 one can write b 1 = ∑ s j=1 c j χ m j where m j ∈ τ 1 ∩ M . Then H = exp(Kb 1 ∂ 1 ) is contained in the product of the root subgroups H r+j ∶= exp(Kχ m j ∂ 1 ), j = 1, . . . , s.

2 .

 2 Due to this theorem, G acts infinitely transitively on its open orbit O G = O G = reg(X). By virtue of Proposition 3.4(c) the same is true for G. 5.3.1. Final remarks. Theorem 5.20 leads to the following questions.5.21. Problem. Let X be a toric affine variety with no toric factor. What is the smallest number of root subgroups (G a -subgroups, respectively) H 1 , . . . , H s of Aut(X) such that G = ⟨H 1 , . . . , H s ⟩ acts on X with an open orbit and is infinitely transitive on this orbit?

  the relations (i)H u+e i ⊂ ⟨H u , H e i ⟩; (ii) H u+e i -e j ⊂ ⟨H u , H e i -e j ⟩ ∀j ≥ 2, j ≠ i; (iii) H 2u+e i -e 1 ⊂ ⟨H u , H e i -e 1 ⟩. Claim. One has H v ⊂ G ∶= ⟨SAff n , H u ⟩ for any root subgroup H v with v ∈ S 1 ∩ Z n .Proof of the Claim. Applying (ii) and (iii) successefully to u and the vectors obtained from u on each step one can get a root u ′ ∈ S 1 ∩ Z n whose coordinates dominate the corresponding coordinates of v and such that H u ′ ⊂ G. Applying now (i) one can conclude.Applying the cyclic permutations of coordinates one can see that the Claim holds as well for any Demazure root v. According to Theorem 2.2 the group G acts infinitely transitivily on its open orbit O G . Being algebraically generated the group G = ⟨SAff n , H u ⟩ does as well, see Proposition 3.

  τ 1 have total coordinates vi = (0, * , . . . , * ) where " * " ≥ 1. We claim that these vectors are linearly independent, that is, v 1 = e 2 + e 3 ∉ span(η 1 , . . . , η n-2 ) =∶ W . Indeed, any η ∈ W has total coordinates (0, 0, * , . . . , * ) whereas ⟨ρ 2 , v 1 ⟩ > 0. Claim. Consider the cone ω ⊂ τ 1 of dimension n-1 with ray generators v 1 , . . . , v n-1 . Consider also the submonoidM 1 = Z ≥0 v 1 + . . . + Z ≥0 v n-1 of ω of rank n -1 and the subgroup G 1 = ⟨H e 1 , H u 1 , H u 2 , . . . , H u n-1 , H e 3 ⟩ ⊂ SAut(X) .

That is, the group SAut(X) acts on X with an open orbit.

The first author was supported by the grant RCF-DFG 16-41-01013. This work was done during a stay of the first and the second authors at the Institut Fourier, Grenoble, France. They would like to thank this institution for hospitality, support, and excellent working conditions. 2010

We claim that the group G = ⟨H e 2 , H u , H v ⟩ acts infinitely transitively on A 2 . Indeed, by (i) one has H u+e 2 ⊂ ⟨H u , H e 2 ⟩ where u + e 2 = (-1, 1) ∈ S 1 . Since SL(2, K) = ⟨H u+e 2 , H v ⟩ and SAff 2 = ⟨SL(2, K), H e 2 ⟩ it follows that ⟨SAff 2 , H u ⟩ ⊂ G for the non-affine group H u .

The rest of the proof proceeds likewise in the case n ≥ 3. Notice first that H e 1 ⊂ SAff 2 ⊂ G. Let u i = (-1, i) ∈ S 1 . We know already that H u i ⊂ G for u 0 = e 1 , u 1 = u + e 2 , and u 2 = u.

Assume by induction that

The latter group acts infinitely transitively in A 2 . Then also G does in view of Proposition 3.4(c).

The following result completes the picture.

Theorem 5.17. For any n ≥ 2 one can find three G a -subgroups

acts infinitely transitively on A n .

Proof. For n = 2 the result follows from Theorem 5.16. Letting n ≥ 3 consider the root vector u = (-1, 2, 0, . . . , 0) ∈ S


V Moutarlier
email: virginie.moutarlier@univ-fcomte.fr

B Neveu

M P Gigandet

Evolution of corrosion protection for sol-gel coatings doped with inorganic inhibitors

Keywords: Sol-gel, Inorganic corrosion inhibitor, Aluminium alloy, GDOES

One the way to improve the self-repair properties of sol-gel films on aluminium alloy is the addition of corrosion inhibitors which could be released from the coating, minimising the corrosion of the unprotected area. Environmentally friendly corrosion inhibitors are studied (chromium III, molybdate, permanganate and cerium III) and compared to the standard corrosion inhibitor, i.e. hexavalent chromium.

Corrosion performances are studied by polarisation resistance (Rp) in chloride medium. Evolution of composition of sol-gel coatings, during the corrosion test, is examined by GDOES (glow discharge optical emission spectroscopy).

The results show that the morphology of sol-gel and the solubility of the additive play a role in the effectiveness of corrosion protection for a long term. Additives such as molybdate and permanganate ions decrease the sol-gel network stability and are too soluble (they are rapidly lost from the sol-gel films, in an aggressive medium), decreasing the power to prevent corrosion. Incorporation of Ce III is not efficient for a long time due to its high solubility. Sol-gel films containing Cr VI and Cr III provide adequate corrosion protection, due to the sol-gel stability and their low solubility.

Introduction

The sol-gel process is a method in which oxide films can be deposited on a substrate at much lower temperatures than traditional ceramic process methods [1]. A variety of metal alkoxides (M(OR) n , where R represents an organic group and M a metal) can be used as precursors for the sol-gel synthesis. Upon deposition, the coating undergoes hydrolysis represented by the Eq. ( 1) and condensation corresponding to Eqs. ( 2) and (3) [START_REF] Brinker | Sol-Gel Science[END_REF]:

Hydrolyse : MðORÞ n þ H 2 O→MðORÞ n-1 ðOHÞ þ ROH ð1Þ
Condensation : 2MðORÞ n-1 ðOHÞ→ðORÞ n-1 M-O-MðORÞ n-1 þ H 2 O ð2Þ
MðORÞ n-1 ðOHÞ þ MðORÞ n →ðORÞ n-1 M-O-MðORÞ n-1 þ ROH ð3Þ
One potential application of the sol-gel coatings is the protection against corrosion of aluminium alloy surface [START_REF] Osborne | [END_REF][4][5][6].

Such use is principally conducted using alkoxides based on silane (M = Si, n = 4) and zirconium (M = Zr, n = 4). The sol-gel films must provide an excellent adhesion to the substrate and a suitable protection against corrosion by creating a chemically inert barrier between the substrate and the aggressive environment [7]. However, these coatings cannot offer an adequate protection for a long term due to the presence of micro-pores, cracks and areas with low cross-link density. These zones favour the diffusion of aggressive species to the coating/substrate interface and are preferential sites of corrosion initiation [8,9]. In order to limit the corrosion process, corrosion inhibitors can be incorporated into the sol-gel films. The doped sol-gel film acts as a reservoir of corrosion inhibitors that, in an aggressive medium, diffuse through the coating and become active for corrosion process. The combination of sol-gel network stability and inhibitor solubility will allow to provide a high corrosion protection for a long term [START_REF] Osborne | [END_REF]10,11].

The most widely used and effective inhibitor is hexavalent chromium [START_REF] Vargel | Corrosion de l'aluminium[END_REF]. However, the use of Cr VI is becoming increasingly restrictive due to its characteristic as human carcinogen and substantial contribution to environmental pollution. Environmental friendly corrosion inhibitors for aluminium alloys, that were found to have a significant inhibiting action on corrosion process, are trivalent chromium [START_REF] Pearlstein | [END_REF], molybdate ions [14], permanganate ions [15] and cerium salts [START_REF] Campestrini | Microstructure-related quality of conversion coatings on aluminium alloys[END_REF].

The objective of this study is to investigate the behaviour during the corrosion process of the doped sol-gel films. These hybrid films are prepared with tetramethoxysilane (TMEOS), methyltriethoxysilane (MTEOS) and gamma-glycidoxypropylmethyldiethoxysilane (GPMDEOS) as a source of hydrolysable silane, tetra-propoxyzirconium (TPOZ) as a source of hydrolysable zirconium, and various inorganic compounds as corrosion inhibitors (Cr VI, Cr III, Ce III, Mo VI and Mn VII). The anticorrosion behaviours of the sol-gel films are characterised by electrochemical measurement: polarisation resistance (Rp). The evolution of the coating composition during the corrosion process is followed by glow discharge optical emission spectroscopy (GDOES).

Experimental

Substrate preparation

The substrate is 2017 aluminium alloy of composition (wt %): 92.3 Al, 4.2 Cu, 0.8 Mg, 0.8 Si, 0.7 Fe, 0.7 Mn, 0.3 Zn, 0.2 Ti [START_REF] Vargel | Corrosion de l'aluminium[END_REF]. The aluminium samples are soaked in an industrial acid cleaner solution, for 4 min, at 45 °C.

Synthesis and deposition of sol-gel films

The synthesis of the sol-gel solutions is performed by mixing reagents from Aldrich: tetramethoxysilane (TMEOS, 99%), methyltriethoxysilane (MTEOS, 90%), gamma-glycidoxypropylmethyldiethoxysilane (GPMDEOS) and tetra-propoxyzirconium (TPOZ, 70% in propanol). The selected inorganic inhibitors are chromium VI used for comparison, chromium III, molybdate, permanganate and cerium III as potential substitutes. The form of the inhibitor salts is listed in Table 1. The toxicity of the alternative inhibitors is lower than that of the hexavalent chromium, as shown in Table 1.

The first solution (solution A) is prepared by combining 6% TMEOS, 60% MTEOS, 20% GPMDEOS and 10% acidified is the quantity that when ingested, injected, or applied to the skin as a single dose will cause death of 50% of test animals. The value is expressed as mg kg -1 of body weight. The sol-gel films are produced by bare-coating. Bare-coating allows to form a thin film by a continuously moving of a bare on the substrate. After coating application, samples remain at room temperature during 30 min and are cured at 100 °C during 15 h.

Experimental techniques

The morphology of the sol-gel films is examined by scanning electron microscopy (SEM) with a JEOL model JSM 5600 operated at 22 kV.

The composition of the sol-gel coatings is studied by glow discharge optical emission spectroscopy (GDOES), using a Horiba Jobin Yvon GD Profiler [START_REF] Shimizu | [END_REF]. The sol-gel coatings are placed as cathode and are sputtered in an argon atmosphere by applying 60 W under 850 Pa. Light emission of characteristic wavelengths, associated with the sputtered species, is monitored throughout the analysis for depth profiling. The detected elements are Al (396 nm), O (130 nm), Si (288 nm), Zr (339 nm), C (156 nm), Cr (425 nm), Mo (386 nm), Mn (403 nm), Ce (413 nm).

To evaluate the anti-corrosive properties of the doped sol-gel films the polarisation resistance method is used. Polarisation resistance values (Rp) are determined during the immersion in 5% NaCl solution. The electrochemical cell is composed of three electrodes: a saturated calomel electrode (SCE) as reference, a titanium sheet as auxiliary electrode and a coated aluminium alloy as working electrode with the exposed area of 2 cm 2 . A PGP 201

. Morphology of doped sol-gel films

The SEM technique is used in order to observe the morphology of the different coatings. The thickness of coatings is estimated by cross-section analysis and is 7 μm ± 1 μm. The surfaces of the solgel films without additive, with Cr VI, Cr III and Ce III seem to be crack-free and without defects. A representative SEM micrograph is shown in Fig. 1a. Thus, in these cases, the organic/inorganic network is stable. On the contrary, particles clearly appear in the sol-gel surface containing Mo VI, as shown in Fig. 1b. The distribution of these particles is inhomogeneous and they have an average diameter around 10 μm. The agglomeration of these particles forms large aggregates (50 μm). According a previous study [18], this inhibitor salt could be crystallised in the sol-gel matrix or/and interfere with the condensation reaction, producing the disorganised sol-gel network. For the sol-gel film formed with Mn VII, few defects with a big size, around 40 μm of diameter, are observed (Fig. 1c). The brown colour of these defects indicates that it could be a manganese oxide precipitate. Thus, permanganate addition also disorganises the sol-gel network.

Composition of doped sol-gel films

The composition of the sol-gel coatings is investigated using the GDOES technique. GDOES profiles present the distribution of the detected elements versus the sputtering time [START_REF] Shimizu | [END_REF]. Fig. 2 shows the general GDOES depth profile of sol-gel films. The sol-gel/substrate interface is defined as the position where the Al signal increases. Thus, the sol-gel/substrate interface is at 60 seconds of analysis. The oxygen (× 2000), carbon (× 5), silicium (× 5) and zirconium (× 50) elements characterise the organic coating. The magnifications are chosen to observe all elements in the same scan. Profile indicates that the distribution of the elements of the hybrid coating is uniform.

Sol-gel coatings are doped with inorganic corrosion inhibitors and the GDOES technique is able to detect these additives. The GDOES profiles in Fig. 3 correspond to the additive signal only. All sol-gel/substrate interfaces are at 60 s of analysis. This confirms that all coatings have the same thickness. For the sol-gel films doped with Cr VI, Cr III and Ce III, the additives Cr and Ce are homogeneously incorporated, as we can see with the profiles in Fig. 3a, b andc, respectively. For the sol-gel film formed in presence of Mo VI, the signal of molybdenum is inhomogeneous (Fig. 3d). This additive appears less incorporated in the outer part of the coating. The distribution of Mn element for the sol-gel film doped with Mn VII is complex since the substrate is composed of manganese (Fig. 3e). The signal of Mn during the first 60 s of analysis corresponds to the additive incorporated into the sol-gel matrix and the last 60 s to the manganese present in the aluminium alloy. Manganese species are incorporated relatively uniformly in the coating.

Corrosion behaviours of doped sol-gel films

Polarisation resistance (Rp) measurements are realised in 5% NaCl solution, to estimate the corrosion behaviours of the different hybrid sol-gel coatings. GDOES technique is also used to followed the evolution of additives (Cr, Ce, Mo, Mn) at different periods of immersion.

Undoped sol-gel films

Fig. 4 shows the evolution of the Rp for the undoped sol-gel film, during the immersion in the aggressive medium. The Rp value is initially around 8 10 5 Ω cm 2 and fast decreases with the immersion time to reach a constant value of 2 10 5 Ω cm 2 . From about 25 h of immersion, pits appear on the surface of the solgel coating.

Sol-gel films doped with Cr VI

Addition of hexavalent chromium demonstrates significant improvement of corrosion protection (Fig. 5a). Indeed, initial Rp value is around 30 10 5 Ω cm 2 and slowly decreases to 8 10 5 Ω cm 2 at 100 h of immersion. Pits are not visualised after the corrosion test.

Fig. 5b corresponds to the GDOES Cr profiles realised at different periods of the corrosion test (1, 50 and 100 h of immersion)

. The Cr signal on the substrate part (after 60 seconds of analysis) is constant. Indeed, the composition of the aluminium alloy remains the same during the corrosion test. Thus, only the evolution of the Cr signal on the sol-gel part will be discussed. Additive signal slowly decreases during the corrosion test. This additive is still present in the sol-gel film after 100 h of immersion. This indicates a low solubility of hexavalent chromium from the hybrid film in an aggressive environment. Coating doped with Cr VI slowly releases the additive which becomes active for corrosion process. This solgel system holds enough additive for a long term protection [START_REF] Osborne | [END_REF].

Sol-gel films doped with Cr III

An improvement of corrosion performance is also observed for coating doped with Cr III (Fig. 6a). This hybrid film shows high initial Rp, above 26 10 5 Ω cm 2 . Then, Rp value decreases to 5 10 5 Ω cm 2 at 100 h that is still higher than the final value for the undoped film. However, these Rp values indicate that this coating is less protective than the one doped with hexavalent chromium. This is confirmed by the presence of pits at 80 h. Fig. 6b shows the evolution of the GDOES Cr signal realised at different periods of immersion. The decrease of the Cr signal is faster than for the coating doped with Cr VI. The trivalent chromium appears easier leached than the hexavalent chromium.

Sol-gel films doped with Ce III

The corrosion behaviour of sol-gel films formed with trivalent cerium is presented in Fig. 7a. At the beginning of corrosion test, the Rp values decrease from 13 to 3 10 5 Ω cm 2 . Then, at 10 h of immersion, Rp values show fast increase until 15 10 5 Ω cm 2 and finally decrease at 45 h to reach 6 10 5 Ω cm 2 . Pits are visualised after 50 h of immersion.

The GDOES profiles of Ce during the immersion time are shown in Fig. 7b. The Ce signal rapidly decreases during the corrosion test, indicating its diffusion through the sol-gel film. The inhibiting power of cerium appears after 10 h of immersion when the Rp values increase. As mentioned in previous works [19][20][21], the cerium inhibiting effect is related to the plugging of corrosion products into the pores, causing the increase of corrosion resistance. However, after 50 h of corrosion test, the most part of cerium ions is already released from the coating. Thus, the corrosion process could occur, explaining the decrease of Rp values and the apparition of pits after 50 h of immersion. The above results demonstrate an weak improvement of corrosion resistance and only for a short term, for the sol-gel film doped with Ce III [19][20][21].

Sol-gel films doped with Mo VI

The sol-gel film doped with Mo VI presents the lowest corrosion resistance (Fig. 8a). Indeed, the Rp values rapidly decrease from 14 to 7 10 3 Ω cm 2 that is significantly lower than for the undoped sol-gel film. Large pits are rapidly initiated in the sol-gel coating.

Fig. 8b shows the GDOES profiles of Mo during the immersion time. The signal of Mo rapidly decreases to be constant at 50 h of immersion. This indicates that Mo has completely disappeared from the sol-gel matrix. The poor corrosion protection could be due to the presence of big defects at the sol-gel surface (see Fig. 1b) and to the high solubility of this additive. The defects are preferential sites of pit initiation and provide pathways for the diffusion of the corrosive species [9]. The high solubility induces that Mo VI could not act as corrosion inhibitor for a long time.

Sol-gel films doped with Mn VII

The evolution of Rp values for the sol-gel film formed with Mn VII (Fig. 9a) indicates an initially corrosion protection (Rp = 11 10 5 Ω cm 2 ) closed to the undoped sol-gel coating, followed to a significant decrease of the corrosion resistance (Rp = 2 10 4 Ω cm 2 at 100 h of immersion). Large pits are observed after 20 h of immersion in the aggressive medium. These pits are probably initiated on the big defects previously observed (see Fig. 1c).

The GDOES profiles of Mn at different immersion times are presented in Fig. 9b. During the immersion, the Mn signal on the sol-gel part rapidly decreases, indicating a strong solubility of this additive. At the end of the corrosion test, the manganese is absent in the sol-gel matrix. As for the molybdenum species, the poor corrosion protection of the sol-gel doped with Mn VII could be explain by the presence of defects and the high solubility of this additive.

Conclusion

In order to improve corrosion protection for a long term, inorganic corrosion inhibitors have been incorporated into the sol-gel matrix. Polarisation resistance measurements have been employed to follow the corrosion behaviour in NaCl solution. GDOES analysis has been used to investigate the evolution of the composition of the sol-gel coatings, during the immersion in the aggressive medium.

The protection of the coating formed with hexavalent chromium, against the corrosive species, was efficient for a long term. Indeed, the absence of pits was noted after the corrosion test. GDOES analysis has shown that the chromium additive slowly released and could act durably on the corrosion process.

Various non-Cr VI corrosion inhibitors have been added to the hybrid coatings. Although the incorporation of chromium III has induced comparable corrosion behaviour than for the Cr VI, this additive has appeared easier leached. Thus, its protection for a long term seems to be slightly weaker. Incorporation of cerium III is not efficient due to its high solubility. Additions of molybdenum VI and magnesium VII did not provide an adequate corrosion protection due to the disorganisation of the sol-gel systems (presence of particles and aggregates in the coating surface) and to the high solubility of these additives. The defects favoured the pit formation. The high solubility of the additive makes that the doped sol-gel films could not act as a reservoir for a prolonged release.

Structure of sol-gel film and solubility of the additive undoubtedly play a role in the effectiveness of corrosion protection for a long term. However, an other parameter must be taken into account: the inhibiting power of each corrosion inhibitor. Thus, addition experiments are necessary to determine the mechanism of corrosion protection of these additives. This is the objective of the future work.

Fig. 1 .

 1 Fig. 1. SEM micrographs (×550) of surfaces of the sol-gel films formed a) without additive, with Cr VI, Cr III, Ce III, b) with Mo VI and c) with Mn VII. Fig. 2. GDOES depth profile of sol-gel film on aluminium alloy.

Fig. 3 .

 3 Fig. 3. GDOES depth profiles of additives from sol-gel films formed with a) Cr VI, b) Cr III, c) Ce III, d) Mo VI and e) Mn VII.

3 .

 3 Results and discussion 3.1. Examination of doped sol-gel films 3.1.1

Fig. 4 .

 4 Fig. 4. Corrosion behaviour of undoped sol-gel film: polarisation resistance during the immersion in 5% NaCl solution.

Fig. 5 .

 5 Fig. 5. Corrosion behaviour of sol-gel film doped with Cr VI: a) polarisation resistance and b) GDOES depth profiles of Cr, during the immersion in 5% NaCl solution.

Fig. 7 .

 7 Fig. 7. Corrosion behaviour of sol-gel film doped with Ce III: a) polarisation resistance and b) GDOES depth profiles of Ce, during the immersion in 5% NaCl solution.

Fig. 6 .

 6 Fig. 6. Corrosion behaviour of sol-gel film doped with Cr III: a) polarisation resistance and b) GDOES depth profiles of Cr, during the immersion in 5% NaCl solution.

Fig. 8 .

 8 Fig. 8. Corrosion behaviour of sol-gel film doped with Mo VI: a) polarisation resistance and b) GDOES depth profiles of Mo, during the immersion in 5% NaCl solution.

Fig. 9 .

 9 Fig. 9. Corrosion behaviour of sol-gel film doped with Mn VII: a) polarisation resistance and b) GDOES depth profiles of Mn, during the immersion in 5% NaCl solution.

Table 1

 1

	Information concerning selected corrosion inhibitors
	Inhibitor	Cr VI Cr III	Mo VI	Mn VII	Ce III
	Compound Toxicity (LD 50 ) ⁎ 55 CrO 3	Cr(NO 3 ) 3 Na 2 MoO 4 KMnO 4 Ce(NO 3 ) 3 1790 570 1090 unknown
	⁎ LD 50