BASS’ TRIANGULABILITY PROBLEM

VLADIMIR L. POPOV

Abstract. Exploring Bass’ Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass’ Triangulability Problem is the affirmative. To this end we prove a theorem on invariant subfields of 1-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy.

1. Introduction

We assume given an algebraically closed field k of arbitrary characteristic which serves as domain of definition for each of the varieties considered below. In this paper, “variety” means “algebraic variety” and it is identified with its set k-rational points.

Recall that the Cremona group (over k) of rank n is the group

C_{n} := Aut_{k} k(A^{n}),

and Aut_{k} k[A^{n}] is the affine Cremona group (over k) of rank n. The group Bir A^{n} of rational self-maps of A^{n} (resp. the group Aut A^{n}) is identified with C_{n} (resp. Aut_{k} k[A^{n}]) by means of the isomorphism φ ↦ (φ^{∗})^{−1}. For n > 1, we identify k(A^{n−1}) with the subfield of k(A^{n}) by means of the natural embedding k(A^{n−1}) ↪ k(A^{n}) determined by the projection A^{n} → A^{n−1}, (α_{1}, …, α_{n}) ↦ (α_{1}, …, α_{n−1}). This makes k[A^{n−1}] the subalgebra of k[A^{n}]. We put k(A^{0}) = k[A^{0}] := k.

Let x_{i}: A^{n} → k, (α_{1}, …, α_{n}) ↦ α_{i}, be the ith standard coordinate function. We have

k(A^{n}) = k(x_{1}, …, x_{n}), k[A^{n}] = k[x_{1}, …, x_{n}].  (1)

The group C_{n−1} is identified with the subgroup of C_{n} by means of the embedding

C_{n−1} ↪ C_{n}, φ ↦ φ̃, where φ̃(x_{i}) := φ(x_{i}) for i < n and φ̃(x_{n}) := x_{n}. (2)

This makes Aut_{k} k[A^{n−1}] the subgroup of Aut_{k} k[A^{n}].

∗ Supported by grants ���� 15-01-02158 and ��-2998.2014.1.

Although the groups C_{n} and Aut_{k} k[A^{n}] are infinite-dimensional for n > 1 (see [Ra 1964], [Po 2014]), the analogies between them and algebraic groups catch the eye: they have the Zariski topology, algebraic subgroups, tori, roots, the Weyl groups, … (see [Po 2013_{1}], [Po 2013_{2}] and references therein). The de Jonquières subgroup

J_{n} := {φ ∈ Aut_{k} k[A^{n}] | φ(x_{i}) = α_{i}x_{i} + h_{i}, α_{i} ∈ k^{×}, h_{i} ∈ k[A^{i−1}]}

is viewed by some authors as an analog of Borel subgroup for Aut_{k} k[A^{n}] (see, e.g., [Ba 1984]). Supporting this viewpoint, [Po 2013_{1}, Thm. 3.1] implies that every algebraic subgroup of J_{n} is affine solvable. Having in mind conjugacy of Borel subgroups in every finite dimensional affine algebraic group, one leads to the question whether it is true that every connected solvable affine algebraic subgroup G of Aut_{k} k[A^{n}] is conjugate in Aut_{k} k[A^{n}] to a subgroup of J_{n}. In particular, Problem III in [Ba 1984] asks whether it is true for unipotent G. In [Ba 1984] Bass answered the latter question in the negative for n = 3 and G = k^{+}, the one-dimensional additive group. In [Po 1987] was then elaborated a simple general method yielding negative answers for G = k^{+} and all n > 2 (this method, in the form of usage of so called replicas, became the crucial instrument in the recent studies on infinite transitivity of automorphism groups of algebraic varieties [Ka 2012]). Given these developments, Bass formulated for char k = 0 the following

Bass’ Triangulability Problem ([Ba 1984, Question 4]). “If a unipotent group G acts on A^{n}, can the action be rationally triangularized, i.e., can we write k(x_{1}, …, x_{n}) = k(y_{1}, …, y_{n}) so that each subfield k(y_{1}, …, y_{i}) is G-invariant?”

Here we explore this problem in the broader context of connected solvable affine algebraic groups G and arbitrary char k. To formulate our results we first introduce two definitions.

Definition 1 (1-extensions). If a field K is a simple transcendental extension of a field L (i.e., K is a purely transcendental field extension of L of degree 1), then, for brevity, we say that K is a 1-extension of L.

Definition 2 (Rationally triangulable subgroups). A subgroup G of C_{n} is called rationally triangulable if there is a flag

k(A^{n}) =: K_{n} ⊃ K_{n−1} ⊃ ⋯ ⊃ K_{1} ⊃ K_{0} := k  (3)

of G-stable subfields of k(A^{n}) such that K_{i}/K_{i−1} is a 1-extension for every i > 0. A rational action of an algebraic group on A^{n} is called rationally triangulable if the image of this group under the homomorphism to C_{n} determined by this action is rationally triangulable.

Now we shall formulate our results.

We start with proving the following theorem that is heavily used in the proofs of our next results.

Theorem 1 (Invariant subfields of 1-extensions). Let Q be a finitely generated field extension of k and let P be a 1-extension of Q. Let G be a connected solvable affine algebraic subgroup of Aut_{k}(P) such that Q is G-stable. Then

(i) P^{G} = Q^{G} if Q^{G} = Q;

(ii) P^{G} is a 1-extension of Q^{G} if Q^{G} ⊊ Q.

Using Theorem 1, we obtain the following triangulability criterion:

Theorem 2 (Triangulability criterion). The following properties of a connected solvable affine algebraic subgroup G of the Cremona group C_{n} are equivalent:

(i) k(A^{n})^{G} is purely transcendental over k; (ii) G is rationally triangulable.

Theorem 2 generalizes [DF 1991, Thm. 3.1], where the claim is proved for one-dimensional unipotent algebraic subgroups of Aut_{k}A^{n} in case char k = 0.

Corollary 1 (Low-dimensional quotients). A connected solvable affine algebraic subgroup G of C_{n} is rationally triangulable in either of the following cases:

(i) tr deg_{k}k(A^{n})^{G} ⩽ 1;

(ii) tr deg_{k}k(A^{n})^{G} = 2 and char k = 0;

(iii) G ⊂ Aut A^{n} and (a) dim G · x ⩾ n − 1 for some point x ∈ A^{n}, or

(b) char k = 0 and dim G · x = n − 2 for some point x ∈ A^{n}.

Corollary 2 (3-dimensional affine space). If char k = 0, then every connected solvable affine algebraic subgroup of Aut A^{3} is rationally triangulable.

Corollary 2 generalizes [DF 1991, Cor. 3.2], where the claim is proved for one-dimensional unipotent algebraic subgroups of Aut A^{3} and char k = 0.

Corollary 3 (Tori). The following properties of an affine algebraic torus T in the Cremona group C_{n} are equivalent:

(i) T is rationally triangulable,

(ii) T is linearizable, i.e., conjugate in C_{n} to a subgroup of GL_{n}.

Next we show that the nontriangulable connected solvable affine algebraic subgroups of C_{n} do exist. In particular, the following theorem implies that in case (ii) of Corollary 1 it is not possible to replace 2 by a bigger integer.

Theorem 3 (Nontriangulable subgroups). Let n be an integer ⩾ 5. Every (n−3)-dimensional connected solvable affine algebraic group G is isomorphic to a rationally nontriangulable algebraic subgroup of the Cremona group C_{n}.

As far as we do not claim that the subgroup in the formulation of Theorem 3 lies in Aut_{k} k[A^{n}], this theorem does not furnish the negative answer to Bass’ Triangulability Problem. However, its proof demonstrates intimate interrelation between triangulability and Zariski’s Cancellation Problem: it shows that if there is a nonrational variety Z such that A^{n} is isomorphic to A^{s} × Z, then the answer to Bass’ Triangulability Problem is negative (in view of this it is worth to recall that in positive characteristic Zariski’s Cancellation Problem is solved in the negative in [Gu 2014]); by Theorems 5, 6, the converse it true on birational level.

On the other hand, in the stable range we do answer Bass’ Triangulability Problem is the affirmative. Namely, the following theorem shows that despite the existence of rationally nontriangulable connected solvable affine algebraic subgroups of C_{n}, every such subgroup is stably rationally triangulable. More precisely, the following holds true.

Theorem 4 (Stable triangulability). Every connected solvable affine algebraic subgroup G of the Cremona group C_{n} is rationally triangulable in the Cremona group C_{m} for every

m ⩾ 2n − tr deg_{k}k(A^{n})^{G}.  (4)

Theorem 4 generalizes [DF 1991, Thm. 3.1], where the statement is proved for one-dimensional unipotent algebraic subgroups of Aut_{k}A^{n}, assuming char k = 0.

Next we obtain a general construction of all rationally triangulable subgroups of C_{n}, see Theorem 6 in Section 3. As an application, it leads to the classification (given below in Corollary 4) of rationally triangulable onedimensional connected unipotent algebraic subgroups in C_{n} up to conjugacy. In this classification we use the following terminology. A one-dimensi-onal connected unipotent algebraic subgroup G of C_{n}, identified with k^{+} by means of an isomorphism G → k^{+}, is called standard if x_{1}, … x_{n−1} ∈ k[A^{n}]^{G} and, for every t ∈ k^{+}, the following holds:

(i) for char k = 0, we have

t(x_{n}) = x_{n} + t

(in this case G is also called the translation); (ii) for char k = p > 0, we have

t(f_{n}) = f_{n} + c_{1}t^{p^{i_{1}}} + ⋯ + c_{d}t^{p^{i_{d}}} ,

where all c_{j}’s are the nonzero elements of k(x_{1}, …, x_{n−1}), and i_{1} < ⋯ < i_{d} are the nonnegative integers.

Corollary 4 (One-dimensional rationally triangulable unipotent subgroups). A one-dimensional connected unipotent algebraic subgroup of C_{n} is rationally triangulable if and only if it is conjugate in C_{n} to a standard subgroup.

Corollary 4 generalizes [DF 1991, Thm. 2.2], where the claim is proved for char k = 0.

Corollary 5 (Low-dimensional affine spaces). Let U be a one-dimensional connected unipotent algebraic subgroup of C_{n}.

(i) If n = 2, then U is conjugate in C_{2} to a standard subgroup.

(ii) If char k = 0 and n = 3, then U is conjugate in C_{3} to the translation.

By [Re 1968], if char k = 0, n = 2, and U ⊂ Aut A^{2}, then “in C_{2}” in Corollary 5(i) may be replaced by “in Aut A^{2}”.

By [Ka 2004], for k = C, n = 3, U ⊂ Aut A^{3}, if U acts on A^{3} freely, then U is conjugate in Aut A^{3} to the translation. Corollary 5(ii) shows that allowing conjugation in C_{3}, the “if” assumption in this result may be dropped, i.e., conjugacy to the translation becomes true for every U in C_{3}.

The proofs of Theorems 1, 2, 3, 4 and Corollaries 1, 2, 3 are given in Section 2. Theorem 6 is formulated and proved, together with Corollaries 4, 5, in Section 3.

Notation and conventions. We use freely the standard notation and conventions of [PV 1994], [Bo 1991], [Sp 1998] and refer to [Ro 1956], [Ro 1961], [Ro 1963], [PV 1994], [Po 2013_{1}] regarding the definitions and basic properties of rational and regular (morphic) actions of algebraic groups on varieties.

Unless otherwise specified, we will assume that every field appearing below contains k and every embedding of fields restricts to the identity map on k.

2. Proofs of Theorems 1, 2, 3, 4 and Corollaries 1, 2

Proof of Theorem 1.

1. We argue by induction on dim G. Assuming the statement is proved for dim G = 1, the inductive step is proved as follows.

Since G is connected solvable affine algebraic group, it contains a closed connected normal subgroup N such that dim G/N = 1, see [Gr 1956, p. 6–03, Cor. 1]. We have

(P^{N})^{G/N} = P^{G} and (Q^{N})^{G/N} = Q^{G}  (5)

By the inductive assumption one of the following holds:

(a) P^{N} = Q^{N};

(b) P^{N} is a 1-extension of Q^{N}.

If (a) is fulfilled, then P^{G} = Q^{G} in view of (5), i.e., (i) holds.

If (b) is fulfilled, then, since G/N is a one-dimensional connected solvable affine algebraic group, (i) follows from (5) and the assumption that the statement is proved for one-dimensional groups.

2. Thus the proof is reduced to the case where dim G = 1. We now assume that this equality holds.

The assumptions on Q and P imply that there is an irreducible variety X such that for

\begin{matrix}Y := A^{1} × X, \\ ρ: Y → X, (a, x) ↦ x, \end{matrix}  (6)

we may (and shall) identify P with k(Y):

P = k(Y),  (7)

and Q with the image of embedding ρ^{∗}: k(X) ↪ k(Y):

Q = ρ^{∗}(k(X)).  (8)

The actions of G on Q and P determine the rational actions of G on X and Y. The action on Y is faithful and the morphism ρ is equivariant with respect to these actions.

By Weil’s regularization theorem [We 1955] (see also [Ro 1956, Thm. 1]) there are

— irreducible algebraic varieties X̃ and Ỹ;

— nonempty open subsets X_{0} and Y_{0} in, respectively, X and Y;

— open embeddings ι_{1} : X_{0} ↪ X̃, ι_{2} : Y_{0} ↪ Ỹ such that the induced rational actions of G on X̃ and Ỹ are regular (morphic). The action of G on Ỹ is faithful since that on Y is. We identify X_{0} and Y_{0} with the images of, respectively, ι_{1} and ι_{2}, that yields the natural identifications

k(X) = k(X̃), k(Y) = k(Ỹ).  (9)

By Rosenlicht’s theorem on generic quotients [Ro 1956, Thm. 2], replacing X̃, X_{0}, Ỹ, and Y by the appropriate open subsets, we may (and shall) assume that for the actions of G on X̃ and Ỹ respectively there are the geometric quotients

π_{X̃,G}: X̃ → X̃/G, π_{Ỹ ,G}: Ỹ → Ỹ/G.

In particular, π_{X̃,G} and π_{Ỹ ,G} are equidimensional morphisms, their fibers are G-orbits, and, in view of (7), (8), (9),

ρ^{∗} ∘ π_{X̃,G}^{∗}: k(X̃/G) ⟶^{≃} Q^{G},  π_{Ỹ ,G}^{∗}: k(Ỹ/G) ⟶^{≃} P^{G}.  (10)

Since dim G = 1 and G acts on Ỹ faithfully, every fiber of π_{Ỹ ,G} is one-di-mensional; in view of (6), this yields

dim Ỹ/G = dim Y − 1 = dim X.  (11)

The morphism ρ induces a G-equivariant dominant rational map

ρ̃: Ỹ −−→ X̃.

Since its domain of definition is G-stable, replacing the varieties again by the appropriate open subsets we may (and shall) assume that ρ̃ is a Gequivariant surjective morphism.

Thus we obtain the following commutative diagram

\xymatrix{Y \ar[d]_{ρ} & Y_{0} \ar@{^{(}->}[l]_{ι_{4}} \ar@{^{(}->}[r]^{ι_{2}} \ar[d]^{ρ_{0}} & Ỹ \ar[d]^{ρ̃} \ar[r]^{π_{Ỹ ,G}} & Ỹ/G \\ X & X_{0} \ar@{^{(}->}[l]_{ι_{3}} \ar@{^{(}->}[r]^{ι_{1}} & X̃ \ar[r]^{π_{X̃,G}} & X̃/G} ,  (12)

where ρ_{0} := ρ|_{Y_{0}} = ρ̃|_{Y_{0}} and ι_{3}, ι_{4} are the identical embeddings. Finally, replacing X_{0} and Y_{0} by the appropriate open subsets, we may (and shall) assume that ρ_{0}(Y_{0}) = X_{0}.

Now we shall consider separately two arising possibilities.

3. First, consider the case

Q^{G} = Q.  (13)

By (8), condition (13) is equivalent to triviality of the action of G on X̃. ¿From Q ⊂ P and (13) we obtain

Q ⊆ P^{G} ⊂ P,  (14)

¿From (7), (9), (11) we infer tr deg_{P^{G}}P = 1, and (6), (7), (9) yield tr deg_{Q}P = 1. Whence by (14) we obtain tr deg_{Q}P^{G} = 0. Since P is a 1-extension of Q, by Lüroth’s theorem ([Lü 1876], see also, e.g., [vdW 1971, §73]) the latter equality implies that P^{G} = Q. Thus, by (13), in the case under consideration we have P^{G} = Q^{G}. This proves claim (i) of Theorem 1.

4. Now consider the case Q^{G} ⊊ Q, i.e., G acts on X̃ nontrivially. Every fiber of π_{X̃,G} is then a one-dimensional G-orbit; whence

dim X̃/G = dim X − 1.  (15)

In view of (7), (8), (9), (11), (15), we have

tr deg_{Q^{G}}P^{G} = 1.  (16)

Since G is a connected solvable affine algebraic group, by Rosenlicht’s cross-section theorem [Ro 1956, Thm. 10] there is a rational section

σ: X̃/G −−→ X̃

of π_{X̃,G}, i.e. a rational map such that

π_{X̃,G} ∘ σ = idX̃.  (17)

Since g ∘ σ for every element g ∈ G is also a rational section of π_{X̃,G}, we

may (and shall) assume that σ(X̃/G) ∩ X_{0} ≠ ∅. This implies that there is a locally closed irreducible subvariety S ⊂ X_{0} such that

π_{X̃,G}|_{S}: S → X̃/G is an open embedding.  (18)

In view of (10), this means that

k(X̃)^{G} → k(S), f ↦ f|_{S}, is a well-defined isomorphsim.  (19)

In particular, in view of (15), we have

dim S = dim X − 1.  (20)

From (6) we obtain that

ρ^{−1}(S) = A^{1} × S.  (21)

Consider in Ỹ the locally closed irreducible subvariety

Z := ρ^{−1}(S) ∩ Y_{0}.  (22)

¿From (21) and (22) we infer that

k(Z) is a 1-extension of K := ρ|_{Z}^{∗}(k(S)),  (23)

and from (11), (20) that

dim Z = dim X = dim Ỹ/G.  (24)

We claim that the morphism

ζ := π_{Ỹ ,G}|_{Z}: Z → Ỹ/G  (25)

is dominant. In view of (24), to prove this, it suffices to show that, for every point z ∈ Z, the fiber ζ^{−1}(ζ(z)) is finite. Assume the contrary. Since ζ^{−1}(ζ(z)) = Z ∩ 𝒪, where 𝒪 := π_{Ỹ ,G}^{−1}(π_{Ỹ ,G}(z) is a certain G-orbit, we then infer from dim 𝒪 = 1 that

dim(Z ∩ 𝒪) = 1.  (26)

Since ρ̃ is a G-equivariant morphism, 𝒪′ := ρ̃(𝒪) is also a G-orbit. Hence dim 𝒪′ = 1, because all G-orbits in X̃ are one-dimensional. The latter equality and (26) imply that ρ̃(Z ∩ 𝒪) is an open subset of 𝒪′; in particular, it is infinite. On the other hand, (22) yields that ρ̃(Z ∩ 𝒪) lies in S ∩ 𝒪′. Since, by (18), the latter is a single point, we obtain a contradiction. This proves the claim.

Thus, since ζ is dominant, it defines an embedding of fields ζ^{∗}: k(Ỹ/G) ↪ k(Z). In view of (7), (9), this means that

τ: P^{G} ↪ k(Z), f ↦ f|_{Z}, is a well-defined embedding,  (27)

and (8), (9), (19), (23) imply that

τ: Q^{G} ↪ K is an isomorphism.  (28)

Thus (27), (28) yield the following commutative diagram

\xymatrix{k(Z) & τ(P^{G}) \ar@{^{(}->}[l]_{id} & K \ar@{^{(}->}[l]_{id} \\ & P^{G} \ar[u]^{τ}_{≃} & Q^{G} \ar@{^{(}->}[l]_{id} \ar[u]^{τ}_{≃}} .

In view of (16), (23), this diagram and Lüroth’s theorem imply that P^{G} is a 1-extension of Q^{G}. This completes the proof of claim (ii) of Theorem 1. □

Proof of Theorem 2.

(i)⇒(ii) Assume that k(A^{n})^{G} is a purely transcendental field extension of k. Then there is a flag

k(A^{n})^{G} =: I_{t} ⊃ I_{t−1} ⊃ ⋯ ⊃ I_{1} ⊃ I_{0} := k  (29)

of the subfields of k(A^{n})^{G} such that I_{i}/I_{i−1} is a 1-extension for every i > 0.

Since G is a connected solvable affine algebraic group, there is a flag of its closed connected normal subgroups

G =: G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{s−1} ⊃ G_{s} := {e}  (30)

such that dim G_{i−1}/G_{i} = 1 for every i > 0, see [Gr 1956, p. 6–03, Cor. 1]. ¿From (30) we obtain the following flag of G-stable subfields of k(A^{n}):

k(A^{n}) = K_{s} ⊇ K_{s−1} ⊇ ⋯ ⊇ K_{1} ⊇ K_{0} = k(A^{n})^{G}, K_{i} := k(A^{n})^{G_{i}}. (31)

By construction,

K_{i−1} = k(A^{n})^{G_{i−1}} = (k(A^{n})^{G_{i}})^{G_{i−1}/G_{i}} = K_{i}^{G_{i−1}/G_{i}} for every i > 0. (32)

If the action of G_{i−1}/G_{i} on K_{i} is trivial, then (32) yields K_{i} = K_{i−1}. If it is nontrivial, then, since G_{i−1}/G_{i} is a one-dimensional connected solvable affine algebraic group, we infer from (32) and [Po 2015, Thm. 1] (or [Ma 1963, Thm. 1] if char k = 0) that K_{i}/K_{i−1} is a 1-extension. Therefore, once repetitions in flag (31) are eliminated, we obtain a flag

k(A^{n}) =: L_{d} ⊃ L_{d−1} ⊃ ⋯ ⊃ L_{1} ⊃ L_{0} := k(A^{n})^{G}  (33)

of G-stable subfields of k(A^{n}) such that L_{i}/L_{i−1} is a 1-extension for every i > 0. From (33) and (29) we then obtain the flag

k(A^{n}) =: L_{d} ⊃ L_{d−1} ⊃ ⋯ ⊃ L_{1} ⊃ L_{0} =I_{t} ⊃ I_{t−1} ⊃ ⋯ ⊃ I_{1} ⊃ I_{0} := k

of G-stable subfields of k(A^{n}) whose “levels” are 1-extensions. By Definition 2 the group G is then rationally triangulable.

(ii)⇒(i) Conversely, assume that the group G is rationally triangulable and consider flag (3) of G-stable subfields of k(A^{n}). Passing to the Ginvariant subfields, we then obtain the following flag of subfields of k(A^{n})^{G}:

k(A^{n})^{G} = K_{n}^{G} ⊇ K_{n−1}^{G} ⊇ ⋯ ⊇ K_{1}^{G} ⊇ K_{0}^{G} = k.  (34)

By Theorem 1 every “level” of this flag is either the equality or a 1-extension. Therefore, (34) implies that k(A^{n})^{G} is a purely transcendental extension of k. This completes the proof. □

Proof of Corollary 1. In cases (i) and (ii), the claim follows, in view of Theorem 2, from, respectively, Lüroth’s theorem and Castelnuovo’s theorem (see, e.g., [Ha 1977, Chap. V, 6.2.1]). In case (iii), it follows from (i) and (ii), since, by [Ro 1956, Thm. 2], tr deg_{k}k(A^{n})^{G} = n − max_{a∈A^{n}} G · a. □

Proof of Corollary 2. This follows from Corollary 1. □

Proof of Corollary 3. In view of Theorem 2, this follows from [Po 2013_{1}, Thm. 2.4]. □

Proof of Theorem 3. By [S-B 2004] there exists a nonrational threefold X such that A^{2} ×X is birationally isomorphic to A^{5}. Hence we may (and shall)

fix a birational isomorphism A^{n−3} × X −−→ A^{n}.  (35)

Since the underlying variety of G is rational (see [Gr 1958, p. 5-02, Cor.]), we also may (and shall) fix a birational isomorphism between it and A^{n−3}. We then obtain from the action of G on itself by left translations a faithful rational action of G on A^{n−3} such that

k(A^{n−3})^{G} = k.  (36)

In turn, the latter action yields a faithful rational action of G on A^{n−3} × X via the first factor. By [Bo 1991, Cor. of Prop. II.6.6] and (36), for this action,

k(A^{n−3} × X)^{G} and k(X) are isomorphic.  (37)

Thus, given (35), we obtain a faithful rational action of G on A^{n} such that the field k(A^{n})^{G} is isomorphic to k(X), and hence is not purely transcendental over k according to the choice of X.

By Theorem 2 we then conclude that the algebraic subgroup of C_{n} determined by this action is isomorphic to G and rationally nontriangulable. □

Proof of Theorem 4. First, we shall introduce some notation. Given a rational action of an algebraic group H on an irreducible algebraic variety Z, we denote by Z---H a rational quotient of this action, i.e., an irreducible variety (uniquely defined up to birational isomorphism) such that there exists an isomorphism k(Z---H) → k(Z)^{H}. We shall write X ≈ Y if X and Y are birationally isomorphic irreducible varieties.

By [Po 2015, Thm. 1] (or [Ma 1963, Thm. 1] if char k = 0) we have

A^{n} ≈ A^{n}---G × A^{s}, where s := n − tr deg_{k}k(A^{n})^{G}.  (38)

Take an integer m such that (4) holds and consider G as a subgroup of C_{m} via embedding (2). The arising rational action of G on A^{m} = A^{n} × A^{m−n} is that through the first factor. Therefore

\begin{matrix}A^{m}---G & = (A^{n}---G) × A^{m−n} \\ & =^{(4)} (A^{n}---G) × A^{s} × A^{m−n−s} \\ & ≈^{(38)} A^{n} × A^{m−n−s} = A^{m−s} \end{matrix}\} .  (39)

By (39) the field k(A^{m})^{G} is purely transcendental over k. Hence by Theorem 2 the group G is rationally triangulable in C_{m}. □

3. Construction ⊛

Now we shall give a general construction of all rationally triangulable subgroups of C_{n}. It is prompted by the following result from [Po 2015] that yields a general construction of all connected solvable affine algebraic subgroups of C_{n}:

Theorem 5 (Standard invariant open subsets [Po 2015, Thm. 3]). Let X be an irreducible variety endowed with a regular action of a connected solvable affine algebraic group S. Then for the restriction of this action on a certain S-stable nonempty open subset Q of X there exist — the geometric quotient π_{S,Q}: Q → Q/S;

— an isomorphism φ: Q → A^{r,u} × (Q/S), where

A^{r,u} := {(α_{1}, …, α_{r+u}) ∈ A^{r+u} | α_{i} ≠ 0 for every i ⩽ r}, r ⩾ 0, u ⩾ 0,

such that the natural projection A^{r,s} × (Q/S) → Q/S is the geometric quotient of the regular action of S on A^{r,u} × (Q/S) induced by φ.

Theorem 5 leads to the following Construction ⊛

Let S be a connected solvable affine algebraic group and let Z be an irreducible variety such that, for some nonnegative integers r, u,

the variety Z × A^{r,u} is rational.  (40)

Consider a map

φ: S × Z → Aut A^{r,u}

that has the following properties:

(i) φ is an algebraic family [Ra 1964], [Po 2014], i.e.,

S × Z × A^{r,u} → A^{r,u}, (s, z, a) ↦ φ(s, z)(a),

is a morphism; (ii) for every point z ∈ Z, the algebraic family

φ_{z}: S → Aut A^{r,u}, s ↦ φ(s, z),

is a homomorphism determining a transitive action of S on A^{r,u}. These data determine the regular action of S on Z × A^{r,u} by the formula

S × (Z × A^{r,u}) → Z × A^{r,u}, (s, (z, a)) ↦ (z, φ(s, z)(a)).  (41)

By (ii), the orbits of this action are the fibers of the projection

π: Z × A^{r,u} → Z, (z, a) ↦ z,

and, more precisely, for every point z ∈ Z, the variety A^{r,u} endowed with the G-action determined by φ_{z} is G-isomorphic to the fiber π^{−1}(z). By [Bo 1991, Prop. II.6.6] this implies that k(Z × A^{r,u})^{S} ≃ k(Z).

In view of (40), for n = dim Z + r + u, we may (and shall) fix a birational isomorphism

γ: Z × A^{r,u} −−→ A^{n}.

Then γ and action (41) determine a rational action of S on A^{n}. The image of the homomorphism S → C_{n} determined by this rational action is a connected solvable affine algebraic subgroup G of C_{n}, and for this G we have

k(A^{n})^{G} ≃ k(Z).  (42)

We say that G is a subgroup of C_{n} obtained by Construction ⊛.

Theorem 5 (combined with Weil’s regularization theorem [We 1955]) implies that this construction is universal, i.e., every connected solvable affine algebraic subgroup of C_{n} is obtained by Construction ⊛.

Example 1 (One-dimensional connected unipotent subgroups of C_{n}). Let G be the one-dimensional additive group k^{+}. In view of (ii), we then have r = 0, u = 1, i.e., φ_{z}: G = k^{+} → Aut A^{1} for every z ∈ Z. Since Aut A^{1} = T ⋉ N, where T is a one-dimensional torus and N ≃ k^{+} is the subgroup consisting of all translations A^{1} → A^{1}, a ↦ a + t, t ∈ k^{+}, this means that every φ_{z} may be identified with a surjective homomorphism k^{+} → k^{+}. What happens next depends on char k, see [Sp 1998, Lemma 3.3.5].

Namely, a map k^{+} → k^{+} is a homomorphism if and only if it has the following form:

(i) case char k = 0: t ↦ ct, where c is a fixed element of k, (ii) case char k = p > 0: t ↦ α_{1}t^{p^{i_{1}}} +⋯+α_{d}t^{p^{i_{d}}} , where α_{1}, …, α_{d} are the

nonzero elements of k and i_{1}, …, i_{d} is an increasing sequence of nonnegative integers.

Since every one-dimensional connected unipotent algebraic group is isomorphic to k^{+} (see, e.g., [Sp 1998, Thm. 3.4.9]), this yields the following general method of constructing connected one-dimensional unipotent algebraic subgroups of C_{n}.

Take an irreducible variety Z such that Z × A^{1} and A^{n} are birationally isomorphic. If char k = 0, fix a nonzero regular function f ∈ k[Z]. If char k = p > 0, fix a sequence of nonnegative integers i_{1} < … < i_{d} and a sequence of nonzero regular functions f_{1}, …, f_{d} ∈ k[Z]. Consider the action of S = k^{+} on Z × A^{1} defined by the formula

S×(Z×A^{1}) → Z×A^{1},
(s, (z, a)) ↦{ ; (z, a+f(z)s)  if char k =0, (z, a+f_{1}(z)s^{p^{i_{1}}} +⋯+f_{d}(z)s^{p^{i_{d}}}) if char k =p>0. (43)

Then action (43) and a fixed birational isomorphism γ: Z × A^{1} −−→ A^{n} determine a one-dimensional connected unipotent algebraic subgroup G of C_{n}, and every such subgroup is obtained this way.

Theorem 6 (Structure theorem). The following properties of a connected solvable affine algebraic subgroup G of the Cremona group C_{n} are equivalent:

(i) G is rationally triangulable; (ii) G is obtained by Construction ⊛, in which the variety Z is rational.

Proof. This follows from (42) and Theorem 2. □

Proof of Corollary 4. Let G be a rationally triangulable one-dimensional connected unipotent algebraic subgroup of C_{n}. By Theorem 6 and Example 1, G is obtained from the action of S = k^{+} on Z × A^{1} defined by formulas (43), where Z is a rational variety. Therefore there are functions f_{1}, …, f_{n} ∈ k(A^{n}) such that k(f_{1}, …, f_{n}) = k(A^{n}), f_{1}, …, f_{n−1} ∈ k(A^{n})^{G}, and t(f_{n}) for every element t ∈ S is the following function:

(i) if char k = 0, then t(f_{n}) = f_{n} + ct, where c ∈ k(f_{1}, …, f_{n−1}),

(ii) if char k = p > 0, then

t(f_{n}) = f_{n} + b_{1}t^{p^{i_{1}}} + ⋯ + b_{d}t^{p^{i_{d}}} ,

where b_{j} ∈ k(f_{1}, …, f_{n−1}), b_{j} ≠ 0 for every j and i_{1} < ⋯ < i_{d} are the nonzero integers.

In case (i), replacing f_{n} by f_{n}/c we may (and shall) assume that c = 1. Then, conjugating S by means of φ ∈ C_{n} such that φ(f_{i}) = x_{i} for every i (see (1)), we obtain a standard subgroup.

The converse (that standard subgroups are rationally triangulable) is clear. □

Proof of Corollary 5. This follows from Corollaries 1, 2, and 4. □

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Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia

National Research University, Higher School of Economics, Myasnitskaya 20, Moscow 101000, Russia

   E-mail address: popovvl@mi.ras.ru