BIRATIONAL SPLITTING AND ALGEBRAIC GROUP ACTIONS

VLADIMIR L. POPOV^{∗}

Abstract. According to the classical theorem, every algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and P^{s} with positive s. We show that the classical proof of this theorem actually works only in characteristic 0 and we give a characteristic free proof of it. To this end we prove and use a characterization of connected linear algebraic groups G with the property that every rational action of G on an irreducible algebraic variety is birationally equivalent to a regular action of G on an affine algebraic variety.

1. Throughout this note k stands for an algebraically closed field of arbitrary characteristic which serves as domain of definition for each of the algebraic varieties considered below. Each algebraic variety is identified with its set of k-rational points. We use freely the standard notation and conventions of [PV 94], [Sp 98] and refer to [Ro 56], [Ro 61], [Ro 63], [PV 94], [Po 13] regarding the definitions and basic properties of rational and regular (morphic) actions of algebraic groups on algebraic varieties. Given a rational action of such a group G on an irreducible algebraic variety X, we denote by π_{G,X}: X −−→ X---G a rational quotient of this action; the latter means that X---G and π_{G,X} are respectively an irreducible variety and a dominant rational map such that π_{G,X}^{∗}(k(X---G)) = k(X)^{G}.

2. Up to a change of notation and terminology, the following statement appeared in classical paper [Ma 63, Thm. 1]:

Theorem 1. Assume that a connected linear algebraic group G acts rationally and nontrivially on an irreducible algebraic variety X, and let B be a Borel subgroup of G. Then X is birationally isomorphic to P^{s} ×X---B, where π_{B,X}: X −−→ X---B is a rational quotient of the natural rational action of B on X and 0 < s ⩽ dim B.

In [Ma 63] no restriction on char k is imposed, but actually the brief argument given there in support of Theorem 1 works only if char k = 0. We reproduce it below in order to pinpoint where the restriction char k = 0 is implicitly used.

Argument from [Ma 63] supporting Theorem 1. Since G is generated by its Borel subgroups and since all Borel subgroups are conjugate to each other, B acts on X nontrivially. Since B is a connected solvable linear algebraic group, there is a chain of connected subgroups

∗ Supported by grants ���� �����������, ��������������, and the granting program Contemporary Problems of Theoretical Mathematics of the Mathematics Branch of the Russian Academy of Sciences.

B = B_{0} ⊃ B_{1} ⊃ ⋯ ⊃ B_{n} = {e}

such that all B_{i} are normal in B and dim B_{i} = dim B − i. If d is the largest index i such that the action of B_{i} on X is not trivial, then let

π_{B_{d},X}: X −−→ X---B_{d} =: X_{d}  (1)

be a rational quotient of X with respect to B_{d}. By the cross-section theorem ([Ro 56]) we find that X is birationally equivalent to P^{1} × X_{d}. The factor group B/B_{d} acts on X_{d} and we can repeat the same argument. □

3. The assumption char k = 0 is actually implicitly used in the penultimate phrase of this argument. Indeed, it purports the following. Let be a section of π_{B_{d},X}, i.e., a rational map such that π_{B_{d},X} ∘ σ = id. Since B_{d+1} lies in the kernel of the action of B_{d} on X, this action is reduced to that of the one-dimensional connected linear algebraic group B_{d}/B_{d+1}. This action is nontrivial, hence the B_{d}/B_{d+1}-stabilizers of points of a dense open subset of X are finite; in particular, the kernel K of this action is finite. The action of C_{d} := (B_{d}/B_{d+1})/K on X is faithful, and (1) is its rational quotient.

Being a connected one-dimensional linear algebraic group, C_{d} is isomorphic to either k^{×} (the multiplicative group of k) or k^{+} (the additive group of k; see, e.g., [Sp 98, Thm. 3.4.9].

If it is isomorphic to k^{×}, then faithfulness of its action on X implies that this action is locally free (i.e., C_{d}-stabilizers of points of a dense open subset of X are trivial); see [Po 13, Lemma 2.4]. Therefore, the dominant rational map γ: C_{d} ×X_{d} −−→ X, (c, b) ↦ c·σ(b), is bijective over a dense open subset of X. If char k = 0, the latter implies that γ is a birational isomorphism and hence X is birationally isomorphic to P^{1} × X_{d} because the group variety of C_{d} is rational. But if char k > 0, one can only say that γ is either a birational isomorphism or purely inseparable. The following example shows that the latter indeed may occur.

Example 1. Let char k = p > 0. Consider the locally free action of G = B = k^{×} on X = k \ {0}, given by b · x := b^{p}x. We have n = 1, d = 0, C_{0} = G, and X_{0} is a point. Therefore, C_{0} × X_{0} is naturally identified with G. Let σ maps X_{0} to 1. Then γ^{∗}(k(X)) = k(t^{p}) ⊊ k(t) = k(G), where t is the standard coordinate function on G. Thus γ is not a birational isomorphism.

If C_{d} is isomorphic to k^{+}, the same argument works if we know that the action of C_{d} on X is locally free. If char k = 0, then local freeness indeed holds because in this case there are no nontrivial finite subgroups in k^{+}. However, if char k > 0, it may happen that the action of C_{d} on X is not locally free; therefore, γ is not bijective over a dense open subset of X, a fortiori is not a birational isomorphism. The example below is a generalization of the one I first learned from G. Kemper, whom I thank for it. It is based on the idea going back to [PV 94, 7.1, Example 1^{∘}] and Corollary of Proposition 1 below.

Example 2. Let char k = p > 0 and let G = B = k^{+}. Let x ∈ k[G] be the standard coordinate function on G. Every homomorphism of algebraic groups f: G → G defines a regular action of G on X := k^{2} by the formula

u · (a, b) := (a + ub + f(u), b), where u ∈ G, (a, b) ∈ X.  (2)

¿From (2) we infer that an element u ∈ G lies in the G-stabilizer of a point (a, b) ∈ X if and only if u is a root of the polynomial f + bx. By [Sp 98, Lemma 3.3.5] there are nonzero elements α_{1}, …, α_{s} ∈ k and an increasing sequence of nonnegative integers n_{1}, …, n_{s} such that

f = α_{1}x^{p^{n_{1}}} + ⋯ + α_{s}x^{p^{ns}};  (3)

any α_{i} and n_{j} may occur in the right-hand side (3) for an appropriate f. One of the roots of f is 0. Now take f with n_{s} ⩾ 1. By (3) the polynomial (f + bx)/x has degree p^{n_{s}} − 1 ⩾ 1 and does not vanish at 0 if b ≠ 0 and b ≠ −α_{1}. Whence if these inequalities hold, the G-stabilizer of (a, b) is nontrivial. Therefore, the action is not locally free.

Moreover, if even the action of C_{d} on X is locally free, and hence γ is bijective over a dense open subset of X, it may happen that γ is purely inseparable. The corresponding example is similar to Example 1.

Example 3. Let char k = p > 0. Consider the locally free action of G = B = k^{+} on X = k, given by b · x := b^{p} + x. Then n = 1, d = 0, C_{0} = G, X_{0} is a point, C_{0} × X_{0} is naturally identified with G, and if σ maps X_{0} to 0, then γ^{∗}(k(X)) = k(t^{p}) ⊊ k(t) = k(G), where t is the standard coordinate function on G. Thus γ is not a birational isomorphism.

4. Below we shall give a characteristic free proof of Theorem 1. For this, we need the following characterization of connected linear algebraic groups G with the property that every rational action of G on an irreducible algebraic variety is birationally equivalent to a regular action of G on an affine algebraic variety.

Definition. We say that a linear algebraic group G has property (A) if for every rational action of G on an irreducible algebraic variety X, there exist an irreducible affine algebraic variety Y and a birational isomorphism

φ: X −−→ Y  (4)

such that the rational action of G on Y induced by φ is regular.

Theorem 2. Let G be a linear algebraic group and let G^{0} be the connected component of the identity in G.

(i) If G^{0} is solvable, then G has property (A).

(ii) If G is connected and has property (A), then G is solvable.

Proof. (i)⇒(ii): Let G^{0} be solvable. Consider a rational action of G on an irreducible algebraic variety X. By [Ro 56, Thm. 1], there exists an irreducible algebraic variety X_{1} and a birational isomorphism

α_{1} : X −−→ X_{1}

such that the rational action of G on X_{1} induced by α_{1} is regular.

Let ν: X_{2} → X_{1} be the normalization of X_{1}. Then the rational action of G on X_{2} induced by the birational isomorphism

α_{2} := ν^{−1} : X_{1} −−→ X_{2}

is regular, see [Ii 82, Thm. 2.25].

By [Su 74, Lemma 8], since X_{2} is a normal algebraic variety, it contains a nonempty G^{0}-stable quasi-projective open subset U. Hence, by [Su 74, Thm. 1], for some positive integer n, there exist a regular action of G^{0} on the projective space P^{n} and a G^{0}-equivariant embedding of the algebraic variety U into P^{n} ,

ι: U ↪ P^{n}.

We may (and shall) assume that n is minimal possible with this property. Since Aut P^{n} = PGL_{n}, this action of G^{0} on P^{n} induces an action of G^{0} on the dual projective space P̌^{n}. By the Borel fixed-point theorem [Sp 98, Thm. 6.2.6], the assumption that G^{0} is a connected solvable linear algebraic group implies that in P̌^{n} there is a fixed point of this action. This means that P^{n} contains a G^{0}-stable hyperplane H. Hence P^{n} \ H is a G^{0}-stable affine open subset of P^{n}. Therefore, the minimality assumption on n implies that ι(U) ∩ (P^{n} \ H) is a nonempty G^{0}-stable open quasiaffine subset of ι(U). This proves that X_{2} contains a nonempty G^{0}-stable open quasiaffine subset V. Normality of G^{0} in G then implies that g · V for every element g ∈ G is a nonempty G^{0}-stable open quasiaffine subset of X_{2}.

By [BS 64, Lemma 5.11 and the footnote to its proof], there exists a finite subgroup F of G that intersects every connected component of G. Put

X_{3} := ⋂_{g∈F} g · V.

Then X_{3} is G^{0}- and F -stable and, therefore, G-stable. Let

α_{3} : X_{2} −−→ X_{3}

be the birational isomorphism inverse to the identity embedding X_{3} ↪ X_{2}. Thus we have proved that X_{3} is a quiasiaffine algebraic variety such that the rational action of G on X_{3} induced by α_{3} is regular.

Finally, by [Ro 61, Lemma 2] (see also [PV 94, Thm. 1.6]) quasiaffiness of X_{3} implies that there exist an irreducible affine algebraic variety X_{4} endowed with a regular action of G and a G-equivariant birational embedding

α_{4} : X_{3} ↪ X_{4}.

This shows that we may take Y := X_{4} and φ := α_{4} ∘ α_{3} ∘ α_{2} ∘ α_{1}.

(ii)⇒(i): Let the group G be connected and has property (A). Assume that it is non-solvable. Then it contains a proper parabolic subgroup P; see [Sp 98, Prop. 6.2.5]. Let X be G/P endowed with the natural action of G. We have dim X > 0. Let Y and φ be respectively an irreducible affine algebraic variety endowed with a regular action of G and a birational isomorphism (4), whose existence is ensured by property (A). Since φ is G-equivariant and the action of G on X is transitive, φ is a morphism. Therefore, completeness and irreducibility of X implies that φ(X) is a complete G-stable closed irreducible subset in Y; see [Sp 98, Prop. 6.1.2(iii)]. Since Y is affine, this yields that φ(X) is a point; see [Sp 98, Prop. 6.1.2(vi)]. But dim φ(X) = dim X > 0 because φ is a birational isomorphism — a contradiction. □

5. We also need the following

Proposition 1. Let G = k^{+} and let X be an irreducible affine algebraic variety endowed with a nontrivial regular action of G. Then there exists an irreducible affine variety Y with the following properties:

(a) there is an isomorphism ϕ of G × Y onto an open subvariety of X;

(b) there is an morphism ψ: G × Y such that for all a, b ∈ k, y ∈ Y,

ψ(a + b, y) = ψ(a, y) + ψ(b, y), a · ϕ(b, y) = ϕ(ψ(a, y) + b, y).

Proof. See [Sp 98, Prop. 14.2.2] and an earlier result [Mi 78, Lemma 1.5]. □

Corollary. Maintain the notation of Proposition 1. Then the formula

a · (b, y) := (ψ(a, y) + b, y), a, b ∈ G, y ∈ Y

defines a regular action of G on G × Y such that

— the natural projection pr_{2} : G × Y → Y is its rational quotient;

— the isomorphism ϕ is G-equivariant.

In particular, Y and X---G are birationally isomorphic.

6. We now turn to a characteristic free proof of Theorem 1.

Characteristic free proof of Theorem 1. We retain the argument in [Ma 63], except its part referring to the cross-section theorem of [Ro 56] that works, as we have explained, only if char k = 0. This part is replaced by the following characteristic free argument.

By Theorem 2, we may assume that X is affine and the action of the one-dimensional connected linear algebraic group H := B_{d}/B_{d+1} on X is nontrivial and regular. There are two possibilities: H is isomorphic to either k^{+} or k^{×}.

Let H be isomorphic to k^{+}. Then by Corollary of Proposition 1 the variety X is H-equivariantly birationally isomorphic to the variety P^{1} × X_{d}, on which H acts rationally via the first factor so that the second projection pr_{2} : P^{1} × X_{d} → X_{d} is a rational quotient of this action.

It remains to show that the same is true if H is isomorphic to k^{×}. The group X (H) of characters of H (i.e., algebraic homomorphisms H → k^{×}) is isomorphic to Z. Let χ be its generator. For every s ∈ Z, put

k(X)_{s} := {f ∈ k(X) | h · f = χ^{s}(h)f for every h ∈ H};  (5)

in particular, k(X)_{0} = k(X)^{H}. Since H is a torus, the H-module k[X] is semisimple and its isotypic decomposition has the form

k[X] = ⨁_{s∈Z} k[X]_{s}, where k[X]_{s} := k(X)_{s} ∩ k[X]  (6)

(see, e.g., [Sp 98, 3.2.13]); in particular, k[X]_{0} = k[X]^{H}. Given (6), every element f ∈ k[X] can be uniquely written as the following sum of the nonzero summands:

f = f_{i_{1}} + ⋯ + f_{i_{s}}, where f_{j} ∈ k[X]_{j} for all j.  (7)

We call (7) the canonical decomposition of f. Nontriviality of the action of H on X implies that k[X]^{H} ≠ k[X]. Therefore, the subgroup

Γ := {s ∈ Z | k(X)_{s} ≠ 0}  (8)

of Z is nonzero, i.e., Γ = nZ for some positive integer n.

It follows from (8) that there are a_{1}, …, a_{m} ∈ Z such that in canonical decomposition (7) we have

i_{1} = na_{1}, …, i_{s} = na_{s}.  (9)

Fix a nonzero element t ∈ k(X)_{n}. From (5), (7), and (9) we deduce that

f_{i_{r}}/t^{a_{r}} ∈ k(X)^{G} for every r.  (10)

In turn, (7) and (10) yield that f = ∑_{r=1}^{s} t^{a_{r}}(f_{i_{r}}/t^{a_{r}}) is an element of the subfield k(X)^{H}(t) of X. Hence k[X] lies in this subfield. But k(X) is the field of fractions of k[X] because X is affine. This proves that

k(X) = k(X)^{H}(t).  (11)

The element t is transcendental over k(X)^{H}. Indeed, if not, there is a relation ∑_{i=0}^{m} a_{i}t^{r_{i}} = 0 for some integers 0 ⩽ r_{0} < r_{1} < ⋯ < r_{m} and nonzero elements a_{i} ∈ k(X)^{H}. From (5) we then deduce that

∑_{i=0}^{m} χ^{nr_{i}}(h)a_{i}t^{r_{i}} = 0 for every element h ∈ H.

This contradicts Artin’s theorem on independence of characters, because k^{×} is the subgroup of the multiplicative group k(X)^{×} of k(X), and therefore, we may consider the elements of X (H) as the homomorphisms H → k(X)^{×}.

Given that t is transcendental over k(X)^{H}, we conclude from (11) that X is H-equivariantly birationally isomorphic to the variety P^{1} × X---H, on which H acts rationally via the first factor so that the second projection pr_{2} : P^{1} × X---H → X---H is a rational quotient of this action. This completes the proof. □

7. Combining the given proof of Theorem 1 with Rosenlicht’s theorem [Ro 63, Thm.] on the existence of generic geometric quotient, we obtain the following generalization of the result of [GP 93, Sect. 1] on “trivial quotient” (my attention was drawn to this result by G. Kemper, whom I thank).

Theorem 3. Let X be an irreducible algebraic variety endowed with a regular action of a solvable connected linear algebraic group G. Then for the restriction of this action on a certain G-stable dense open subset U of X there exist

— the geometric quotient π_{G,U}: U → U/G;

— an isomorphism φ: U → A^{r,s} × (U/G), where

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

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

As a corollary we obtain Rosenlicht’s cross-section theorem:

Corollary 1 ([Ro 56, Thm. 10]). Let X be an irreducible algebraic variety endowed with a regular action of a solvable connected linear algebraic group G. Let π_{G,X}: X −−→ X---G be a rational quotient of this action. Then there

is a rational map σ: X---G −−→ X such that π_{G,X} ∘ σ = id_{X---G}.

References

[BS 64] A. Borel, J-P. Serre, Théorèmes de finitude en cohomogie galoisienne, Comm.

Math. Helv. 39 (1964), 111–164. [GP 93] G.-M. Greuel, G. Pfister, Geometric quotients of unipotent group actions, Proc.

London Math. Soc. (3) 76 (1993), 75–105. [Ii 82] S. Iitaka, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 76, Sprin-

ger-Verlag, New York, 1982. [Ma 63] H. Matsumura, On algebraic groups of birational transformations, Atti Accad.

Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 34 (1963), 151–155. [Mi 78] M. Miyanishi, Lectures on Curves on Rational and Unirational Surfaces, Tata

Institute of Fundamental Research Lectures on Mathematics and Physics, Vol. 60, Tata Institute of Fundamental Research, Bombay, Narosa Publishing House, New Delhi 1978.

[Po 13] V. L. Popov, Some subgroups of the Cremona groups, in: Affine Algebraic Geometry, Proceedings of the conference on the occasion of M. Miyanishi’s 70th birthday (Osaka, Japan, 3–6 March 2011), World Scientific, Singapore, 2013, pp. 213–242.

[PV 94] V. L. Popov, E. B. Vinberg, Invariant theory, in: Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284.

[Ro 56] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78

(1956), 401–443. [Ro 61] M. Rosenlicht, On quotient varieties and the affine embedding of certain homoge-

neous spaces, Trans. Amer. Math. Soc. 101 (1961), 211–223. [Ro 63] M. Rosenlicht, A remark on quotient spaces, Anais Acad. Brasil. Ciênc. 35 (1963),

487–489. [Sp 98] T. A. Springer, Linear Algebraic Groups, 2nd edition, Progress in Mathematics,

Vol. 9, Birkhäuser, Boston, 1998. [Su 74] H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.

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

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

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