Birational splitting and algebraic group actions
Abstract
According to a 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 \(\mathbf{P}^s\) with strictly 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.
Keywords
Algebraic group Algebraic variety Action Rational quotientMathematics Subject Classification
20B271. 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 [7, 11] and refer to [6, 7, 8, 9, 10] 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 Open image in new window a rational quotient of this action; the latter means that Open image in new window and \(\pi _{G, X}\) are respectively an irreducible variety and a dominant rational map such that Open image in new window .
2. Up to a change of notation and terminology, the following statement appeared in the classical paper [4], Theorem 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 Open image in new window , where Open image in new window is a rational quotient of the natural rational action of \(B\) on \(X\) and \(0<s\leqslant \dim B\).
In [4] no restriction on \({\mathrm{char}}\,k\) is imposed, but actually the brief argument given there in support of Theorem 1 works only if \({\mathrm{char}}\,k=0\). We reproduce it below in order to pinpoint where the restriction \({\mathrm{char}}\,k=0\) is implicitly used.
Proof
3. The assumption \({\mathrm{char}}\,k=0\) is actually implicitly used in the penultimate phrase of this argument. Indeed, it purports the following. Consider a section of \(\pi ^{\ }_{B_d, X}\), i.e., a rational map \(\sigma :X_d\dashrightarrow X\) such that \(\pi _{B_d, X}{\circ }\sigma =\mathrm{id}\). Since \(B_{d+1}\) lies in the kernel of the action of \(B_d\) on \(X\), this action is reduced to that of the onedimensional 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 onedimensional linear algebraic group, \(C_d\) is isomorphic to either \(k^{\times }\) (the multiplicative group of \(k\)) or \(k^+\) (the additive group of \(k\)); see, e.g., [11], Theorem 3.4.9].
If it is isomorphic to \(k^{\times }\), 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 [6], Lemma 2.4]. Therefore, the dominant rational map \(\gamma :C_d{\times } X_d\dashrightarrow X\), \((c, b)\mapsto c{\cdot } \sigma (b)\), is bijective over a dense open subset of \(X\). If \({\mathrm{char}}\,k=0\), the latter implies that \(\gamma \) is a birational isomorphism and hence \(X\) is birationally isomorphic to \(\mathbf{P}^1{\times } X_d\) because the group variety of \(C_d\) is rational. But if \({\mathrm{char}}\,k>0\), one can only say that \(\gamma \) is either a birational isomorphism or purely inseparable. The following example shows that the latter indeed may occur.
Example 1
Let \({\mathrm{char}}\,k=p>0\). Consider the locally free action of \(G=B=k^{\times }\) on \(X= k{\setminus } \{0\}\), given by \(b{\cdot } x=b^px\). We have \(n=1\), \(d=0\), \(C_0=G\), and \(X_0\) is a point. Therefore, \(C_0{\times } X_0\) is naturally identified with \(G\). Let \(\sigma \) map \(X_0\) to \(1\). Then \(\gamma ^*(k(X))=k(t^p)\varsubsetneq k(t)=k(G)\), where \(t\) is the standard coordinate function on \(G\). Thus \(\gamma \) 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 \({\mathrm{char}}\,k=0\), then local freeness indeed holds because in this case there are no nontrivial finite subgroups in \(k^+\). However, if \({\mathrm{char}}\,k>0\), it may happen that the action of \(C_d\) on \(X\) is not locally free; therefore, \(\gamma \) is not bijective over a dense open subset of \(X\), and a fortiori is not a birational isomorphism. The example below is a generalization of the one the author first learned from G. Kemper, whom we thank for it. It is based on the idea going back to [7], 7.1, Example \(1^\circ \)] and Corollary of Proposition 1 below.
Example 2
Moreover, if even the action of \(C_d\) on \(X\) is locally free, and hence \(\gamma \) is bijective over a dense open subset of \(X\), it may happen that \(\gamma \) is purely inseparable. The corresponding example is similar to Example 1.
Example 3
Let \({\mathrm{char}}\,k=p>0\). Consider the locally free action of \(G=B=k^+\) on \(X=k\), given by \(b{\cdot } x=b^p+x\). Then \(n=1\), \(d=0\), \(C_0=G\), \(X_0\) is a point, \(C_0{\times } X_0\) is naturally identified with \(G\), and if \(\sigma \) maps \(X_0\) to \(0\), then \(\gamma ^*(k(X))=k(t^p)\varsubsetneq k(t)=k(G)\), where \(t\) is the standard coordinate function on \(G\). Thus \(\gamma \) 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
Theorem 2
 (i)
If \(G^0\) is solvable, then \(G\) has property \(\mathrm{(A)}\).
 (ii)
If \(G\) is connected and has property \(\mathrm{(A)}\), then \(G\) is solvable.
Proof
(ii) Let the group \(G\) be connected and has property (A). Assume that it is nonsolvable. Then it contains a proper parabolic subgroup \(P\); see [11], Proposition 6.2.5]. Let \(X\) be \(G/P\) endowed with the natural action of \(G\). We have \(\dim X>0\). Let \(Y\) and \(\varphi \) 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 \(\varphi \) is \(G\)equivariant and the action of \(G\) on \(X\) is transitive, \(\varphi \) is a morphism. Therefore, completeness and irreducibility of \(X\) implies that \(\varphi (X)\) is a complete \(G\)stable closed irreducible subset in \(Y\); see [11], Proposition 6.1.2 (iii)]. Since \(Y\) is affine, this yields that \(\varphi (X)\) is a point; see [11], Proposition 6.1.2 (vi)]. But \(\dim \,\varphi (X)=\dim \,X>0\) because \(\varphi \) is a birational isomorphism—a contradiction.\(\square \)
5. We also need the following statement, see [11], Proposition 14.2.2] and an earlier result [5], Lemma 1.5].
Proposition 1
 (a)
there is an isomorphism \(\phi \) of \(G{\times } Y\) onto an open subvariety of \(X\);
 (b)there is a morphism \(\psi :G{\times } Y\) such that for all \(a, b\in k\), \(y\in Y\),$$\begin{aligned} \psi (a{+}b, y)=\psi (a, y)+\psi (b, y),\qquad a{\cdot }\phi (b, y)=\phi (\psi (a, y){+}b, y). \end{aligned}$$
Corollary

the natural projection \(\mathrm{pr}_2:G{\times } Y\rightarrow Y\) is its rational quotient;

the isomorphism \(\phi \) is \(G\)equivariant.
6. We now turn to a characteristic free proof of Theorem 1.
Proof
(Characteristic free proof of Theorem 1) We retain the argument in [4], except its part referring to the crosssection theorem of [8] that works, as we have explained, only if \({\mathrm{char}}\,k=0\). 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 onedimensional 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 \(k^{\times }\).
Let \(H\) be isomorphic to \(k^+\). Then by Corollary of Proposition 1 the variety \(X\) is \(H\)equivariantly birationally isomorphic to the variety \(\mathbf{P}^1{\times } X_d\), on which \(H\) acts rationally so that the second projection \(\mathrm{pr}_2:\mathbf{P}^1{\times } X_d\rightarrow X_d\) is a rational quotient of this action.
Given that \(t\) is transcendental over \(k(X)^H\), we conclude from (11) that \(X\) is \(H\)equivariantly birationally isomorphic to the variety Open image in new window , on which \(H\) acts rationally via the first factor so that the second projection Open image in new window is a rational quotient of this action. This completes the proof.\(\square \)
7. Combining the given proof of Theorem 1 with Rosenlicht’s theorem on the existence of generic geometric quotient [10], Theorem], we obtain the following generalization of the result of [2], Section 1] on “trivial quotient” (our attention was drawn to this result by G. Kemper, whom we thank).
Theorem 3

the geometric quotient \(\pi _{G, U}:U\rightarrow U/G\);
 an isomorphism \(\varphi :U\rightarrow \mathbf{A}^{r, s}{\times } (U/G)\), where$$\begin{aligned} \mathbf{A}^{r, s}=\bigl \{(\alpha _1,\ldots , \alpha _{r+s})\in \mathbf{A}^{r+s}: \alpha _i\ne 0 \text{ for } \text{ every } i\leqslant r \bigr \},\qquad r\geqslant 0,\quad s\geqslant 0, \end{aligned}$$
Theorem 3 immediately implies the crosssection theorem.
Corollary
([8], Theorem 10]) Let \(X\) be an irreducible algebraic variety endowed with a regular action of a solvable connected linear algebraic group \(G\). Let Open image in new window be a rational quotient of this action. Then there is a rational map Open image in new window such that \(\pi _{G, X} {\circ } \sigma =\mathrm{{id}}\).
Notes
Acknowledgments
The author is indebted to the referee for thorough reading and remarks.
References
 1.Borel, A., Serre, J.P.: Théorèmes de finitude en cohomologie galoisienne. Comment. Math. Helv. 39, 111–164 (1964)zbMATHMathSciNetCrossRefGoogle Scholar
 2.Greuel, G.M., Pfister, G.: Geometric quotients of unipotent group actions. Proc. Lond. Math. Soc. 67(1), 75–105 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
 3.Iitaka, S.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 76. NorthHolland Mathematical Library, vol. 24. Springer, New York (1982)Google Scholar
 4.Matsumura, H.: On algebraic groups of birational transformations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur 34, 151–155 (1963)zbMATHMathSciNetGoogle Scholar
 5.Miyanishi, M.: Lectures on Curves on Rational and Unirational Surfaces. Tata Institute of Fundamental Research Lectures on Mathematics, vol. 60. Narosa, New Delhi (1978)Google Scholar
 6.Popov, V.L.: Some subgroups of the Cremona groups. In: Masuda, K., Kojima, H., Kishimoto, T. (eds.) Affine Algebraic Geometry (Osaka, 2011), pp. 213–242. World Scientific, Singapore (2013)CrossRefGoogle Scholar
 7.Popov, V.L., Vinberg, E.B.: Invariant theory. In: Parshin, A.N., Shafarevich, I.R. (eds.) Algebraic Geometry IV. Encyclopaedia of Mathematical Sciences, vol. 55, pp. 123–284. Springer, Berlin (1994)Google Scholar
 8.Rosenlicht, M.: Some basic theorems on algebraic groups. Amer. J. Math. 78(2), 401–443 (1956)zbMATHMathSciNetCrossRefGoogle Scholar
 9.Rosenlicht, M.: On quotient varieties and the affine embedding of certain homogeneous spaces. Trans. Amer. Math. Soc. 101(2), 211–223 (1961)zbMATHMathSciNetCrossRefGoogle Scholar
 10.Rosenlicht, M.: A remark on quotient spaces. An. Acad. Brasil. Ciênc. 35, 487–489 (1963)zbMATHMathSciNetGoogle Scholar
 11.Springer, T.A.: Linear Algebraic Groups. Progress in Mathematics, vol. 9, 2nd edn. Birkhäuser, Boston (1998)CrossRefGoogle Scholar
 12.Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ. 14(1), 1–28 (1974)zbMATHMathSciNetGoogle Scholar