On stable conjugacy of finite subgroups of the plane Cremona group, I
aa r X i v : . [ m a t h . AG ] J u l ON STABLE CONJUGACY OF FINITE SUBGROUPSOF THE PLANE CREMONA GROUP, I
FEDOR BOGOMOLOV AND YURI PROKHOROV
Abstract.
We discuss the problem of stable conjugacy of fi-nite subgroups of Cremona groups. We show that the group H ( G, Pic( X )) is a stable birational invariant and compute thisgroup in some easy cases. Introduction
Let k be an algebraically closed field of characteristic 0. The Cre-mona group Cr n ( k ) over k is the group of birational automorphisms ofthe projective space P n , or, equivalently, the group of k -automorphismsof the field k ( x , x , . . . , x n ) of rational functions in n independent vari-ables. Note that embeddings adding new variables one gets a tower ⊂ Cr n ( k ) ⊂ Cr n +1 ( k ) ⊂ · · · . Subgroups G , G ⊂ Cr n ( k ) are said tobe stably conjugate if they are conjugate in some Cr m ( k ) ⊃ Cr n ( k ).Stable conjugacy of Cremona groups is an analog of stable birationalequivalence (see [BCSS85], [Vos77]) of varieties over non-closed fields.We are interested in stable conjugacy of subgroups of Cr n ( k ). Anexample of a subgroup G ⊂ Cr n ( k ) (for n = 3) that is not stablyconjugate to any subgroup induced by a linear action was constructedby V. Popov [Pop11]. The construction is based on the example ofM. Artin and D. Mumford [AM72] and uses non-triviality of torsionsof H ( Y, Z ) for the quotient variety Y = P /G . In this paper we useanother, very simple approach.Recall that any finite subgroup G ⊂ Cr n ( k ) is induced by a bireg-ular action on a non-singular projective rational variety X (see 2.1).One can easily show that the group H ( G, Pic( X )) is stable birationalinvariant and so it does not depend on the choice of X (see Corollary2.5.1). In particular, if G ⊂ Cr n ( k ) is stably conjugate to a linear ac-tion, then H ( G, Pic( X )) = 0. This fact is not surprising: in the arith-metic case it was known for a long time (see e.g. [Man66], [Vos77]). An Both authors were partially supported by AG Laboratory SU-HSE, RF govern-ment grant ag. 11.G34.31.0023. The second author was partially supported by thegrants RFBR-11-01-00336-a, N.Sh.-5139.2012.1, and Simons-IUM fellowship. nteresting fact is that in the geometric case the group H ( G, Pic( X ))typically admits a good description. Note that in [LPR06] the authorsused a similar approach to construct non-stably conjugate embeddingsof certain groups to Cr n ( k ), n ≥ ( k ). Inthis case the classification of finite subgroups has a long history. It wasstarted in works of E. Bertini and completed recently by I. Dolgachevand V. Iskovskikh [DI09] (however even in this case some questionsremain open). We prove the following. Let a finite cyclic group G of prime order p act on anon-singular projective rational surface X . Assume that G fixes ( point-wise ) a curve of genus g > . Then (1.1.1) H ( G, Pic( X )) ≃ ( Z /p Z ) g . Taking the results of [BB00], [dF04] and [BB04] into account (seeTheorem 2.3.1) we get the following.
In the notation of the following are equivalent: (i) H ( G, Pic( X )) = 0 , (ii) G does not fix point-wise a curve C of positive genus, (iii) ( X, G ) is conjugate to a linear action on P , (iv) ( X, G ) is stably conjugate to a linear action on P n for some n . In particular, we see that classical de Jonqui`eres, Bertini and Geiserinvolutions are not stably linearizable. Another application of theabove corollary is that the action of the simple Klein group of order 168on del Pezzo surface of degree 2 is also not stably linearizable. Moreapplication will be given in the forthcoming second part of the paper.
Acknowledgments.
This work was completed while the authors werevisiting University of Edinburgh. They are grateful to this institutionand personally to Ivan Cheltsov for invitation and hospitality. Theauthors would like to thank A. Beauville, J.-L. Colliot-Th´el`ene, V. L.Popov, and C. Shramov for constructive suggestions and criticism.2.
Preliminaries G -varieties. Let G be a finite group. A G -variety is a pair( X, ρ ), where X is a projective variety and ρ is an isomorphismfrom G to Aut( X ). A morphism (resp. rational map ) of the pairs( X, ρ ) → ( X ′ , ρ ′ ) is defined to be a morphism f : X → X ′ (resp.rational map f : X X ′ ) such that ρ ′ ( G ′ ) = f ◦ ρ ( G ) ◦ f − . n particular, two subgroups of Aut( X ) define isomorphic (resp. G -birationally equivalent) G -varieties if and only if they are conjugateinside of Aut( X ) (resp. Bir( X )). If no confusion is likely, we willdenote a G -variety by ( X, G ) or even by X .Now let X be an algebraic variety and let G ⊂ Bir( X ) be a finitesubgroup. By shrinking X we may assume that G acts on X by bireg-ular automorphisms. Then replacing X with the normalization of X/G in the field k ( X ) we may assume that X is projective. Finally we canapply an equivariant resolution of singularities and replace X with itsnon-singular (projective) model. Thus G ⊂ Bir( X ) is induced by abiregular action of G on a non-singular projective G -variety. In partic-ular, this construction can be applied to finite subgroups of Cr n ( k ). G -surfaces. Let X be a projective non-singular G -surface, where G is a finite group. It is said to be G -minimal if any birational G -equivariant morphism f : X → Y is an isomor-phism. It is well-known that for any G -surface there is a minimal(projective non-singular) model (see e.g. [Isk80]). If the surface X isadditionally rational, then one of the following holds [Isk80]: • X is a del Pezzo surface whose invariant Picard numberPic( X ) G is of rank 1, or • X admits a structure of G -conic bundle, that is, there existsa surjective G -equivariant morphism f : X → P such that f ∗ O X = O P , − K X is f -ample and rk Pic( X ) G = 2. The classification of elements ofprime order in the space Cremona group can be summarized as follows. [BB00] , [dF04] , [BB04] ). Let G = h δ i ⊂ Cr ( k ) is cyclic subgroup of prime order p and let ( X, G ) be its non-singularprojective model. Then the following hold. (i) The action ( X, G ) is conjugate to a linear action on P if andonly if the fixed point locus X G does not contain any curve ofpositive genus. (ii) If X G contains a curve C of genus g > , then other irreduciblecomponents of X G are either points or rational curves. In thiscase the minimal model ( X min , G ) is unique up to isomorphismand there are the following possibilities: g K X X min δ ≥ − g conic bundle de Jonqui`eres involution2 3 2 del Pezzo surface Geiser involution2 4 1 del Pezzo surface Bertini involution3 1 3 del Pezzo surface [dF04, A1]3 2 1 del Pezzo surface [dF04, A2]5 1 1 del Pezzo surface [dF04, A3] Let (
X, G ) and (
Y, G ) be G -varieties. Wesay that ( X, G ) and (
Y, G ) are stably birational if for some n and m there exists an equivariant birational map X × P n Y × P m , whereactions on P n and P m are trivial. This is equivalent to the conjugacyof the embeddings G ⊂ k ( X )( x , . . . , x n ) and G ⊂ k ( X )( x , . . . , x m ),i.e. stable conjugacy of G ⊂ k ( X ) and G ⊂ k ( X ).The following fact is well-known in the arithmetic case (see e.g.[Vos77]). Since we were not able to find a good reference for the present,geometric form, we provide a complete proof. [Man66, 2.2] ). Let X and X be projectivenon-singular G -varieties. Assume that ( X , G ) and ( X , G ) are stablybirational. Then there are permutation G -modules Π and Π such thatthe following isomorphism of G -modules holds Pic( X ) ⊕ Π ≃ Pic( X ) ⊕ Π . The proof below is quite standard and depends on the resolution ofsingularities. There is more sophisticated but similar proof due to L.Moret-Bailly which works in positive characteristic as well (see [MB86,6.2], [CTS87, 2.A.1]).
Proof.
By our assumption, for some n and m , there exists a G -birational map X × P n X × P m , where the action of G on P n and P m is trivial. Replacing X and X with X × P n and X × P m respec-tively we may assume that there exists a G -birational map X X .Consider a common G -equivariant resolution (see e.g. [AW97]) W f ! ! ❇❇❇❇❇❇❇❇ f } } ⑤⑤⑤⑤⑤⑤⑤⑤ X / / ❴❴❴❴❴❴❴ X Then the maps f ∗ and f ∗ induce isomorphisms Pic( W ) ≃ Pic( X ) ⊕ Π ≃ Pic( X ) ⊕ Π , where Π (resp. Π ) is a free Z -module whose basisis formed by the prime f -exceptional (resp. f -exceptional) divisors. ince f and f are G -equivariant, the group G permutes these divisors,so Π and Π are permutation modules. (cid:3) In the notation of Proposition we have H ( G, Pic( X )) ≃ H ( G, Pic( X )) .Proof. By Shapiro’s lemma H ( G, Π ) = H ( G, Π ) = 0 (see e.g.[Ser63, Ch. 1, § (cid:3) If in the notation of Proposition
X, G ) is stablylinearizable, then H ( H, Pic( X )) = 0 for any subgroup H ⊂ G . Proof of Theorem δ ∈ G be a generator and let C be a (smooth) curve of fixedpoints with g := g ( C ) >
0. We replace (
X, G ) with its minimalmodel. First we consider the case where X is a del Pezzo surfacewith rk Pic( X ) G = 1. We start with more general settings. Let X be a del Pezzo surface and let d := K X .Let G ⊂ Aut( X ) be any finite subgroup such that Pic( X ) G ≃ Z .Denote Q := { x ∈ Pic( X ) | x · K X = 0 } . In the above notation there exists the following naturalexact sequence → Z /d Z −→ H ( G, Q ) −→ H ( G, Pic( X )) −→ . Proof.
By our assumption g >
0, we have d ≤ X ) G = Z · [ K X ] (see [Isk80]). Then the assertionfollows from the exact sequence of G -modules(3.1.2) 0 −→ Q −→ Pic( X ) · K X −→ Z −→ H ( G, Z ) = 0 for a finite group G . (cid:3) In the notation of assume that G be a cyclicgroup generated by δ ∈ G . Then the order of H ( G, Pic( X )) equalsto | χ δ,Q (1) | /d , where χ δ,Q ( t ) is the characteristic polynomial of the ac-tion of δ on Q . The proof uses the following easy observation.
Let G be a cyclic group of order n generated by δ ∈ G . Let Π be a Z -torsion free Z [ G ]-module. Denote N := 1 + δ + · · · + δ n − , η := 1 − δ ∈ Z [ G ] . Then H ( G, Π) = ker( N ) /η (Π) . roof of Lemma . Apply the above fact with Π = Q . We getker( N ) = Q (because N ( Q ) ⊂ Q δ = 0). Hence [ Q : η ( Q )] = | det( η ) | . (cid:3) In the notation of , let G be a cyclic group ofprime order p . Then d = p j , where j = 0 or , and H ( G, Pic( X )) ≃ ( Z /p Z ) (9 − d ) / ( p − − j . Proof.
Since Q has no δ -invariant vectors, the only Q -irreducible factorof χ δ,Q ( t ) is the cyclotomic polynomial t p − + · · · + t + 1. Hence we have χ δ,Q ( t ) = ( t p − + · · · + t + 1) s , s = rk( Q ) / ( p −
1) = (9 − d ) / ( p − . On the other hand, H ( G, Pic( X )) is a p -torsion group because G ≃ Z /p Z and Z /d Z is a subgroup of Z /p Z by (3.1.2). Hence d = p j with j = 0 or 1 and H ( G, Pic( X )) ≃ ( Z /p Z ) s − j . (cid:3) By the classification theorem 2.3.1, for ( p, d, g ) we have one of thefollowing possibilities: (2 , , , , , , , , , , Now assume that X has a structure of G -equivariant conic bundle f : X → P . Then again by Theorem 2.3.1 δ is a de Jonqui`eres involution of genus g >
0. Recall that this is anelement δ ∈ Cr ( k ) of order 2 induced by an action on a ( G -equivariant)relatively minimal conic bundle f : X → P with 2 g + 2 degeneratefibers so that the locus of fixed points is a hyperelliptic curve C ofgenus g (elliptic curve if g = 1) and the restriction f | C : C → P is adouble cover (see [BB00], [DI09] for details). Let F be a typical fiber and let F i = F ′ i + F ′′ i , i = 1 , . . . , g + 2be all the degenerate fibers. Let Q ⊂ Pic( X ) be the Z -submodule ofrank 2 g + 3 generated by the components of degenerate fibers. It hasa Z -basis consisting of F and F ′ i , i = 1 , . . . , g + 2. The action of δ isgiven by δ : F ′ i F − F ′ i , F F. Apply 3.1.4 with Π = Q . We haveker( N ) = n αF + X α i F ′ i ∈ Q (cid:12)(cid:12)(cid:12) α + X α i = 0 o and η ( Q ) is generated by the classes of 2 F ′ i − F . Therefore, H ( G, Q ) = ( Z / Z ) g +1 . Since H ( G, Z ) = 0, from the exact sequence(3.3.1) 0 −→ Q −→ Pic( X ) · F −→ Z −→ e get0 → Q G −→ Pic( X ) G · F −→ Z −→ H ( G, Q ) −→ H ( G, Pic( X )) → . Note that the image of Pic( X ) G in Z is generated by K X · F = 2.Therefore, H ( G, Pic( X )) = ( Z / Z ) g .This proves Theorem 1.1. Proof of Corollary . The implication (i) ⇒ (ii) follows by Theorem1.1, implications (iii) ⇒ (iv) ⇒ (i) are obvious, and (ii) ⇒ (iii) followsby Theorem 2.3.1. (cid:3) References [AW97] D. Abramovich and J. Wang. Equivariant resolution of singularitiesin characteristic 0.
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