aa r X i v : . [ m a t h . AG ] A ug CREMONA MAPS AND INVOLUTIONS
JULIE DÉSERTIA
BSTRACT . We deal with the following question of Dolgachev : is the Cremona groupgenerated by involutions ? Answer is yes in dimension 2 ( see [6]). We give an upperbound of the minimal number n ϕ of involutions we need to write a birational self map ϕ of P C .We prove that de Jonquières maps of P C and maps of small bidegree of P C can bewritten as a composition of involutions of P C and give an upper bound of n ϕ for suchmaps ϕ . We get similar results in particular for automorphisms of ( P C ) n , automor-phisms of P n C , tame automorphisms of C n , monomial maps of P n C , and elements of thesubgroup generated by the standard involution of P n C and PGL( n + C ).
1. I
NTRODUCTION
This article is motivated by the following question:
Question (Dolgachev) . Is the n dimensional Cremona group generated by involutions ?Answer is yes in dimension 2; more precisely: Proposition 1.1 ([6]) . For any ϕ in Bir( P C ) there exist A , A , . . . , A k in Aut( P C ) such that ϕ = ³ A ◦ σ ◦ A − ´ ◦ ³ A ◦ σ ◦ A − ´ ◦ . . . ◦ ³ A k ◦ σ ◦ A − k ´ where σ denotes the standard involution of P C σ : ( z : z : z ) ( z z : z z : z z ).Let us note that since Bir( P R ) is generated by PGL(3, R ) and some involutions ([21]),any element of Bir( P R ) can be written as a composition of involutions.If ϕ is an element of G, then n ( ϕ , H) is the minimal number of involutions of H ⊃ G weneed to write ϕ . In dimension 2 we get the following result: Theorem A. If ϕ is an automorphism of P C × P C , then n ¡ ϕ , Aut( P C × P C ) ¢ ≤ .If ϕ is an automorphism of P C , then n ¡ ϕ , Aut( P C ) ¢ ≤ .If ϕ belongs to the Jonquières subgroup J ⊂ Bir( P C ) , then n ¡ ϕ , J ¢ ≤ .If ϕ is a birational self map of P C of degree d , then n ¡ ϕ , Bir( P C ) ¢ ≤ d − . One can be more precise for the well-known subgroup Aut( C ) of polynomial auto-morphisms of C of Bir( P C ): Theorem B.
Let ϕ be an element of Aut( C ) of degree d . Then n ¡ ϕ , Bir( P C ) ¢ ≤ + d .More precisely, • if ϕ is affine, then n ¡ ϕ , Aut( P C ) ¢ ≤ ; • if ϕ is elementary, then n ¡ ϕ , J ¢ ≤ ; • if ϕ is generalized Hénon map, then either it is of jacobian and n ¡ ϕ , Aut( C ) ¢ = or n ¡ ϕ , Bir( P C ) ¢ ≤ ; Date : August 7, 2017. REMONA MAPS AND INVOLUTIONS 2 • if d is prime, then n ¡ ϕ , Bir( P C ) ¢ ≤ . What happens in higher dimension ? A first result is the following:
Proposition C. • If ϕ is an automorphism of ( P C ) n , then ϕ can be written as a com-position of involutions of ( P C ) n , and n ¡ ϕ , Aut ¡ P C ¢ n ¢ ≤ n. • If ϕ is an automorphism of P n C , then ϕ can be written as a composition of involu-tions of P n C , and n ¡ ϕ , Aut( P n C ) ¢ ≤ n + . Since any element ofG n ( C ) = 〈 σ n = ³ n Y i = i z i : n Y i = i z i : . . . : n Y i = i n z i ´ , Aut( P n C ) 〉 can be written as a composition of conjugate involutions ([7]) one gets that: Theorem D.
Let n ≥ . Any element of the normal subgroup generated by G n ( C ) in Bir( P n C ) can be written as a composition of involutions of P n C . Furthermore one can give an upper bound of n ¡ ϕ , Bir( P n C ) ¢ when ϕ belongs to the sub-group of tame automorphisms of C n : Theorem E.
Let n ≥ . Let ϕ be a tame automorphism of C n of degree d . Then ϕ can bewritten as a composition of involutions of P n C . Moreover, • if ϕ is affine, then n ¡ ϕ , Aut( P n C ) ¢ ≤ n + ; • if ϕ is elementary, then n ¡ ϕ , Bir( P n C ) ¢ ≤ n + ; • otherwise n ¡ ϕ , Bir( P n C ) ¢ ≤ d (2 n + + n + . Let us recall ( see [15]) that the Jonquières subgroup J O (1, P C ) of Bir( P C ) is given in theaffine chart z = © ϕ = ¡ ϕ ( z , z , z ), ψ ( z , z ) ¢ | ϕ ∈ PGL(2, C [ z , z ]), ψ ∈ Bir( P C ) ª .Denote by Mon( n , C ) the group of monomial maps of P n C , and finally set J n = PGL(2, C ( z , z ,... , z n − )) × PGL(2, C ( z , z ,... , z n − )) × ... × PGL(2, C ( z n − )) × PGL(2, C ) ⊂ Bir( P n C ). Theorem F.
Assume that ≤ ℓ ≤ , and n ≥ . • If ϕ ∈ Bir( P C ) is of bidegree (2, ℓ ) , then ϕ can be written as a composition of invo-lutions of P C , and n ¡ ϕ , Bir( P C ) ¢ ≤ + ℓ . • Any element ϕ of J O (1; P C ) of degree d can be written as a composition of involu-tions of P C , and n ¡ ϕ , Bir( P C ) ¢ ≤ d + . • If ϕ belongs to Mon( n , C ) , then ϕ can be written as a composition of involutionsof P n C , and n ¡ ϕ , Mon( n , C ) ¢ ≤ n + . • Any element ϕ of J n can be written as a composition of involutions of P n C , and n ¡ ϕ , J n ¢ ≤ n − . If H is a subgroup of G let us denote by N(H; G) the normal subgroup generated by Hin G.
Corollary G.
Any element of 〈 N ¡ PGL(4, C );Bir( P C ) ¢ , N ¡ J O (1; P C );Bir( P C ) ¢ , N ¡ Mon(3, C );Bir( P C ) ¢ ,N ¡ G ( C );Bir( P C ) ¢ ,N ¡ 〈 ϕ ,... , ϕ k 〉 ;Bir( P C ) ¢ | ϕ i ∈ Bir( P C ) of bidegree (2, ℓ ) , ≤ ℓ ≤ 〉 can be written as a composition of involutions of P C . REMONA MAPS AND INVOLUTIONS 3
For any n ≥ , any element of 〈 N ¡ PGL( n + C );Bir( P n C ) ¢ , N ¡ J n ;Bir( P n C ) ¢ , N ¡ Mon( n , C );Bir( P n C ) ¢ , N ¡ G n ( C );Bir( P n C ) ¢ 〉 can be written as a composition of involutions of P n C . Remark 1.2.
An other motivation for studying birational maps of P n C that can be writtenas a composition of involutions is the following. The group of birational maps of P n C thatcan be written as a composition of involutions is a normal subgroup of Bir( P n C ). So ifthe answer to Dolgachev Question is no, we can give a negative answer to the followingquestion asked by Mumford ([12]): is Bir( P n C ) a simple group ? Acknowledgments.
I would like to thank D. Cerveau for his constant availability andkindness. Thanks to S. Zimmermann for pointing out Proposition 4.2.2. R
ECALLS AND DEFINITIONS
Polynomial automorphisms of C n . A polynomial automorphism ϕ of C n is a bijec-tive map from C n into itself of the type( z , z , . . . , z n − ) ¡ ϕ ( z , z , . . . , z n − ), ϕ ( z , z , . . . , z n − ), . . . , ϕ n − ( z , z , . . . , z n − ) ¢ with ϕ i ∈ C [ z , z , . . . , z n − ]. The set of polynomial automorphisms of C n form a groupdenoted Aut( C n ).Let A n be the group of affine automorphisms of C n , and let E n be the group of elemen-tary automorphisms of C n . In other words A n is the semi-direct product of GL( n , C ) withthe commutative unipotent subgroup of translations. Furthermore E n is formed withautomorphisms ( ϕ , ϕ , . . . , ϕ n − ) of C n where ϕ i = ϕ i ( z i , z i + , . . . , z n − )depends only on z i , z i + , . . ., z n − . The subgroup Tame n of Aut( C n ), called the groupof tame automorphisms of C n , is the group generated by A n and E n . For n = = Aut( C ), more precisely: Theorem 2.1 ([11]) . The group
Aut( C ) has a structure of amalgamated product Aut( C ) = A ∗ S E with S = A ∩ E . Nevertheless Tame Aut( C ) ( see [19]).2.2. Birational maps of P n C . A rational self map of P n C is a map of the type( z : z : . . . : z n ) ¡ ϕ ( z , z , . . . , z n ) : ϕ ( z , z , . . . , z n ) : . . . : ϕ n ( z , z , . . . , z n ) ¢ where the ϕ i ’s denote homogeneous polynomials of the same degree without commonfactor (of positive degree).A birational self map ϕ of P n C is a rational map of P n C such that there exists a rationalself map ψ of P n C with the following property ϕ ◦ ψ = ψ ◦ ϕ = id where id : ( z : z : . . . : z n ) ( z : z : . . . : z n ).The degree of ϕ ∈ Bir( P n C ) is the degree of the ϕ i ’s. For n =
2, one has deg ϕ = deg ϕ − ;for n = bidegree of ϕ which is (deg ϕ , deg ϕ − ).The group of birational self maps of P n C is denoted Bir( P n C ) and called Cremona group .The groups Aut( P n C ) = PGL( n + C ) and Aut( C n ) are subgroups of Bir( P n C ).Let us mention that contrary to Aut( C ) the Cremona group in dimension 2 does notdecompose as a non-trivial amalgam (appendix of [4]). REMONA MAPS AND INVOLUTIONS 4
Birational involutions in dimension . Let us first describe some involutions: • Consider an irreducible curve C of degree ν ≥ p ;assume furthermore that p is an ordinary multiple point with multiplicity ν − C , p ) we can associate a birational involution I J which fixes pointwise C andpreserves lines through p as follows. Let m be a generic point of P C r C ; let r m , q m and p be the intersections of the line ( mp ) with C ; the point I J ( m ) is definedby: the cross ratio of m , I J ( m ), q m and r m is equal to −
1. The map I J is a deJonquières involution of P C . A birational involution is of de Jonquières type if it isbirationally conjugate to a de Jonquières involution of P C . • Let p , p , . . ., p be eight points of P C in general position. Consider the set ofsextics S = S ( p , p , . . . , p ) with double points at p , p , . . ., p . Take a point m in P C . The pencil given by the elements of S having a double point at m has atenth base double point point m ′ . The involution which switches m and m ′ isa Bertini involution . A birational involution is of Bertini type if it is birationallyconjugate to a Bertini involution. • Let p , p , . . ., p be seven points of P C in general position. Denote by L thelinear system of cubics through the p i ’s. Consider a generic point p in P C anddefine by L p the pencil of elements of L passing through p . The involution whichswitches p and the ninth base-point of L p is a Geiser involution . A birationalinvolution is of Geiser type if it is birationally conjugate to a Geiser involution.Birational involutions of P C have been classified: Theorem 2.2 ([2]) . A non-trivial birational involution of P C is either of de Jonquières type,or of Bertini type, or of Geiser type. Birational involutions in higher dimension.
There are no classification in higherdimension; in [17] the author gives a first nice step toward a classification in dimension 3.Let us give some examples: • the involution σ n = ³ n Y i = i z i : n Y i = i z i : . . . : n Y i = i n z i ´ • the involutions of PGL( n + C ); • the involutions of Mon( n , C ) induced by the involutions of GL( n , Z ); • the de Jonquières involutions: consider a reduced hypersurface H of degree ν in P n C that contains a linear subspace of points of multiplicity ν −
2. Assumethat p is a singular point of H of multiplicity ν −
2. Take a general point m of H .Denote by ℓ p the line passing through p and m . The intersection of ℓ p with H contains p with multiplicity ν −
2, and the residual intersection is a set of twopoints r m and q m in ℓ p . Define I J ( m ) to be the point on ℓ p such that the crossratio of m , I J ( m ), q m and r m are equal to −
1. The map I J is a de Jonquièresinvolution of P n C .3. A UTOMORPHISMS OF ( P C ) n AND OF P n C Lemma 3.1.
Any non-trivial homography is either an involution, or the composition oftwo involutions of
PGL(2, C ) . REMONA MAPS AND INVOLUTIONS 5
In particular if ϕ belongs to Aut ¡ P C × P C × . . . × P C | {z } n times ¢ , then n ¡ ϕ , Aut ¡ P C × P C × . . . × P C | {z } n times ¢¢ ≤ n . Remark 3.2.
The homography ν ∈ PGL(2, C ) is a non-trivial involution if and only if thereexists p ∈ P C r Fix( ν ) such that ν ( p ) = p , where Fix( ν ) denotes the set of fixed points of ν .Indeed assume that there exists p ∈ P C r Fix( ν ) such that ν ( p ) = p , then Fix( ν ) P C and so ν id. If m ∈ { p , ν ( p )}, then ν ( m ) = m . If p { p , ν ( p )}, the cross ratio of p , ν ( p ), ν ( m ), m is equal to the cross ratio of p , ν ( p ), ν ( m ) and ν ( m ). This implies that ν ( m ) = m . Lemma 3.3.
Let ν ∈ PGL(2, C ) be an homography. Consider three points a, b, c of P C suchthat a, b, c are distinct, a Fix( ν ) , b Fix( ν ) and b ν ( a ) .There exist two involutions ι , ι ∈ PGL(2, C ) such that ν = ι ◦ ι .Proof. Let us first prove that there exists two unique homographies ι , ι ∈ PGL(2, C ) suchthat ½ ι ( a ) = ν ( b ), ι ( b ) = ν ( a ), ι ( ν ( a )) = b ; ι ( ν ( a )) = ν ( b ), ι ( ν ( b )) = ν ( a ), ι ( ι ( c )) = ν ( c ).Note that by assumptions a , b , ν ( a ) (resp. ν ( b ), ν ( a ), b ) are pairwise distinct. Hencethere exists a unique homography ι ∈ PGL(2, C ) that sends a , b , ν ( a ) onto ν ( b ), ν ( a ), b .The points ν ( a ), ν ( b ) and ι ( c ) are distinct. Assume by contradiction that ι ( c ) = ν ( a ),then ι ( c ) = ι ( b ). By injectivity of ι , one has c = b : contradiction. Similarly ι ( c ) ν ( b ) = ι ( a ) and ν ( a ) ν ( b ). Since a , b , c are distinct, ν ( a ), ν ( b ) and ν ( c ) also. As a consequencethere exists a unique homography ι ∈ PGL(2, C ) that sends ν ( a ), ν ( b ), ι ( c ) onto ν ( b ), ν ( a ), ν ( c ).By assumption b and ν ( a ) are distinct so b does not belong to Fix( ι ). But ι ( b ) = ι ( ν ( a )) = b . According to Remark 3.2 the homography ι is thus an involution. Similarly ν ( a ) and ν ( b ) are distinct but ν ( a ) and ν ( b ) are switched by ι hence ι is an involution(Remark 3.2).Since ν ( p ) = ι ◦ ι ( p ) for p ∈ { a , b , c } one gets ν = ι ◦ ι . (cid:3) Proof of Lemma 3.1.
Let ν be an homography. If ν = id, then ν = ι ◦ ι for any involution ι .Assume now that ν id; then ν has at most two fixed points. Let us choose a , b in P C r Fix( ν ). If a ν ( b ) or if b ν ( a ), then ν can be written as a composition of two involutions(Lemma 3.3). If b = ν ( a ) and a = ν ( b ), then ν ( a ) = a with a Fix( ν ); Remark 3.2 thusimplies that ν is an involution. (cid:3) Lemma 3.4.
Let n ≥ be an integer. (1) Let k be a commutative ring of any characteristic. If ϕ is an element of SL( n , k ) ,then n ¡ ϕ , SL( n , k ) ¢ ≤ n + . (2) Assume that k is an algebraically closed field, and that ϕ belongs to PGL( n , k ) .Then n ¡ ϕ , PGL( n , k ) ¢ ≤ n + . (3) If ϕ is an element of PGL(2, C [ z , z , . . . , z n − ]) , then n ¡ ϕ , Bir( P n C ) ¢ ≤ .Proof. (1) Let us recall that an element of SL( n , k ) can be written as a compositionof ≤ n + n , k ) can be written as a composition of ≤ n + k is algebraically closed, then PSL( n , k ) ≃ PGL( n , k ) and one gets the result. REMONA MAPS AND INVOLUTIONS 6 (3) Let g be an element of PGL(2, C [ z , z , . . . , z n − ]); denote by P ( z , z , . . . , z n − ) itsdeterminant and by h a scaling of scale factor P ( z , z ,..., z n − ) . Then h ◦ g belongs toSL(2, C ( z , z , . . . , z n − )) and hence, according to the first assertion, can be writtenas a composition of ≤ h is as a composition of two involutions:1 z P ( z , z , . . . , z n − ) ◦ z .As a result n ¡ ϕ , Bir( P n C ) ¢ ≤ (cid:3)
4. D
IMENSION
The real Cremona group.
There is an analogue to Proposition 1.1 for the real Cre-mona group.
Theorem 4.1.
Any element of
Bir( P R ) can be written as a composition of involutions of P R . Theorem 4.1 directly follows from the simplicity of PGL(3, R ) and the following state-ment: Proposition 4.2 ([21]) . The group
Bir( P R ) is generated by PGL(3, R ) , the set of standardquintic involutions and the two following quadratic involutions ( z z : z z : z z ) ( z z : z z : z + z ).4.2. The de Jonquières subgroup.
An element of Bir( P C ) is a de Jonquières map if it pre-serves a rational fibration, i.e. if it is conjugate to an element ofJ = PGL(2, C ( z )) ⋊ PGL(2, C ).We will denote by e J the subgroup of birational maps that preserves fiberwise the fibra-tion z = constant, i.e. e J = PGL(2, C ( z )). Lemma 4.3. If ϕ = ( ϕ , ϕ ) belongs to e J , then n ¡ ϕ , e J ¢ ≤ .Furthermore if det ϕ = ± , then n ¡ ϕ , e J ¢ ≤ .Proof. Lemma 3.4 implies the first assertion, and the last assertion follows from [9]. (cid:3)
Corollary 4.4.
Any de Jonquières map of P C can be written as a composition of ≤ Cre-mona involutions of P C .Proof. Let us remark that any Jonquières map ϕ of P C can be written as ψ ◦ j ◦ ψ − where ψ denotes an element of Bir( P C ) and j an element of J . But j = µ a ( z ) z + b ( z ) c ( z ) z + d ( z ) , α z + βγ z + δ ¶ = µ z , α z + βγ z + δ ¶ ◦ µ a ( z ) z + b ( z ) c ( z ) z + d ( z ) , z ¶ As a consequence ϕ = µ ψ ◦ µ z , α z + βγ z + δ ¶ ◦ ψ − ¶ ◦ µ ψ ◦ µ a ( z ) z + b ( z ) c ( z ) z + d ( z ) , z ¶ ◦ ψ − ¶ Then one concludes with Lemmas 3.1 and 4.3. (cid:3)
REMONA MAPS AND INVOLUTIONS 7
Subgroup of polynomial automorphisms of C . Note that there is no analogue toProposition 1.1 in the context of polynomial automorphisms of C . For instance the au-tomorphism (2 z , 3 z ) cannot be written as a composition of involutions in Aut( C ).According to Lemma 3.4 and Corollary 4.4 one has the following result: Lemma 4.5.
Let ϕ be a polynomial automorphism of C .If ϕ is an affine automorphism, then n ¡ ϕ , Aut( P C ) ¢ ≤ .If ϕ is an elementary automorphism, then n ¡ ϕ , J ¢ ≤ . An element ϕ ∈ Aut( C ) is a generalized Hénon map if ϕ = ( z , P ( z ) − δ z )where δ belongs to C ∗ and P is an element of C [ z ] of degree ≥
2. Note that δ = jac( ϕ ). Lemma 4.6.
Let ϕ ∈ Aut( C ) be a generalized Hénon map. • If ϕ has jacobian , then n ¡ ϕ , Aut( C ) ¢ ≤ ; • otherwise n ¡ ϕ , Bir( P C ) ¢ ≤ .Proof. Any generalized Hénon map of jacobian 1 can be written ¡ z , P ( z ) − z ¢ and so isthe composition of two involutions: ( z , P ( z ) − z ) = ( z , z ) ◦ ¡ P ( z ) − z , z ¢ .Let ϕ be a generalized Hénon map; then ϕ = ( z , P ( z ) − δ z ) = ( z , z ) ◦ ¡ P ( z ) − δ z , z ¢ .Note that ¡ P ( z ) − δ z , z ¢ is an elementary automorphism; therefore n ¡ ϕ , Bir( P C ) ¢ ≤ + =
11 (Lemma 4.5). (cid:3)
Friedland and Milnor proved that any polynomial automorphism of degree d with d prime is conjugate via an affine automorphism either to a generalized Hénon map or toan elementary automorphism ([8, Corollary 2.7]). Since any generalized Hénon map isthe composition of ( z , z ) ∈ A with an elementary map one gets that any polynomialautomorphism of degree d with d prime can be written as a e a with a i ∈ A and e ∈ E .Lemmas 4.5 and 4.6 thus imply: Lemma 4.7. If ϕ ∈ Aut( C ) is of degree d with d prime, then n ¡ ϕ , Bir( P C ) ¢ ≤ . A sequence ( ϕ , ϕ , . . . , ϕ k ) of length k ≥ ϕ = ϕ k ◦ ϕ k − ◦ . . . ◦ ϕ if • each factor ϕ i belongs to either A or E but not to the intersection A ∩ E , • and no two consecutive factors belong to the same subgroup A or E .It follows from Theorem 2.1 that every element of Aut( C ) can be expressed as such areduced word, unless it belongs to the intersection S = A ∩ E . The degree of any re-duced word ϕ = ϕ k ◦ ϕ k − ◦ . . . ◦ ϕ is equal to the product of the degree of the factor ϕ i ( see [8, Theorem 2.1]). Hence take ϕ ∈ Aut( C ) of degree d ≥
2, then ϕ is a reduced word ϕ k ◦ ϕ k − ◦ . . . ◦ ϕ and • either there exists only one ϕ i of degree >
1, then ϕ = ϕ ◦ ϕ ◦ ϕ with deg ϕ > ϕ = deg ϕ =
1; as a result n ¡ ϕ , Bir( P C ) ¢ ≤
26 (Lemma 4.5), • or there exits at least two ϕ i ’s of degree >
1, then n ¡ ϕ , Bir( P C ) ¢ ≤ d +
44. Indeed let( a , e , a , e , a , . . . , e k , a k ) be a reduced word representing ϕ . Any e i has degree ≥ ϕ = deg e deg e Q ki = deg e i hence Q ki = deg e i ≤ d and so 2( k − ≤ d .As a result k ≤ d + n ¡ ϕ , Bir( P C ) ¢ ≤ ( k + n ¡ a i , Bir( P C ) ¢ + k n ¡ e i , Bir( P C ) ¢ ≤ µ d + ¶ n ¡ a i , Bir( P C ) ¢ + µ d + ¶ n ¡ e i , Bir( P C ) ¢ . REMONA MAPS AND INVOLUTIONS 8
One can thus state
Theorem 4.8.
Let ϕ be a polynomial automorphism of C of degree d . • If ϕ is affine, n ¡ ϕ , Aut( P C ) ¢ ≤ ; • if ϕ is elementary, then n ¡ ϕ , J ¢ ≤ ; • if ϕ is generalized Hénon map, then either it is of jacobian and n ¡ ϕ , Aut( C ) ¢ = or n ¡ ϕ , Bir( P C ) ¢ ≤ ; • if d is prime, then n ¡ ϕ , Bir( P C ) ¢ ≤ ; • otherwise n ¡ ϕ , Bir( P C ) ¢ ≤ d + . Corollary 4.9. If ϕ is a polynomial automorphism of C of degree d , then n ¡ ϕ , Bir( P C ) ¢ ≤ d + . Birational maps.Theorem 4.10. If ϕ ∈ Bir( P C ) is of degree d , then n ¡ ϕ , Bir( P C ) ¢ ≤ d − . Before proving Theorem 4.10 let us give a first and "bad" bound. Let ϕ be a birationalself map of P C of degree d . The number of base points of ϕ is ≤ d − ϕ can be written with ≤ d −
1) blow ups. Since a blow up can be written as A ◦ σ ◦ A ◦ σ ◦ A with A i ∈ PGL(3, C ) the map ϕ can be written with 4(2 d −
1) involution σ and4(2 d − + C ). As a consequence ϕ can be written as a compositionof ≤ d − + ¡ d − + ¢ = d −
28 involutions.
Proof of Theorem 4.10.
Let us recall that if ϕ is a birational self map of P C of degree d ,then there exists a de Jonquières map ψ of P C such that deg( ϕ ◦ ψ ) < d ( see [5], [1, Theo-rem 8.3.4]).As a result any ϕ ∈ Bir( P C ) of degree d ≥ A ◦ ¡ ψ ◦ j ◦ ψ − ¢ ◦ ¡ ψ ◦ j ◦ ψ − ¢ ◦ . . . ◦ ¡ ψ k ◦ j k ◦ ψ − k ¢ with A in PGL(3, C ), ψ ℓ ∈ Bir( P C ), j l in J and k ≤ d − (cid:3)
5. D
IMENSION de Jonquières maps in dimension . Let us recall that a de Jonquières map ϕ of P C of degree d is a plane Cremona map satisfying one of the following equivalent conditions: • there exists a point O ∈ P C such that the restriction of ϕ to a general line passingthrough O maps it birationally to a line passing through O ; • ϕ has homaloidal type ( d ; d −
1, 1 d − ), i.e. ϕ has 2 d − d − d − • up to projective coordinate changes (source and target) ϕ = ¡ z g d − + g d : ( z q d − + q d − ) z : ( z q d − + q d − ) z ¢ with g d − , g d , q d − , q d − ∈ C [ z , z ] of degree d −
1, resp. d , resp. d −
2, resp. d − P n C , n ≥
3. More precisely they study elements of the Cremona groupBir( P n C ) satisfying a condition akin to the first alternative above: for a point O ∈ P n C anda positive integer k they consider the Cremona transformations that map a general k -dimensional linear subspace passing through O onto another such subspace. Fixing thepoint O these maps form a subgroup J O ( k ; P n C ) of Bir( P n C ). For any k ≤ ℓ the followinginclusion holds ([15]) J O ( ℓ ; P n C ) ⊂ J O ( k ; P n C ) REMONA MAPS AND INVOLUTIONS 9
Let us recall the following characterization of elements of J O (1; P n C ): Proposition 5.1 ([13]) . Fix O = (0 : 0 : . . . : 0 : 1) . A Cremona map ϕ ∈ Bir( P n C ) belongs to J O (1; P n C ) if and only if ϕ = ¡ z g d − + g d : ( z q ℓ − + q ℓ ) t : ( z q ℓ − + q ℓ ) t : . . . : ( z q ℓ − + q ℓ ) t n ¢ where • g d , g d − , q ℓ , q ℓ − , t , . . . , t n ∈ C [ z , z , . . . , z n ] , • deg g d − = d − , deg g d = d , deg q ℓ − = ℓ − , deg q ℓ = ℓ , • deg t i = d − ℓ for i ∈ {1, . . . , n } , • ( t : t : . . . : t n ) ∈ Bir( P n − C ) . Theorem 5.2.
Let ϕ be an element of J O (1, P C ) of degree d ; then n ¡ ϕ , Bir( P C ) ¢ ≤ d + . If H is a subgroup of G let us denote by N(H; G) the normal subgroup generated by Hin G.
Corollary 5.3.
Any birational map of N ¡ J O (1, P C ); Bir( P C ) ¢ is a composition of involutionsof Bir( P C ) .Proof of Theorem 5.2. Any ϕ in J O (1, P C ) can be written in the affine chart z = ϕ = µ z A ( z , z ) + B ( z , z ) z C ( z , z ) + D ( z , z ) , ψ ( z , z ) ¶ where z A ( z , z ) + B ( z , z ) z C ( z , z ) + D ( z , z ) ∈ PGL(2, C [ z , z ]), ψ ∈ Bir( P C ).Let us note that ϕ = ¡ z , ψ ( z , z ) ¢ ◦ µ z A ( z , z ) + B ( z , z ) z C ( z , z ) + D ( z , z ) , z , z ¶ .The map ψ can be written as a composition of ≤ d − z A ( z , z ) + B ( z , z ) z C ( z , z ) + D ( z , z ) ∈ PGL(2, C [ z , z ]) can be written as a composition of ≤ ϕ is a composition of 10 d + (cid:3) Maps of small bidegrees. If ϕ is a birational self map of P C , then the bidegree of ϕ is the pair (deg ϕ , deg ϕ − ). Let us recall that deg ϕ − ≤ ¡ deg ϕ ¢ . The left-right conjugacyis the following onePGL(4, C ) × Bir( P C ) × PGL(4, C ) ( A , ϕ , B ) A ϕ B − .Pan, Ronga and Vust give birational self maps of P C of bidegree (2, · ) up to left-rightconjugacy, and show that there are only finitely many biclasses ([14, Theorems 3.1.1,3.2.1, 3.2.2, 3.3.1]). In particular they show that the smooth and irreducible variety ofbirational self maps of P C of bidegree (2, · ) has three irreducible components of dimen-sion 26, 28, 29. More precisely the component of dimension 26 (resp. 28, resp. 29) cor-responds to birational maps of bidegree (2, 4) (resp. (2, 3), resp. (2, 2)). Let us denote by O ( ϕ ) the orbit of ϕ under the left-right conjugacy. Proposition 5.4.
Let ϕ be a birational self map of P C of bidegree (2, 2) . Then ϕ can bewritten as a composition of involutions of P C . Furthermore n ¡ ϕ , Bir( P C ) ¢ ≤ . REMONA MAPS AND INVOLUTIONS 10
Proof. If ϕ is a birational self map of P C of bidegree (2, 2), then up to left-right conjugacy ϕ is one of the following ([14]) f = ¡ z z : z z : z z : z − z z ¢ f = ¡ z z : z z : z z : z z ¢ f = ¡ z z : z z : z z : z ¢ f = ¡ z z : z z : z : z z − z + z z ¢ f = ¡ z z : z z : z : z z − z z ¢ f = ¡ z z : z z : z : z z − z ¢ f = ¡ z z − z z : z z : z z : z ¢ f = ¡ z z : z z : z z : z ¢ Note that f = id, and that f , f , f are involutions. Any element ψ in O ( f i ), i ∈ {1, 2, 3, 8}, satisfies n ¡ ψ , Bir( P C ) ¢ ≤
21. The other f i are de Jonquières maps of P C soaccording to Theorem 5.2 can be written as compositions of involutions. Nevertheless tofind a better bound for n ¡ ϕ , Bir( P C ) ¢ we will give explicit decompositions.First f = ¡ z : z + z : − z : z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z − z z : z z : z ¢ ◦ ¡ z : z : − z : z + z ¢ hence for any ψ ∈ O ( f ) one has n ¡ ψ , Bir( P C ) ¢ ≤
23 (which corresponds to two elementsin PGL(4, C ) and three involutions).Second f = ( z : z − z : z : z ) ◦ ( z z : z z : z : z z ) ◦ ( z z : z z : z : z z ) ◦ ( z : z : z − z : z ). As a consequence n ¡ ψ , Bir( P C ) ¢ ≤
22 for any ψ ∈ O ( f ).Third f = ¡ z : z : z : − z ¢ ◦ ¡ z z : z z : z − z z : z ¢ ◦ ¡ z : z : z : z ¢ . Therefore for any ψ ∈ O ( f ) one has the inequality n ¡ ψ , Bir( P C ) ¢ ≤ f = ¡ − z : z : z : z ¢ ◦ ¡ − z z + z z : z z : z z : z ¢ . So n ¡ ψ , Bir( P C ) ¢ ≤
21 for any ψ ∈ O ( f ). (cid:3) Proposition 5.5.
Any birational self map ϕ of P C of bidegree (2, 3) can be written as acomposition of involutions of P C ; moreover n ¡ ϕ , Bir( P C ) ¢ ≤ .Proof. If ϕ is a birational self map of P C of bidegree (2, 3), then ϕ ∈ O ( f i ) where f i is oneof the following map ([14]) f = ¡ − z z + z z : z z : − z z + z z : z z ¢ f = ¡ z z : z z − z z : z z : z z ¢ f = ¡ z z − z z : z z : z z : z z ¢ f = ¡ z z : z z : z z − z z : z ¢ f = ¡ z z : z z : z : z z ¢ f = ¡ z z : z z − z : z z : z z ¢ f = ¡ z z : z z − z z : z z : z ¢ f = ¡ z z − z : z z : z z : z z ¢ f = ¡ z : z z : z z : z z − z ¢ f = ¡ z : z z : z z : z z − z ¢ f = ¡ z z + z : z : z z : z z − z z ¢ REMONA MAPS AND INVOLUTIONS 11
Let us give for any of these maps a decomposition with involutions and elements ofPGL(4, C ): f = ¡ − z + z : − z + z : z : z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z : z − z : z : z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ − z + z : z : z : z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z : z z : z z : z z ¢ ◦ ¡ z z : z z : z z z : z z z ¢ f = ¡ z : z : z : z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z + z : z : z : z ¢ ◦ ¡ z : z z : z z : z z ¢ ◦ ¡ z : z z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z − z : z : z : z ¢ f = ¡ − z : z : z : z ¢ ◦ ¡ − z z + z z : z z : z z : z ¢ ◦ ¡ z : z z : z z : z z ¢ ◦ ¡ z z : z z : z z z : z z z ¢ f = ¡ z + z : z : z : z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z − z : z : z : z ¢ ◦ ¡ z z : z : z z : z z ¢ f = ¡ z : z : z : z ¢ ◦ ¡ z z : z z z : z z : z z ¢ ◦ ¡ z z : z z z : z z z : z z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z z : z z : z z : z ¢ f = ¡ z : − z : z : z ¢ ◦ ¡ z z : z z : z z − z z : z ¢ ◦ ¡ z z : z z : z z : z z ¢ f = ¡ z : z : z : z ¢ ◦ ¡ z : z z : z z : z z ¢ ◦ ¡ z : z z : z z : z z ¢ ◦ ¡ z : z : z − z : z ¢ ◦ ¡ z : z z : z z : z z ¢ ◦ ¡ z : z z : z z : z z ¢ f = ¡ z : z : z : z ¢ ◦ ¡ z z : z z − z z : z z : z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ f = ¡ z : z : z : − z ¢ ◦ ¡ z : z z : z z : − z z + z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ f = ¡ z : z : z : − z ¢ ◦ ¡ z z : z z : z : − z z + z z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ f = ¡ − z : z : z : − z ¢ ◦ ¡ z : z z : − z z − z : z z ¢ ◦ ¡ z : z z : z z : − z z + z z ¢ (cid:3) Proposition 5.6.
Let ϕ be a birational self map of P C of bidegree (2, 4) . Then ϕ can bewritten as a composition of involutions of P C . Furthermore n ¡ ϕ , Bir( P C ) ¢ ≤ .Proof. If ϕ is a birational self map of P C of bidegree (2, 4), then ϕ ∈ O ( f i ) where f i is oneof the following map ([14]) f = ¡ z z : z z : z z : z z − z − z ¢ f = ¡ z − z z : z z : z : z z ¢ f = ¡ z z : z z : z : z z − z ¢ f = ¡ z : z z : z ( z − z ) : z z − z ¢ f = ¡ z z : z z : z : z z − z ¢ f = ¡ z z : z z : z : z z − z − z ¢ f = ¡ z : z z : z − z z : z z ¢ f = ¡ z : z z : z − z z : z z − z ¢ f = ¡ z : z z : z − z z : z z − z ¢ f = ¡ z − z z : z − z z : z z : z z ¢ f = ¡ z − z z : z z : z z : z z − z ¢ REMONA MAPS AND INVOLUTIONS 12
Let us give for any of these maps a decomposition with involutions and elements ofPGL(4, C ): f = ¡ z : z : z : − z ¢ ◦ ¡ − z z z + z z + z z : z z : z z : z z z ¢ ◦ ¡ z z z : z z : z z : z z z ¢ ◦ ¡ z z : z z : z z : z z z ¢ f = ¡ z : z : z : z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z : z : z : z − z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z z : z z z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z z z : z z z : z z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ f = ¡ z : z : z : − z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ − z + z : z : z : z ¢ ◦ ¡ z z : z z : z z : z z ¢ ◦ ¡ z z : z z : z z : z z ¢ f = ¡ z : z − z : z : z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ − z : z : z : z − z ¢ ◦ ¡ − z z + z : z : z z : z z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z z : z z : z z : z ¢ f = ¡ z : z : z : z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ − z : z : z : z ¢ ◦ ¡ − z z + z : z : z z : z z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z z : z z : z z : z ¢ f = ¡ − z : z : z : − z − z ¢ ◦ ¡ z z : − z z + z : z z : z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ − z z + z : z z : z z : z ¢ ◦ ¡ z : z + z : z : z ¢ f = ¡ z : z : z : z ¢ ◦ ¡ z z : z z : z : z − z z ¢ ◦ ¡ z z z : z z z : z z : z z ¢ ◦ ¡ z z : z z z : z z : z z ¢ f = ¡ z : z : z : z ¢ ◦ ¡ z z : z : − z z + z : z z ¢ ◦ ¡ z z : z : z z z : z z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ − z : z : z : z ¢ ◦ ¡ − z z + z : z z : z z : z ¢ f = ¡ z : z : z : − z ¢ ◦ ¡ z z : z z : − z z + z : z ¢ ◦ ¡ − z z + z : z z : z z : z ¢ f = ¡ z − z : z − z : z : z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z : z : z : z + z ¢ ◦ ¡ z z : z : z z : z z ¢ ◦ ¡ z : z : z : z − z ¢ ◦ ¡ z z z : z z : z z z : z z ¢ ◦ ¡ z z : z z : z z z : z z ¢ f = ¡ z : z : z − z : − z ¢ ◦ ¡ z z : z z : z : z z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ − z + z : z : z : z − z ¢ ◦ ¡ z z : z z : z z : z ¢ ◦ ¡ z z : z z z : z z : z z ¢ ◦ ¡ z z z : z z z : z z : z z ¢ (cid:3) REMONA MAPS AND INVOLUTIONS 13
6. D
IMENSION ≥ The group generated by the automorphisms of P n C and the Cremona involution. Pan has proved that, as soon as n ≥
3, the subgroup generated by Aut( P n C ) and the invo-lution σ n is a strict subgroup G n ( C ) of Bir( P n C ). This subgroup has been studied in [3, 7],and in particular: Proposition 6.1 ([7]) . For any ϕ in G n ( C ) there exist A , A , . . . , A k in Aut( P n C ) such that ϕ = ³ A ◦ σ n ◦ A − ´ ◦ ³ A ◦ σ n ◦ A − ´ ◦ . . . ◦ ³ A k ◦ σ n ◦ A − k ´ Corollary 6.2.
Any element of N ¡ G n ( C ); Bir( P n C ) ¢ can be written as a composition of invo-lutions of P n C . The group of tame automorphisms.
As we already mentioned it, Tame does notcoincide with Aut( C ) ( see §2.1): the Nagata automorphism N = ¡ z + z ( z z − z ) + z ( z z − z ) , z + z ( z z − z ), z ¢ is not tame ([19]). Note that since the Nagata automorphism is contained in G ( C ) ( see [3]), it can also be written as a composition of involutions (Proposition 6.1). Since G n ( C )contains the group of tame polynomial automorphisms of C n ( see [7]) one gets that Proposition 6.3.
Any element of
N(Tame n , Aut( C n )) is a composition of involutions of P n C . Can we give an upper bound for n ( ϕ , Bir( P n C ) ¢ when ϕ ∈ Tame n ?Set H = n³ α z + p ( z ), n − X i = a i z i + γ , n − X i = a i z i + γ , . . . , n − X i = a n − i z i + γ n − ´¯¯¯ p ∈ C [ z ], α , a i , j , γ i ∈ C , α det( a i j ) o ,H = n³ α z + β z + γ , δ z + n − X i = a i z i + γ , n − X i = a i z i + γ , . . . , n − X i = a n − i z i + γ n − ´¯¯¯ α , β , γ , δ , a i , j , γ i ∈ C , det M ( α , β , γ , δ , a i , j ) o where M ( α , β , γ , δ , a i , j ) = α β γ δ a i , j ...0 One can check thatH ∩ H = n³ α z + β z + γ , n − X i = a i z i + γ , n − X i = a i z i + γ , . . . , n − X i = a n − i z i + γ n − ´¯¯¯ α , β , γ , δ , a i , j , γ i ∈ C , det M ( α , β , γ , a i , j ) o REMONA MAPS AND INVOLUTIONS 14 where M ( α , β , γ , a i , j ) = α β γ a i , j ...0 Proposition 6.4.
Let ϕ = ϕ k ◦ ϕ k − ◦ . . . ◦ ϕ be a reduced word in the amalgamated product H ∗ H ∩ H H .The degree of ϕ is equal to the product of the degree of the factors ϕ i .Proof. We follow the proof of [8, Theorem 2.1].Let ψ = ( ψ , ψ , . . . , ψ n − ) be an element of H ∗ H ∩ H H which satisfy the conditiondegrees d = deg ψ ≥ deg ψ i for any 0 ≤ i ≤ n − ϕ = ( ϕ , ϕ , . . . , ϕ n − ) an element of H r ( H ∩ H ) of degree d ; in par-ticular ϕ = α z + p ( z ) with d = deg p ≥
2. Denote by f ϕ i the components of ϕ ◦ ψ . Onehas d d = deg f ϕ > deg f ϕ i for any 1 ≤ i ≤ n − φ in H r ( H ∩ H ), and set φ ◦ ϕ ◦ ψ = ( e φ , f φ , . . . , ‚ φ n − ). Then d d = deg f φ ≥ deg f φ i for any i ≥ H r ( H ∩ H ) followed with anelement of H r ( H ∩ H ) the degree will be multiply by d . The statement follows byinduction. (cid:3) Let us now remark that 〈 H , H 〉 contains both A n and ( z + z , z , z , . . . , z n − ). SinceTame n = 〈 A n , ( z + z , z , z , . . . , z n − ) 〉 ( see [20, Chapter 5.2]) any tame automorphism isa reduced word in H ∗ H ∩ H H . Following what we did in §4.3 one obtains: Theorem 6.5.
Let ϕ be a tame automorphism of C n , n ≥ , of degree d . • If ϕ is affine, then n ¡ ϕ , Aut( P n C ) ¢ ≤ n + ; • if ϕ is elementary, then n ¡ ϕ , Bir( P n C ) ¢ ≤ n + ; • otherwise n ¡ ϕ , Bir( P n C ) ¢ ≤ d (2 n + + n + . Remark 6.6.
We cannot use this strategy to get a more precise statement for G n ( C ). In-deed using similar arguments as in the appendix of [4] one can prove that G n ( C ) hasproperty ( F R ); in particular, according to [18] one has:
Proposition 6.7.
The group G n ( C ) does not decompose as a non-trivial amalgam. More precisely if G n ( C ) is contained in an amalgam G ∗ A G , then G n ( C ) is containedin a conjugate of either G or G ( see [18]).6.3. Monomial maps in any dimension.
Let A n C be the affine space of dimension n . Themultiplicative group G nm can be identified to the Zariski open subset ( A C r {0}) n of P n C .Hence Bir( P n C ) contains the group of all algebraic automorphisms of the group G nm , i.e. the group Mon( n , C ) of monomial maps GL( n , Z ). Theorem 6.8 ([10]) . Let n ≥ be an integer. Any element ϕ of GL( n , Z ) can be written as acomposition of involutions of GL( n , Z ) , and n ¡ ϕ , GL( n , Z ) ¢ ≤ n + . Corollary 6.9.
Let ϕ be an element of Mon( n , C ) , with n ≥ . Then ϕ can be written as acomposition of involutions of Mon( n , C ) , and n ¡ ϕ , Mon( n , C ) ¢ ≤ n + . Remark 6.10. If n is even, then Mon( n , C ) ⊂ G n ( C ) ( see [3]) ; Proposition 6.1 thus alreadysays that any monomial map of Bir( P n C ) can be written as a composition of involutionsbut here we get two more informations: REMONA MAPS AND INVOLUTIONS 15 • a bound for the minimal number of involutions, • and the fact that the involutions belong to Mon( n , C ).Furthermore Proposition 6.1 gives nothing for Mon( n , C ) for n odd since Mon( n , C ) G n ( C ) as soon as n is odd ([3]). Corollary 6.11.
Any element of N ¡ Mon( n , C ); Bir( P n C ) ¢ is a composition of involutions of P n C . Subgroups J n . Let us introduce J n the subgroup of Bir( P n C ) formed by the maps ofthe type µ ϕ , ϕ , . . . , ϕ n − , α z n − + βγ z n − + δ ¶ with ϕ i = µ z i A i ( z i + , z i + ,... , z n − ) + B i ( z i + , z i + ,... , z n − ) z i C i ( z i + , z i + ,... , z n − ) + D i ( z i + , z i + ,... , z n − ) ¶ ∈ PGL(2, C ( z i + , z i + ,... , z n − )) and α z n − + βγ z n − + δ ∈ PGL(2, C ).According to the proof of Lemma 4.3 one gets: Proposition 6.12.
Let ϕ = ( ϕ , ϕ , . . . , ϕ n − ) be an element of J n .Assume ≤ i ≤ n − . If det ϕ i = ± , then n ¡ ϕ i , PGL(2, C ( z i + , z i + , . . . , z n − )) ¢ ≤ oth-erwise n ¡ ϕ i , PGL(2, C ( z i + , z i + , . . . , z n − )) ¢ ≤ .In particular n ¡ ϕ , J n ¢ ≤ n − . Corollary 6.13.
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