Cremona maps and involutions

Cremona maps and involutions

Julie Déserti
July 19, 2019
Abstract.

We deal with the following question of Dolgachev : is the Cremona group generated by involutions ? Answer is yes in dimension (see [CerveauDeserti]). We give an upper bound of the minimal number of involutions we need to write a birational self map  of .

We prove that de Jonquières maps of and maps of small bidegree of can be written as a composition of involutions of and give an upper bound of for such maps . We get similar results in particular for automorphisms of , automorphisms of , tame automorphisms of , monomial maps of , and elements of the subgroup generated by the standard involution of and .

1. Introduction

Question (Dolgachev).

Is the dimensional Cremona group generated by involutions ?

Answer is yes in dimension ; more precisely:

Proposition 1.1 ([CerveauDeserti]).

For any in there exist , , , in such that

 φ=(A0∘σ2∘A−10)∘(A1∘σ2∘A−11)∘…∘(Ak∘σ2∘A−1k)

where denotes the standard involution of

 σ2:(z0:z1:z2)⇢(z1z2:z0z2:z0z1).

Let us note that since is generated by and some involutions ([Zimmermann]), any element of can be written as a composition of involutions.

If is an element of , then is the minimal number of involutions of we need to write . In dimension we get the following result:

Theorem A.

If is an automorphism of , then .

If is an automorphism of , then .

If belongs to the Jonquières subgroup , then .

If is a birational self map of of degree , then .

One can be more precise for the well-known subgroup of polynomial automorphisms of of :

Theorem B.

Let be an element of of degree . Then .

More precisely,

• if is affine, then ;

• if is elementary, then ;

• if is generalized Hénon map, then either it is of jacobian and or ;

• if is prime, then .

What happens in higher dimension ? A first result is the following:

Proposition C.
• If is an automorphism of , then can be written as a composition of involutions of , and .

• If is an automorphism of , then can be written as a composition of involutions of , and .

Since any element of

 Gn(C)=⟨σn=(n∏\lx@stackreli=0i≠0zi:n∏\lx@stackreli=0i≠1zi:…:n∏\lx@stackreli=0i≠nzi),Aut(PnC)⟩

can be written as a composition of conjugate involutions ([Deserti:reg]) one gets that:

Theorem D.

Let . Any element of the normal subgroup generated by in can be written as a composition of involutions of .

Furthermore one can give an upper bound of when belongs to the subgroup of tame automorphisms of :

Theorem E.

Let . Let be a tame automorphism of of degree . Then can be written as a composition of involutions of . Moreover,

• if is affine, then ;

• if is elementary, then ;

• otherwise .

Let us recall (see [PanSimis]) that the Jonquières subgroup of is given in the affine chart by

 {φ=(φ0(z0,z1,z2),ψ(z1,z2))∣∣φ0∈PGL(2,C[z1,z2]),ψ∈Bir(P2C)}.

Denote by the group of monomial maps of , and finally set

 Jn=PGL(2,C(z1,z2,…,zn−1))×PGL(2,C(z2,z3,…,zn−1))×…×PGL(2,C(zn−1))×PGL(2,C)⊂Bir(PnC).
Theorem F.

Assume that , and .

• If is of bidegree , then can be written as a composition of involutions of , and .

• Any element of of degree can be written as a composition of involutions of , and .

• If belongs to , then can be written as a composition of involutions of , and .

• Any element of can be written as a composition of involutions of , and .

If is a subgroup of let us denote by the normal subgroup generated by in .

Corollary G.

Any element of

 ⟨N(PGL(4,C);Bir(P3C)),N(JO(1;P3C);Bir(P3C)),N(Mon(3,C);Bir(P3C)), N(G3(C);Bir(P3C)),N(⟨φ1,…,φk⟩;Bir(P3C))|φi∈Bir(P3C) of bidegree (2,ℓ), 2≤ℓ≤4⟩

can be written as a composition of involutions of .

For any , any element of

can be written as a composition of involutions of .

Remark 1.2.

An other motivation for studying birational maps of that can be written as a composition of involutions is the following. The group of birational maps of that can be written as a composition of involutions is a normal subgroup of . So if the answer to Dolgachev Question is no, we can give a negative answer to the following question asked by Mumford ([Mumford]): is a simple group ?

Acknowledgments

I would like to thank D. Cerveau for his constant availability and kindness. Thanks to S. Zimmermann for pointing out Proposition 4.2.

2. Recalls and definitions

2.1. Polynomial automorphisms of Cn

A polynomial automorphism of is a bijective map from into itself of the type

 (z0,z1,…,zn−1)↦(φ0(z0,z1,…,zn−1),φ1(z0,z1,…,zn−1),…,φn−1(z0,z1,…,zn−1))

with . The set of polynomial automorphisms of form a group denoted .

Let be the group of affine automorphisms of , and let be the group of elementary automorphisms of . In other words is the semi-direct product of with the commutative unipotent subgroup of translations. Furthermore is formed with automorphisms of where

 φi=φi(zi,zi+1,…,zn−1)

depends only on , , , . The subgroup of , called the group of tame automorphisms of , is the group generated by and . For one has , more precisely:

Theorem 2.1 ([Jung]).

The group has a structure of amalgamated product

 Aut(C2)=A2∗S2E2

with .

Nevertheless (see [ShestakovUmirbaev]).

2.2. Birational maps of PnC

A rational self map of is a map of the type

 (z0:z1:…:zn)⇢(φ0(z0,z1,…,zn):φ1(z0,z1,…,zn):…:φn(z0,z1,…,zn))

where the ’s denote homogeneous polynomials of the same degree without common factor (of positive degree).

A birational self map of is a rational map of such that there exists a rational self map  of with the following property where .

The degree of is the degree of the ’s. For , one has ; for such an equality does not necessary hold, we thus speak about the bidegree of  which is .

The group of birational self maps of is denoted and called Cremona group.

The groups and are subgroups of .

Let us mention that contrary to the Cremona group in dimension does not decompose as a non-trivial amalgam (appendix of [CantatLamy]).

2.3. Birational involutions in dimension 2

Let us first describe some involutions:

• Consider an irreducible curve of degree with a unique singular point ; assume furthermore that is an ordinary multiple point with multiplicity . To we can associate a birational involution which fixes pointwise and preserves lines through as follows. Let be a generic point of ; let , and be the intersections of the line with ; the point is defined by: the cross ratio of , , and is equal to . The map is a de Jonquières involution of . A birational involution is of de Jonquières type if it is birationally conjugate to a de Jonquières involution of .

• Let , , , be eight points of in general position. Consider the set of sextics with double points at , , , . Take a point  in . The pencil given by the elements of having a double point at has a tenth base double point point . The involution which switches and is a Bertini involution. A birational involution is of Bertini type if it is birationally conjugate to a Bertini involution.

• Let , , , be seven points of in general position. Denote by the linear system of cubics through the ’s. Consider a generic point in and define by  the pencil of elements of passing through . The involution which switches and the ninth base-point of is a Geiser involution. A birational involution is of Geiser type if it is birationally conjugate to a Geiser involution.

Birational involutions of have been classified:

Theorem 2.2 ([Bertini]).

A non-trivial birational involution of is either of de Jonquières type, or of Bertini type, or of Geiser type.

2.4. Birational involutions in higher dimension

There are no classification in higher dimension; in [Prokhorov:involution] the author gives a first nice step toward a classification in dimension .

Let us give some examples:

• the involution

 σn=(n∏\lx@stackreli=0i≠0zi:n∏\lx@stackreli=0i≠1zi:…:n∏\lx@stackreli=0i≠nzi)
• the involutions of ;

• the involutions of induced by the involutions of ;

• the de Jonquières involutions: consider a reduced hypersurface of degree in  that contains a linear subspace of points of multiplicity . Assume that  is a singular point of of multiplicity . Take a general point of . Denote by the line passing through and . The intersection of with contains  with multiplicity , and the residual intersection is a set of two points  and  in . Define to be the point on such that the cross ratio of , , and are equal to . The map is a de Jonquières involution of .

3. Automorphisms of (P1C)n and of PnC

Lemma 3.1.

Any non-trivial homography is either an involution, or the composition of two involutions of .

In particular if belongs to , then

 n(φ,Aut(P1C×P1C×…×P1Cn times))≤2n.
Remark 3.2.

The homography is a non-trivial involution if and only if there exists such that , where denotes the set of fixed points of .

Indeed assume that there exists such that , then and so . If , then . If , the cross ratio of , , , is equal to the cross ratio of , , and . This implies that .

Lemma 3.3.

Let be an homography. Consider three points , , of such that , , are distinct, , and .

There exist two involutions , such that .

Proof.

Let us first prove that there exists two unique homographies , such that

 {ι1(a)=ν(b),ι1(b)=ν(a),ι1(ν(a))=b;ι2(ν(a))=ν(b),ι2(ν(b))=ν(a),ι2(ι1(c))=ν(c).

Note that by assumptions , , (resp. , , ) are pairwise distinct. Hence there exists a unique homography that sends , , onto , , .

The points , and are distinct. Assume by contradiction that , then . By injectivity of , one has : contradiction. Similarly and . Since , , are distinct, , and also. As a consequence there exists a unique homography that sends , , onto , , .

By assumption and are distinct so does not belong to . But . According to Remark 3.2 the homography is thus an involution. Similarly and are distinct but and are switched by hence is an involution (Remark 3.2).

Since for one gets . ∎

Proof of Lemma 3.1.

Let be an homography. If , then for any involution . Assume now that ; then has at most two fixed points. Let us choose , in . If or if , then can be written as a composition of two involutions (Lemma 3.3). If and , then with ; Remark 3.2 thus implies that is an involution. ∎

Lemma 3.4.

Let be an integer.

1. Let be a commutative ring of any characteristic. If is an element of , then .

2. Assume that is an algebraically closed field, and that belongs to . Then .

3. If is an element of , then .

Proof.
1. Let us recall that an element of can be written as a composition of transvections ([Perrin]). But a transvection is a composition of two involutions so any an element of can be written as a composition of involutions.

2. If is algebraically closed, then and one gets the result.

3. Let be an element of ; denote by its determinant and by a scaling of scale factor . Then belongs to and hence, according to the first assertion, can be written as a composition of involutions. But is as a composition of two involutions:

 1z0P(z1,z2,…,zn−1)∘1z0.

As a result .

4. Dimension 2

4.1. The real Cremona group

There is an analogue to Proposition 1.1 for the real Cremona group.

Theorem 4.1.

Any element of can be written as a composition of involutions of .

Theorem 4.1 directly follows from the simplicity of and the following statement:

Proposition 4.2 ([Zimmermann]).

The group is generated by , the set of standard quintic involutions and the two following quadratic involutions

 (z1z2:z0z2:z0z1) (z0z2:z1z2:z20+z21).

4.2. The de Jonquières subgroup

An element of is a de Jonquières map if it preserves a rational fibration, i.e. if it is conjugate to an element of

 J2=PGL(2,C(z1))⋊PGL(2,C).

We will denote by the subgroup of birational maps that preserves fiberwise the fibration constant, i.e. .

Lemma 4.3.

If belongs to , then .

Furthermore if , then .

Proof.

Lemma 3.4 implies the first assertion, and the last assertion follows from [GustafsonHalmosRadjavi]. ∎

Corollary 4.4.

Any de Jonquières map of can be written as a composition of Cremona involutions of .

Proof.

Let us remark that any Jonquières map of can be written as where  denotes an element of and an element of . But

 j=(a(z1)z0+b(z1)c(z1)z0+d(z1),αz1+βγz1+δ)=(z0,αz1+βγz1+δ)∘(a(z1)z0+b(z1)c(z1)z0+d(z1),z1)

As a consequence

 φ=(ψ∘(z0,αz1+βγz1+δ)∘ψ−1)∘(ψ∘(a(z1)z0+b(z1)c(z1)z0+d(z1),z1)∘ψ−1)

Then one concludes with Lemmas 3.1 and 4.3. ∎

4.3. Subgroup of polynomial automorphisms of C2

Note that there is no analogue to Proposition 1.1 in the context of polynomial automorphisms of . For instance the automorphism cannot be written as a composition of involutions in .

According to Lemma 3.4 and Corollary 4.4 one has the following result:

Lemma 4.5.

Let be a polynomial automorphism of .

If is an affine automorphism, then .

If is an elementary automorphism, then .

An element is a generalized Hénon map if

 φ=(z1,P(z1)−δz0)

where belongs to and is an element of of degree . Note that .

Lemma 4.6.

Let ) be a generalized Hénon map.

• If has jacobian , then ;

• otherwise .

Proof.

Any generalized Hénon map of jacobian can be written and so is the composition of two involutions: .

Let be a generalized Hénon map; then

 φ=(z1,P(z1)−δz0)=(z1,z0)∘(P(z1)−δz0,z1).

Note that is an elementary automorphism; therefore (Lemma 4.5). ∎

Friedland and Milnor proved that any polynomial automorphism of degree with  prime is conjugate via an affine automorphism either to a generalized Hénon map or to an elementary automorphism ([FriedlandMilnor, Corollary 2.7]). Since any generalized Hénon map is the composition of with an elementary map one gets that any polynomial automorphism of degree with prime can be written as with and . Lemmas 4.5 and 4.6 thus imply:

Lemma 4.7.

If is of degree with prime, then .

A sequence of length is a reduced word, representing the group element if

• each factor belongs to either or but not to the intersection ,

• and no two consecutive factors belong to the same subgroup or .

It follows from Theorem 2.1 that every element of can be expressed as such a reduced word, unless it belongs to the intersection . The degree of any reduced word is equal to the product of the degree of the factor (see [FriedlandMilnor, Theorem 2.1]). Hence take of degree , then is a reduced word and

• either there exists only one of degree , then with and ; as a result (Lemma 4.5),

• or there exits at least two ’s of degree , then . Indeed let be a reduced word representing . Any has degree and hence and so . As a result and

 n(φ,Bir(P2C)) ≤ (k+1)n(ai,Bir(P2C))+kn(ei,Bir(P2C)) ≤

One can thus state

Theorem 4.8.

Let be a polynomial automorphism of of degree .

• If is affine, ;

• if is elementary, then ;

• if is generalized Hénon map, then either it is of jacobian and or ;

• if is prime, then ;

• otherwise .

Corollary 4.9.

If is a polynomial automorphism of of degree , then .

4.4. Birational maps

Theorem 4.10.

If is of degree , then .

Before proving Theorem 4.10 let us give a first and "bad" bound. Let be a birational self map of of degree . The number of base points of is and the map can be written with blow ups. Since a blow up can be written as with the map can be written with involution and elements of . As a consequence can be written as a composition of involutions.

Proof of Theorem 4.10.

Let us recall that if is a birational self map of of degree , then there exists a de Jonquières map of such that (see [Castelnuovo], [AlberichCarraminana, Theorem 8.3.4]).

As a result any of degree can be written as follows

 A∘(ψ1∘j1∘ψ−11)∘(ψ2∘j2∘ψ−12)∘…∘(ψk∘jk∘ψ−1k)

with in , , in and .

The statement follows from Lemma 3.4 and Corollary 4.4. ∎

5. Dimension 3

5.1. de Jonquières maps in dimension 3

Let us recall that a de Jonquières map of of degree is a plane Cremona map satisfying one of the following equivalent conditions:

• there exists a point such that the restriction of to a general line passing through maps it birationally to a line passing through ;

• has homaloidal type , i.e. has base points, one of multiplicity and of multiplicity ;

• up to projective coordinate changes (source and target)

 φ=(z0gd−1+gd:(z0qd−2+qd−1)z1:(z0qd−2+qd−1)z2)

with , , ,