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 , , ,