Variety of birational maps of degree d of Pn

A natural and simple question asked is: Does the Cremona group Cr(n) admit a structure that is of an algebraic group of infinite dimension. This is still an open question because we don’t know if the set Cr≤d(n) of birational maps of degree ≤ d admits a structure of the algebraic variety.

Mathematical and Physical Sci., 2013, Vol 58, No 7, pp 50-58

This paper is available online at http://stdb.hnue.edu.vn

VARIETY OF BIRATIONAL MAPS OF DEGREE d ofPn

k

Nguyen Dat Dang

Faculty of Mathematics, Hanoi National University of Education

Abstract. Let S d = k[x0, , x n]d be the k-vector space of homogeneous

polynomials of degree d in (n + 1)-variables x0, , x nand the zero polynomial

over an algebraically closed fieldk of characteristic 0 In this paper, we show that

the birational maps of degree d of the projective spacePn

k form a locally closed subvariety of the projective spaceP(S n+1

d ) associated with S d n+1, denoted Crd (n).

We also prove the existence of the quotient variety PGL(n + 1)Cr d (n) that

parametrize all the birational maps of degree d of P(S n+1

d ) modulo the projective

linear group PGL(n + 1) on the left.

Keywords:Birational map, Cremona group, Grassmannian

1 Introduction

Let Cr(n) = Bir(Pn

k) denote the set of all birational maps of projective space Pn

k.

It is clear that Cr(n) is a group under composition of dominant rational maps; called the Cremona group of order n This group is naturally identified with the Galois group of k-automorphisms of the field k(x1, , x n ) of rational fractions in n-variables x1, , x n

It was studied for the first time by Luigi Cremona (1830 - 1903), an Italian mathematician Although it has been studied since the 19th centery by many famous mathematicians, it

is still not well understood For example, we still don’t know if it has the structure of an algebraic group of infinite dimension

The first important result is the theorem of Max Noether (1871): The Cremona

group Bir(P2

C) of the complex projective plane P2

C is generated by its subgroup PGL(3)

and the standard quadratic transformation ω = [x0x1 : x1x2 : x2x0], as an abstract

group This theorem was proved completely by Castelnuovo in 1901 This statement is

only true if the dimension n = 2 The case n > 2, Ivan Pan proved a result following

Hudson’s work on the generation of the Cremona group (see [6])

Received September 10, 2013 Accepted October 30, 2013.

Contact Nguyen Dat Dang, e-mail address: dangnd@hnue.edu.vn

One of the approachs in the study of the Cremona group is based on the knowledge

of its subgroups These studies were started by Bertini, Kantor and Wiman in the 1890s Many important results have stemmed from this approach For example, in 1893, Enriques determined the maximal connected algebraic subgroups of Bir(P2

C) In 1970, Demazure

classified all the algebraic subgroups of rank maximal of Cr(n) with the aid of Enriques

systems (see [2]) More recently, in 2000, Beauville and Bayle gave the classification of birational involutions up to birational conjugation And then, in 2006, Blanc, Dolgachev and Iskovskikh also gave the classification of finite subgroups of Bir(P2

C) (see [1]) In

2009, Serge Cantat showed that Bir(P2

C) is simple as an abstract group.

In the 1970s, Shafarevich published an article (see [8]) with the title: On some

infinite dimensional groups, in which he showed that the group G = Aut(An

C) of all

polynomial automorphisms of the affine spaceAn

Cadmits a structure of an algebraic group

of infinite dimension with the natural filtration: G = ∪ ∞

d=1 G ≤d where G ≤d is the affine algebraic variety of all the polynomial automorphisms of degree≤ d of the affine space

An

C He also calculated its Lie algebra So, he proved that the group Aut(An

C) is not simple

as an abstract group

A natural and simple question asked is: Does the Cremona group Cr(n) admit a

structure that is of an algebraic group of infinite dimension This is still an open question

because we don’t know if the set Cr≤d (n) of birational maps of degree ≤ d admits a

structure of the algebraic variety However, the answer is "yes" for the set Crd (n) of all birational maps of degree d of the projective spacePn

k and PGL(n + 1)Crd (n) These

results are related to my PhD thesis (see [5]) that was successfully defended in 2009 at the Université de Nice (in France) but which has not yet been published in any journal

2 Subvariety Crd(n) ⊂ P (Sd n+1)

In classic algebraic geometry, we know that a rational map of the projective space

Pn

k is of the form:

Pn

k ∋ [x0 : : x n ] = x 99K φ(x) =[P0(x) : : P n (x)]

∈ P n

k,

where P0, , P n are homogeneous polynomials of same degree in (n + 1)-variables

x0, , x n and are mutually prime The common degree of P i is called the degree of

φ; denoted deg φ In the language of linear systems; giving a rational map such as φ is

equivalent to giving a linear system without fixed components ofPn

k

φ ⋆ |OPn(1)| =

{ n

∑

i=0

λ i P i |λ i ∈ k

}

.

Clearly, the degree of φ is also the degree of a generic element of φ ⋆ |OPn(1)| and

the undefined points of φ are exactly the base points of φ ⋆ |OPn(1)|.

Note that a rational map φ :Pn

k 99K Pn

kis not in general a map of the setPn

ktoPn

k; it

is only the map defined in its domain of definition Dom(φ) =Pn

k\V (P0, , P n) We say

that φ is dominant if its image φ(Dom(φ)) is dense inPn

k By the Chevalley theorem, the

image φ(Dom(φ)) is always a constructible subset ofPn

k, hence, it is dense inPn

k if and

only if it contains a non-empty Zariski open subset ofPn

k(see page 94, [4]) In general, we

can not compose two rational maps However, the composition ψ ◦ φ is always defined if

φ is dominant so that the set of all the dominant rational maps φ :Pn

k 99K Pn

kis identified

with the set of injective field homomorphisms φ ⋆ of the field of all the rational fractions

k(x1, , x n ) in n-variables x1, , x n We say that a rational map φ : Pn

k 99K Pn

k is

birational (a birational automorphism) if there exists a rational map ψ :Pn

k 99K Pn

k such

that ψ ◦ φ = idPn = φ ◦ ψ as rational maps Clearly, if such a ψ exists, then it is unique

and is called the inverse of φ Moreover, φ and ψ are both dominant If we denote by Cr(n) = Bir(Pn

k) the set of all birational maps of the projective space Pn

k, then Cr(n) is

a group under composition of dominant rational maps and is called the Cremona group

of order n This group is naturally identified with the Galois group ofk-automorphisms

of the fieldk(x1, , x n ) of rational fractions in n-variables x1, , x n We immediately have the two following propositions:

Proposition 2.1 A rational map φ :Pn

k 99K Pn

k is birational if and only if it is dominant

and there exists a rational map ψ :Pn

k 99K Pn

ksuch that ψ ◦ φ = idPn Proof of Proposition 2.1. The necessary condition is obvious Conversely, by assumption,

we have an injective field homomorphism φ ⋆ : k(x1, , x n) → k(x1, , x n) If there

exists a rational map ψ : Pn

k 99K Pn

k verifying ψ ◦ φ = idPn , then ψ is dominant Hence

ψ ⋆ :k(x1, , x n)→ k(x1, , x n) is also injective Moreover, we have:

φ ⋆ ◦ ψ ⋆

= (ψ ◦ φ) ⋆

=(

idPn

)⋆

= idk(x1, ,xn).

Consequently, φ ⋆ is also surjective, hence an automorphism In other words, φ is

birational

Proposition 2.2 If φ :Pn

k 99K Pn

k is a birational map, then deg φ −1 6 (deg φ) n −1 .

Proof of Proposition 2.2 Denote X the locus of undefined points of φ −1 If Z is a generic

linear subvariety ofPn

k, we denote eZ := φ −1 (Z r X) the Zariski closure, and we call it the strict transform of Z by φ By definition, the degree of φ is also the degree of the strict transform of a generic hyperplane H ∈ |OPn(1)|, that is, deg φ = deg e H We will

show that the degree of φ −1 is equal to the degree of the strict transform of a generic line:

deg φ −1 = deg eL Indeed, consider the subvariety of incident lines

C1(X) ={

L ∈ G(1,Pn)

|L ∩ X ̸= ∅}

is the grassmannian of all the lines inPn We have

dim C1(X) = 1.(n − 1) + dim X < 2(n − 2) = dim G(1,Pn)

.

Hence, C1(X) $ G(1,Pn)

Consequently, there exists a generic line L ⊂ P n such that

L ∩X = ∅ The restriction of φ −1 to L is described by a linear system without base points

and deg φ −1 = deg eL Since L is a generic line, we can write: L = H1 ∩ · · · ∩ H n −1 as

the complete intersection of the generic hyperplanes ofPn Therefore

deg φ −1 = deg eL = deg

n∩−1

i=1

e

H i 6(deg eH i)n −1

=(

deg φ)n −1

.

If V is a k-vector space, we denote by P(V ) = (V − {0})/k ∗ the projective space

associated with V , whose points are one-dimensional vector subspaces of V In particular, when V = S d n+1 is thek-vector space of (n + 1)-uples of d-forms in (n + 1)-variables, dim S d n+1 = (n+d

d

)

.(n + 1), then dim P(S n+1

d ) = (n+d

d

)

.(n + 1) − 1 The homogeneous

coordinates of each point φ = [

P0 : : P n

]

∈ P(S d n+1)

are the coefficients of the

polynomials P0, , P n Now, we present the most important result of this section:

Theorem 2.3 The set Cr d (n) of all birational maps of degree d of the projective spacePn

k

is a locally closed subvariety of the projective space P(S n+1

d ).

In order to prove Theorem 2.3, we need the following lemmas:

Lemma 2.4 A rational map φ : Pn

k 99K Pn

k is dominant if and only if its jacobian

determinant is not zero.

Proof of Lemma 2.4 The necessary condition: If φ is a dominant rational map, then

φ : Dom(φ) → P n is a dominant morphism of integral schemes of finite type overk

According to Proposition 10.4, in [4], page 270-273, there is a nonempty open subset

U ⊂ Dom(φ) ⊂ P n such that φ : U → P nis a smooth morphism of relative dimension

dim(U ) − n = 0, that is, an étale morphism By definition, its tangent linear map

T x φ : T x U → T ∼ xPn is an isomorphism of vector spaces, for all x Hence, its determinant, which is also the jacobian determinant of φ must be not zero: Jac(φ)(x) = det(T x φ) ̸= 0,

for all x ∈ U.

The sufficient condition: If φ(x) = [

P0(x) : : P n (x)]

is a non-dominant rational

map, we will prove that its jacobian determinant Jac(φ) = det

(

∂P i

∂x j

)

≡ 0 Indeed, since φ(Dom(φ)) ⊂ φ(Dom(φ)) and the Zariski closure φ(Dom(φ)) is a proper closed subset

ofPn

k, hence, the image of φ must be contained in some hypersurface F (x0, , x n) = 0,

that is, F (P0(x), , P n (x)) = 0 We can suppose F is a homogeneous polynomial of

least degree such that F (P0(x), , P n (x)) = 0 Hence, we have for all x



















0 = ∂F (P0(x), , P n (x))

n

∑

i=0

∂F

∂x i (P0(x), , P n (x))

∂P i

∂x0(x)

.

0 = ∂F (P0(x), , P n (x))

n

∑

i=0

∂F

∂x i (P0(x), , P n (x))

∂P i

∂x n (x).

Since deg(∂F

∂x i ) < deg F for all i and by the hypothesis of the smallest degree of

F , the partial derivatives ∂F

∂x i (P0(x), , P n (x)) must be non-zero Consequently, the

vectors v i =

(

∂P0

∂x i (x), ,

∂P n

∂x i (x)

)

, i = 0, , n of the k(x0, , x n)-vector space

k(x0, , x n)n+1 are linearly dependent Consequently, the determinant of the family of

these vectors must be zero In other words, the jacobian determinant Jac(φ) = 0.

Corollary 2.5 U d = {

φ =[

P0 : : P n]

∈ P(S d n+1)

| φ : P n99K Pn dominant}

is a Zariski open subset of the projective spaceP(S d n+1)

Proof of Corollary 2.5 This is obvious because U d = P(S d n+1)

\ V (Jac) is the

complement of the projective algebraic set V (Jac) in the projective space P(S d n+1)

.

Here V (Jac) is the projective algebraic set inP(S d n+1)

defined by the annulation of all coefficients of the polynomial Jac

Lemma 2.6 V d = {

φ =[

P0 : : P n]

∈ P(S d n+1)

| P i are mutually prime}

is also a Zariski open subset of the projective spaceP(S d n+1)

Proof of Lemma 2.6 We consider the following regular map (a variant of the Segre

embedding)

s d,r : P(S d n+1 −r)

× P(S r

)

−→ P(S d n+1) ([

P0 : : P n]

; P)

7−→ [P0P : : P n P]

.

According to Theorem 3.13, page 38, in [3], the image of this regular map is a Zariski closed subset ofP(S d n+1)

, corresponding to the points[

Q0 : : Q n

]

∈ P(S d n+1)

having

a common divisor of degree r Hence, the complement V d,r := P(S d n+1)

\Ims d,r is a Zariski open subset ofP(S d n+1)

Clearly, the intersection V d=

d∩−1 r=1

V d,ris also open

Lemma 2.7 F d = {

φ =[

P0 : : P n

]

∈ P(S d n+1)

| ∃ ψ ∈ P(S e n+1)

, ψ ◦ φ = idPn

}

is a Zariski closed subset of the projective spaceP(S d n+1)

, where we denote e = d n −1

Proof of Lemma 2.7. We consider the following regular map (the first projection):

p d,e : P(S d n+1)

× P(S e n+1)

−→ P(S d n+1)

where e = d n −1

([

P0 : : P n]

;[

Q0 : : Q n])

7−→ [P0 : : P n]

.

The set∑

d,e of points (φ, ψ) = ([

P0 : : P n

]

;[

Q0 : : Q n

])

∈ P(S d n+1)

×

P(S e n+1)

such that ψ ◦ φ = idPn , that is, (ψ ◦ φ)(x)∧x = 0, ∀ x is a Zariski closed

subset ofP(S d n+1)

× P(S n+1 e

)

It is easy to find that F dis also the image of∑

d,e by p d,e, and that it is a Zariski closed subset ofP(S n+1

d ) by Theorem 3.12, page 38, in [3].

Proof of Theorem 2.3 According to Proposition 2.1, we have:

Crd (n) =





φ =

[

P0 : : P n]

∈ P(S d n+1) (i) φ :Pn 99K Pnis dominant

(ii) the P iare mutually prime

(iii) ∃ ψ ∈ P(S e n+1)

, ψ ◦ φ = idPn







where e is chosen equal to d n −1 because if φ is a birational map of degree d, the deg φ −1 ≤

d n −1by Proposition 2.2 Hence, Cr

d (n) is the intersection Cr d (n) = U d ∩ V d ∩ F dof the

open set U d ∩ V d (by Corollary 2.5 and Lemma 2.6) and of the closed set F d(by Lemma 2.7) Consequently, it is a locally closed subset ofP(S d n+1)

.

3 Subvariety PGL (n + 1) Crd(n) in G ( n + 1, Sd)

While the projective linear group PGL(n + 1) is obviously an algebraic group of the Cremona group Cr(n), it is not normal in Cr(n) Hence, we obtain the two distinct

quotient sets:

PGL(n + 1) Cr(n) = {PGL(n + 1) ◦ φ : φ ∈ Cr(n)} , Cr(n) PGL(n + 1) = {φ ◦ PGL(n + 1) : φ ∈ Cr(n)} Here, we will not speak of them and we will study only the subset PGL(n +

1)Crd (n) of the first

PGL(n + 1)Crd (n) = {PGL(n + 1) ◦ φ : φ ∈ Cr d (n) }

From the viewpoint of algebraic geometry, we know that the algebraic group

PGL(n + 1) acts on the variety Cr d (n) by the left multiplication

PGL(n + 1) × Cr d (n) −→ Cr d (n)

(u, φ) 7−→ u ◦ φ.

We have a natural question: Does the set PGL(n + 1) Cr d (n) of all the distinct

orbits of Cr d (n) by the action of PGL(n + 1) admit the structure of quotient variety? In order to answer this question, we need recall that the k-planes of a given vector space V form an algebraic variety, called the grassmannian of k-planes of V , denotedG(k, V)

In particular, we have the grassmannianG(n + 1, S d)

of (n + 1)-planes of the vector space

S d Evidently, G(n + 1, S d)

is also the grassmannian G(n, |OPn (d) |) of all the linear

subvariety of dimension n of the linear system |OPn (d) |.

Theorem 3.1 The set PGL(n + 1) Cr d (n) of all the distinct orbits of Cr d (n) by the

action of PGL(n + 1) can be identified as a locally closed subvariety of the grassmannian

G(n + 1, S d)

of (n + 1)-planes of the vector space S d

In order to prove this theorem, we need the following lemma:

Lemma 3.2 We have the two following results:

(i) If φ = [

P0 : : P n]

is a birational map of degree d, then the vectors P0, , P n are linearly independent in S d

(ii) The set Ud (n) ={[

P0 : : P n]

∈ P(S d n+1)

: The P iare linearly independent}

is

a Zariski open subset of P(S d n+1)

.

Proof of Lemma 3.2 (i) If they were linearly dependent in S d, without loss of generality,

we could suppose: P0 = λ1P1+· · · + λ n P n with λ i ∈ k Hence, the image of φ would be

contained in the hyperplane x0− λ1x1− · · · − λ n x n= 0, so that it would not be dense in

Pn

k Therefore, φ would not be birational.

(ii) The P i are linearly independent in S d if and only if the rank of the (n + 1) ×

dim(S d )-matrix (P0 P n ) formed by the coefficients of all the P i is equal to n + 1.

If we denote F d (n), the set of all the φ = [

P0 : : P n]

∈ P(S d n+1)

such that the

P i are linearly dependent in S d , that is, rank(P0 P n ) < n + 1, then F d (n) is a closed subvariety defined by the annulation of all the (n + 1)-sub-determinant of the matrix (P0 P n) Hence,Ud (n) =P(S d n+1)

− F d (n) is a Zariski open subset.

Proof of Theorem 3.1 On the Zariski open setUd (n), we have a natural surjective map:

Φn,d : Ud (n)  G(n + 1, S d)

φ =[

P0 : : P n]

7−→ Span(P0, , P n)

where Span(P0, , P n) =

{∑n

i=0

λ i P i : λ i ∈ k

}

is the k-vector space spanned by

P0, , P n This map is also a surjective morphism of schemes Moreover, the morphism

Φn,d : Ud (n)  G(n + 1, S d)

is still the principal bundle with the fiber PGL(n + 1)

and the structural group PGL(n + 1) Therefore, the grassmannian G(n + 1, S d ) ∼= PGL(n + 1)Ud (n) is the quotient of the spaceUd (n) by PGL(n + 1).

If we denoteGd (n) = Φ n,d(

Crd (n))

the image of the locally closed subvariety Crd (n) by

Φn,d, then by the surjectivity of Φn,d, we obtain:

Crd (n) = (Φ n,d)−1(Gd (n))

According to Theorem 2.3, (Φn,d)−1(Gd (n)) = Cr d (n) is a locally closed subvariety of

P(S d n+1)

, and also of Ud (n) By the property of the principal bundle, Gd (n) is also a

locally closed subvariety of the grassmannian G(n + 1, S d)

Hence, the restriction to

Crd (n) of Φ n,dgives us a surjective morphism of schemes, also denoted Φn,d

Φn,d : Crd (n)  Gd (n) ⊂ G(n + 1, S d)

φ =[

P0 : : P n]

7−→ Span(φ).

Then, we obtain the cartesian square

Ud (n) Φn,d //G(n + 1, S d

)

Crd (n)

inclusion

Φn,d //Gd (n)

inclusion

Therefore, Φn,d : Crd (n)  Gd (n) is also a principal bundle with the fiber PGL(n + 1) and the structural group PGL(n + 1) Consequently,Gd (n) ∼ = PGL(n + 1)Cr d (n) is the

quotient of the variety Crd (n) by PGL(n + 1) In summary, we have an isomorphism:

PGL(n + 1)Crd (n) −→ G ∼ d (n) ⊂ G(n + 1, S d)

PGL(n + 1) ◦ φ 7−→ Span(φ).

4 Conclusion

In this paper, the author has proven two main results The first is Theorem 2.3: The set Crd (n) of birational maps of degree d of the projective space Pn

k is a locally closed

subvariety of the projective space P(S n+1

d ) The second is Theorem 3.1, which proves

the existence of the quotient variety PGL(n + 1)Crd (n) that parametrize all birational maps of degree d of P(S n+1

d ) modulo the projective linear group PGL(n + 1) on the left.

In the next publications, the author will continue to give new results on the irreductible

components of the quotient variety PGL(n + 1)Crd (n).

[1] Jérémy Blanc, 2006 Thèse: Finite abelian subgroups of the Cremona group of the

plane En ligne: http://www.unige.ch/cyberdocuments/theses2006/BlancJ/these.pdf.

[2] Michel Demazure, 1970 Sous-groupes algébriques de rang maximum du groupe

de Cremona Ann scient Éc Norm Sup., 4 e série, t 3, p 507 à 588 En ligne

http://archive.numdam.org.

[3] Joe Harris, 1992 Algebraic Geometry Graduate Texts in Mathematics, Springer

Verlag

[4] Robin Hartshorne, 1977 Algebraic Geometry New York Heidelberg Berlin.

Springer Verlag

[5] Dat Dang Nguyen, 2009 Groupe de Cremona Thèse in Université de Nice, in

France

[6] Ivan Pan, 1999 Une remarque sur la génération du groupe de Cremona Sociedade

Brasileira de Matemática, Volume 30, Issue 1, pp 95-98

[7] Ivan Pan and Alvaro Rittatore, 2012 Some remarks about the Zariski topology of the

Cremona group.

Online on http://www.cmat.edu.uy/ ivan/preprints/cremona130218.pdf

[8] I.R Shafarevich, 1982 On some infinitedimensional groups. American Mathematical Society