On the monomial birational maps of the projective space

On the Monomial Birational Maps of the Projective Space

Gérard Gonzalez-Sprinberg^I; Ivan Pan^II

^IInstitut Fourier, Université de Grenoble I UMR 5582 CNRS-UJF 38402 Saint-Martin-d'Hères, France

^IIInstituto de Matemática, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brasil

^{Correspondence}

ABSTRACT

We describe the group structure of monomial Cremona transformations. It follows that every element of this group is a product of quadratic monomial transformations, and geometric descriptions in terms of fans.

Keywords: monomial birational maps, toric varieties.

RESUMO

Descrevemos a estrutura do grupo das transformações de Cremona monomiais. Concluímos que todo elemento deste grupo é um produto de aplicações monomiais quadráticas e damos descrições geométricas em termos de leques.

Palavras-chave: aplicações birracionais monomiais, variedades tóricas.

1. INTRODUCTION

The best known birational map of

^r is maybe the standard Cremona transformation S_r : ^r

^r defined by S_r = (X₀^-1 : ... : X_r^-1). For r = 2 there is a geometric description in the classic references on this subject; moreover a Max Noether's famous theorem shows that every birational map of ² is a composition of automorphisms and S₂, and then that every Cremona transformation is a composition of quadratic ones. If r > 2 there is no analogous to Noether's theorem (Hudson 1927, Katz 1992, Pan 1999). In this note we consider birational maps generalizing S_r. Our approach is based in a toric point of view and the property that S_r stabilizes the open set X₀...X_r 0. We consider birational maps of ^r with this property. In the second paragraph of (Russo and Simis 2001) the birationality of these maps is characterized in terms of certain syzygies as an application of a more general criterion; see also (Simis and Villarreal 2002): compare their Proposition 1.1 with our Proposition 3.1.

2. MONOMIAL CREMONA TRANSFORMATIONS

Let N be a rank r free -module, an algebraically closed field, ^* : = \{0}. Let = _N : = N

^* be the algebraic torus associated to N over . The action of on itself induces a natural inclusion of the torus in the group Aut() of the algebraic variety's automorphisms of ; we consider the subgroup G of Aut() given by the algebraic group's automorphisms of . Note that, with an appropriate choice of a basis of N, we may identify to (^*)^r. Since is algebraically closed an automorphism in Aut() may be written in the form F = (F₁,..., F_r) with

(1)

where l_i

^*, a_ij

, 1 i, j r.

LEMMA 2.1. There is a split exact sequence of groups

where(F) : = (F(1)^-1) o F, for F Aut().

PROOF. One has (F) G, because F_i is a character of the torus, 1 i r; it is easily seen that the sequence is exact. The inclusion of G in Aut() is a section of , hence the sequence splits.

In the following we consider the projective space

^r over as a toric variety, e.g., a compactification of the torus = _N associated to a complete regular fan of N : = N with (r + 1) cones of dimension 1. (One standard reference for toric varieties is (Oda 1988)).

DEFINITION 2.2. A monomial Cremona transformation is a birational map F :

^r

^r defined on and such that F() .

We note

Bir(^r) the group of these transformations; there is a natural isomorphism Bir(^r) Aut().

PROPOSITION 2.3. Fixing a basis of the lattice N, there exists an isomorphism

PROOF. It follows from Lemma 2.1. Given F

Bir(^r), we associate (F(1),(a_ij)), where (a_ij) is the r × r-matrix corresponding to (F) G via the equality (1) and the isomorphism G

GL_r() induced by the choice of the basis of N.

REMARK 2.4. The action of

GL_r() on corresponding to the semidirect product in the proposition is

(a_ij) ^. (,...,) =

,...,

.

COROLLARY 2.5. The center of Bir(^r) is trivial. The center of the subgroup {F : F(1) = 1} is S_r.

As a consequence of Proposition 2.3 we obtain an analogous to the M. Noether's theorem (on the generators of the Cremona group of

²) in arbitrary dimension for the monomial case.

THEOREM 2.6. The group Bir(^r) is generated by a quadratic transformation and the linear monomial automorphisms. Consequently, every birational map is a product of quadratic transformations.

PROOF. It follows from Proposition 2.3 and the fact that

GL_r() is generated by a transvection and two permutations (Coxeter and Moser 1957, Trott 1962) which induce respectively a quadratic transformation and two linear maps in Bir(^r).

3. DEGREES AND MATRICES

Let X₀,..., X_r be homogeneous coordinates in ^r, x_i : = X_i/X₀, 1 i r, the affine coordinates corresponding to the canonical basis of N = ^r. Every non-constant monomial rational map F : ^r

^r satisfying F(1) = 1 may be written uniquely in the form

...

(2)

such that:

(i) (e_j) C₊, 0 jr, (ii)

(e) = deg(F)e, where = = (), e_j, 0 j r, is the canonical basis of ^r+1 , e = e_j , C₊ is the cone

0e_j and C₊ its boundary. The positive integer deg(F) is the degree of F, e.g. deg(F) = ,i.

The map F is birational if and only if there exists an integer (r + 1) × (r + 1)- matrix , satisfying (i), (ii), and v

^r+1 such that:

(iii)

= Id + e ^{. t}v.

In the affine open set X₀...X_r 0 one may write F as

F = (1 : x₁^a₁₁ ... x_r^a_1r : ... : x₁^a_r1 ... x_r^a_rr). (3)

The matrix A : = (a_ij) is the matrix of exponents of (1).

PROPOSITION 3.1. Let F

Bir(^r) be such that F(1) = 1, = , d = deg(F). There exists a unique isomorphism such that the following diagram is commutative and with exact lines.

One has d = | det()| and A is the matrix of in the basis ,...,.

PROOF. It follows from properties (ii) and (iii) of and the form in which is written in the basis e, e₁,..., e_r because (2) and (3) imply a_ij = - , 1 i, j r.

REMARK 3.2.

(a)

^r+1/(^r+1) is cyclic of order d.

(b) We obtain from A as follows:

EXAMPLE 3.3. Let A be the matrix whose lines are (1, 0, 0),(a, 1, 0),(0, a, 1), a 1. Then

F_A = (X₀^a+1 : X₀^aX₁ : X₁^aX₂ : X₂^aX₃)

and

F_A^-1 = (X₀^a2-a+1X₁^aX₂^a : X₀^a2-aX₁^a+1X₂^a : X₀a²X₂^a+1 : X₁^a2+aX₃).

EXAMPLE 3.4. Finite subgroup of

Bir(^r) are obtained by this method. For example the Weyl's group of type W(D_r), r 3, and order 2^r-1r!, as a Cremona subgroup may be represented as the subgroup generated by an involution of degree 3

F = (X₀X_r-1X_r : X₁X_r-1X_r : ... : X_r-2X_r-1X_r : X₀²X_r-1 : X₀²X_r)

and the linear automorphisms F permuting X_i with X_i+1, 1 i < r. In an analogous form one obtains a representation of the group W(B_r), r 2, of order 2^rr!, generated by an involution of degree 2

F = (X₀X_r : X₁X_r : ... : X_r-1X_r : X₀²)

and the preceding r — 1 permutations.

4. GEOMETRIC DESCRIPTION, EXAMPLES

Let F

Bir(^r); we denote by B_F its base-scheme. By composing F with an automorphism induced by an element of the torus we may assume that F(1) = 1. e.g. F = (1, A)

GL_r(), we note F_A = F; the base-scheme is not changed.

The union of the fundamental hyperplanes

^r\ contains the base-points set.

Let e₁,..., e_r the canonical basis for N = ^r, e : = e₁ + ... + e_r, the fan associated to the faces of the simplex = [e₁,..., e_r, - e] and A() the same with respect to the simplex A(). We consider ^r defined by ; let be a fan that is a common subdivision for and A().

PROPOSITION 4.1. One has a commutative diagram

where X is the toric variety associated to , is equivariant and is equivariant relatively to the automorphism of the torus induced by A.

PROOF. From (3), il follows that F induce F^* : [] ® [] which corresponds, by duality, to the lattice's automorphism A : N ® N.

EXAMPLE 4.2. Let be, F_A = (X₀², X₀X₁, X₁X₂). One obtains a (regular) fan that is a subdivision of and A() by blowing-up the closed orbits associated to the cones e₁, e₂ and e₁, - e and then - e₂, - e.

EXAMPLE 4.3. Let A = - Id, then F_A = S_r. One obtains a (regular) fan by the elementary subdivisions of (resp. of - ), successively, of the cones of decreasing dimensions from r to 2. For example, if r = 3, let (resp. ) be the fan obtained by the elementary subdivisions of the 4 maximal cones of (resp. - ), and (resp. ), and the following subdivisions corresponding to the 6 cones of dimension 2 of (resp. - ). One has = = . The toric variety V (resp. V) is the blowing-up of ³ in 4 points and V (resp. V) is the blowing-up of these in the strict transforms of the 6 lines. The induced birational map V

V is composition of 6 flops, e.g., corresponding to small resolutions of singularities of type an affine cone over a smooth quadric, associated to the 6 faces of the convex polyhedron P : = Conv( (- )).

On the other hand, the (normalized) blowing-up of the base-scheme B is the toric variety associated to the fan (B) defined by the faces of the polyhedron P(B) = Conv(P

C), where C is the set of sums of the vertices of a diagonal of each 2-face of P. The toric variety V(B) has 12 singular points of type an affine cone over a smooth quadric. Finally, the fan is a regular subdivision of (B) and the induced morphism V® V(B) is a minimal resolution of V(B).

ACKNOWLEDGMENTS

The second author was a visiting Professor at Institut Fourier and also partially supported by the Cooperação Franco-Brasileira.

Manuscript received on December 18, 2002; accepted for publication on April 30, 2003;

presented by ARON SIMIS

Mathematics Subject Classification: 14E07, 14M25.

• COXETER H AND MOSER W. 1957. Generators and relations for Discrete Groups. Springer-Verlag.
• HUDSON HP. 1927. Cremona Transformations in Plane and Space. Cambridge at the University Press.
• KATZ S. 1992. Recent works on Cremona Transformations. Research Notes in Math. 2: 109-119, Sundance 90.
• ODA T. 1988. Convex Bodies and Algebraic Geometry. Springer-Verlag.
• PAN I. 1999. Une remarque sur la génération du groupe de Cremona. Bol Soc Bras Mat 30: 95-98.
• RUSSO F AND SIMIS A. 2001. On birational maps and Jacobian matrices. Compositio Math 126: 335-358.
• SIMIS A AND VILLARREAL R. 2002. Constraints for the Normality of Monomial Subrings and birationality, to appear in Proc. of the A.M.S.
• TROTT S. 1962. A pair of generators for the unimodular group. Can Math Bull 5: 245-252.

Publication Dates

• Publication in this collection
  31 July 2003
• Date of issue
  June 2003

History

• Accepted
  30 Apr 2003
• Received
  18 Dec 2002