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On the monomial birational maps of the projective space

Abstracts

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.

monomial birational maps; toric varieties


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.

aplicações birracionais monomiais; variedades tóricas


On the Monomial Birational Maps of the Projective Space

Gérard Gonzalez-SprinbergI; Ivan PanII

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 Correspondence to: Ivan Pan E-mail: pan@mat.ufrgs.br/ gonsprin@ujf-grenoble.fr

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 Sr : r
r defined by Sr = (X0-1 : ... : Xr-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 Sr. Our approach is based in a toric point of view and the property that Sr stabilizes the open set X0...Xr 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,..., Fr) with

(1)

where li

*, aij
, 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 Fi 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),(aij)), where (aij) is the r × r-matrix corresponding to (F) G via the equality (1) and the isomorphism G
GLr() induced by the choice of the basis of N.

REMARK 2.4. The action of

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

(aij) . (,...,) =

,...,
.

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

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

GLr() 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,..., Xr be homogeneous coordinates in r, xi : = Xi/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) (ej) C+, 0 jr, (ii)

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

0ej 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 . tv.

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

F = (1 : x1a11 ... xra1r : ... : x1ar1 ... xrarr). (3)

The matrix A : = (aij) 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,..., er because (2) and (3) imply aij = - , 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

FA = (X0a+1 : X0aX1 : X1aX2 : X2aX3)

and

FA-1 = (X0a2-a+1X1aX2a : X0a2-aX1a+1X2a : X0a2X2a+1 : X1a2+aX3).

EXAMPLE 3.4. Finite subgroup of

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

F = (X0Xr-1Xr : X1Xr-1Xr : ... : Xr-2Xr-1Xr : X02Xr-1 : X02Xr)

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

F = (X0Xr : X1Xr : ... : Xr-1Xr : X02)

and the preceding r — 1 permutations.

4. GEOMETRIC DESCRIPTION, EXAMPLES

Let F

Bir(r); we denote by BF 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)
GLr(), we note FA = F; the base-scheme is not changed.

The union of the fundamental hyperplanes

r\ contains the base-points set.

Let e1,..., er the canonical basis for N = r, e : = e1 + ... + er, the fan associated to the faces of the simplex = [e1,..., er, - 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, FA = (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 FA = Sr. 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.

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  • 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.
  • Correspondence to:

    Ivan Pan
    E-mail:
  • Publication Dates

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

    History

    • Accepted
      30 Apr 2003
    • Received
      18 Dec 2002
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