On the Monomial Birational Maps of the Projective Space

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.


INTRODUCTION
The best known birational map of P r is maybe the standard Cremona transformation S r : P r P r defined by S r = (X −1 0 : • • • : X −1 r ).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 P 2 is a composition of automorphisms and S 2 , 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 X 0 • • • X r = 0. We consider birational maps of P 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.

MONOMIAL CREMONA TRANSFORMATIONS
Let N be a rank r free Z-module, K an algebraically closed field, K * := K\{0}.Let T = T N := N ⊗ Z K * be the algebraic torus associated to N over K.The action of T on itself induces a natural inclusion ı of the torus in the group Aut(T) of the algebraic variety's automorphisms of T; we consider the subgroup G T of Aut(T) given by the algebraic group's automorphisms of T. Note that, with an appropriate choice of a basis of N , we may identify T to (K * ) r .Since K is algebraically closed an automorphism in Aut(T) may be written in the form F = (F 1 , . . ., F r ) with where Lemma 2.1.There is a split exact sequence of groups Proof.One has φ(F ) ∈ G T , because λ −1 i F i is a character of the torus, 1 ≤ i ≤ r; it is easily seen that the sequence is exact.The inclusion of G T in Aut(T) is a section of φ, hence the sequence splits.
In the following we consider the projective space P r over K as a toric variety, e.g., a compactification of the torus T = T N associated to a complete regular fan of N R := N ⊗ Z R 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 : P r P r defined on T and such that F (T) ⊂ T.
We note Bir T (P r ) the group of these transformations; there is a natural isomorphism Bir T (P r ) ∼ −→ Aut(T).
Proposition 2.3.Fixing a basis of the lattice N, there exists an isomorphism Proof.It follows from Lemma 2.1.Given F ∈ Bir T (P r ), we associate (F (1 T ), (a ij )), where (a ij ) is the r × r-matrix corresponding to φ(F ) ∈ G T via the equality (1) and the isomorphism G T ∼ −→ GL r (Z) induced by the choice of the basis of N.
Remark 2.4.The action of GL r (Z) on T corresponding to the semidirect product in the proposition is 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 P 2 ) in arbitrary dimension for the monomial case.
Theorem 2.6.The group Bir T (P 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 (Z) is generated by a transvection and two permutations (Coxeter andMoser 1957, Trott 1962) which induce respectively a quadratic transformation and two linear maps in Bir T (P r ).

DEGREES AND MATRICES
Let X 0 , . . ., X r be homogeneous coordinates in P r , x i := X i /X 0 , 1 ≤ i ≤ r, the affine coordinates corresponding to the canonical basis of N = Z r .Every non-constant monomial rational map F : P r P r satisfying F (1 T ) = 1 T may be written uniquely in the form such that: The map F is birational if and only if there exists an integer (r + 1) × (r + 1)-matrix β, satisfying (i), (ii), and v ∈ Z r+1 such that: In the affine open set X 0 • • • X r = 0 one may write F as (3) The matrix A := (a ij ) is the matrix of exponents of (1).
Proposition 3.1.Let F ∈ Bir T (P r ) be such that There exists a unique isomorphism ᾱ such that the following diagram is commutative and with exact lines.Proof.It follows from properties (ii) and (iii) of α and the form in which α is written in the basis e, e 1 , . . ., e r because ( 2) and ( 3) ) is cyclic of order d.
(b) We obtain α 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 and Example 3.4.Finite subgroup of Bir T (P r ) are obtained by this method.For example the Weyl's group of type W (D r ), r ≥ 3, and order 2 r−1 r!, as a Cremona subgroup may be represented as the subgroup generated by an involution of degree 3 and the linear automorphisms F α i 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 r r!, generated by an involution of degree 2 and the preceding r − 1 permutations.

GEOMETRIC DESCRIPTION, EXAMPLES
Let F ∈ Bir T (P 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 T ) = 1 T .e.g.F = (1 T , A) ∈ T GL r (Z), we note F A = F ; the base-scheme is not changed.The union of the fundamental hyperplanes P r \T contains the base-points set.
Let e 1 , . . ., e r the canonical basis for N = Z r , e := e 1 + • • • + e r , the fan associated to the faces of the simplex δ = [e 1 , . . ., e r , −e] and A( ) the same with respect to the simplex A(δ).We consider P 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 , π 1 is equivariant and π 2 is equivariant relatively to the automorphism of the torus induced by A.
Proof.From (3), il follows that F induce F * : K[T] → K[T] which corresponds, by duality, to the lattice's automorphism A : N → N .
Example 4.2.Let A = 1 0 1 1 be, F A = (X 2 0 , X 0 X 1 , X 1 X 2 ).One obtains a (regular) fan that is a subdivision of and A( ) by blowing-up the closed orbits associated to the cones e 1 , e 2 and e 1 , −e and then −e 2 , −e .
Example 4.3.Let A = −I d, 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 0 (resp.0 ) be the fan obtained by the elementary subdivisions of the 4 maximal cones of (resp.− ), and 1 (resp. 1 ), and the following subdivisions corresponding to the 6 cones of dimension 2 of (resp.− ).One has = 1 = 1 .The toric variety V 0 (resp.V 0 ) is the blowing-up of P 3 in 4 points and V 1 (resp.V 1 ) is the blowing-up of these in the strict transforms of the 6 lines.The induced birational map V 0 V 0 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) .

(
ii) α(e) = deg(F )e, where α = α F = (α ij ), e j , 0 ≤ j ≤ r, is the canonical basis of Z r+1 , e = j e j , C + is the cone j R ≥0 e j and ∂ C + its boundary.The positive integer deg(F ) is the degree of F , e.g.deg(F ) = j α ij , ∀ i.
and IVAN PAN One has d = | det(α)| and A is the matrix of ᾱ in the basis ē1 , . . ., ēr .