On the monomial birational maps of the projective space

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.

Combinatorics of Cremona monomial maps

“…One of the peculiarities of the theory is that even if the given monomials are square-free to start with, the inverse map is generally defined by non-square-free monomials. This makes classification in high degrees, if not the structure of the Cremona monomial group itself, a difficult task (see [5,6]). A complete classification of monomial Cremona maps of degree 2 in any number of variables is given in [5].…”

“…Cremona monomial maps have been dealt with in [6] and in [7], but the methods and some of the goals are different and have not been drawn upon here. The group structure of Cremona monomial maps is studied in [6] by means of toric algebra with emphasis on the multiplicative structure, while in the present account we take up the so to say additive side of the problem by stressing the underlying integer arithmetic.…”

“…Cremona monomial maps have been dealt with in [6] and in [7], but the methods and some of the goals are different and have not been drawn upon here. The group structure of Cremona monomial maps is studied in [6] by means of toric algebra with emphasis on the multiplicative structure, while in the present account we take up the so to say additive side of the problem by stressing the underlying integer arithmetic. In [7] the group of plane Cremona monomial maps is described by generators and relations and it is shown that this group (except that they allow certain coefficients) is generated by permutations and two degree 2 maps: hyperbolism and the Steiner involution (see Section 4).…”

On Characteristic Classes of Determinantal Cremona Transformations

“…, e n is the standard basis of R n and e 0 := − n i=1 e i . The central symmetry −Id in R n induces the standard Cremona transformation S n viewed as a monomial birational map ( [10]). A natural way to resolve the indeterminacies of S n is to factorize it through the toric variety X Σ associated to the fan given by the minimal common subdivision of ∆ and −∆.…”

Counting Square free Cremona monomial maps

“…Cremona transformations are birational automorphism of the projective space, they were firstly systematically studied by L. Cremona in the 19th century and remains a major classical topic in Algebraic Geometry. The interest in monomial Cremona transformations, otherwise has flourished more recently as one can see, for example in [V,GP,SV1,SV2,SV3]. Following the philosophy introduced in [SV1,SV2] and shared by [SV3,CS] we study the so called "birational combinatorics" meaning, see loc.…”