Section: New Results

Effective criterions for bigraded birational maps

Participant : Laurent Busé.

A rational map $ℱ:{\mathbb{P}}^m⤏{\mathbb{P}}^n$ between projective spaces is defined by a collection of homogeneous polynomials $𝐟:=(f_0,...,f_n)$ in $m+1$ variables of the same degree. The problem of deciding or providing sufficient conditions for such a map $ℱ$ to be birational have attracted a lot of interest in the past and it is still an active area of research. Methods that are based of some properties of the syzygies of $𝐟$ are definitely the more adapted for computational purposes in the sense that they make the problem of birationality effectively computable in the usual implementation of the Gröbner basis algorithm. The goal of this work is to extend these syzygies-based methods and techniques to the context of rational maps whose source is a product of two projective spaces ${\mathbb{P}}^r\times {\mathbb{P}}^s$ instead.

An important motivation for considering bi-graded rational maps comes from the field of geometric modeling. Indeed, the geometric modeling community uses almost exclusively bi-graded rational maps for parameterizing curves, surfaces or volumes under the name of rational tensor-product Bézier parameterizations. It turns out that an important property is to guarantee the birationality of these parameterizations onto their images. An even more important property is to preserve this birationality property during a design process, that is to say when the coefficients of the defining polynomials are continuously modified. As a first attempt to tackle these difficult problems, we analyze in detail birational maps from ${\mathbb{P}}^1\times {\mathbb{P}}^1$ to ${\mathbb{P}}^2$ in low bi-degree by means of syzygies.

This work is done in the context of the SYRAM project which is funded by the MathAmSud programme. It is a collaboration with N. Botbol (University of Buenos Aires), M. Chardin (University of Paris 6), H. Hassanzadeh (University of Rio de Janeiro), A. Simis (University de Pernambuco) and Q. H. Tran (University of Paris 6). A paper is in preparation.