## Section: New Results

### Effective criteria for bigraded birational maps

Participant : Laurent Busé.

In [2], we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is given in terms of the rank of a couple of matrices that became to be known as *Jacobian dual matrices*. Then, we focus on rational maps from ${\mathbb{P}}^{1}\times {\mathbb{P}}^{1}$ to ${\mathbb{P}}^{2}$ in very low bidegrees and provide new matrix-based birationality criteria by analyzing the syzygies of the defining equations of the map, in particular by looking at the dimension of certain bigraded parts of the syzygy module. Finally, applications of our results to the context of geometric modeling are discussed at the end of the paper.

This is a joint work with N. Botbol (University of Buenos Aires, ARgentina), M. Chardin (UMPC, France), S. H. Hassanzadeh (University of Rio, Brazil), A. Simis (University of Pernambuco, Brazil), Q. H. Tran (UMPC, France). It has been done in the framework of the SYRAM project.