Summary of the research project :
The study of rational maps is of theoretical interest in algebraic geometry and commutative algebra, and of practical importance in geometric modeling. This research proposal focus on rational maps in low dimension, typically parameterizations of curves and surfaces embedded in the projective space of dimension 3, but also dominant rational maps in dimension two and three. The two main objectives amount to unravel geometric properties of these rational maps from the syzygies of their projective coordinates. The first one aims at extending and generalizing the determination of the closed image of a rational map, as well as its geometric features, whereas the second one will focus on the study of dominant rational maps, in particular on the characterization of those that are generically one-to-one.
Project goals :
More specifically, our aim is to: 1. Development of new representations of parameterized algebraic curves and surfaces for applications in Computer-Aided Geometric Design. 2. Extension of results on Cremona transformations for the characterization of birational maps in low dimension and degree.
Research domain :
Applied algebraic geometry, computational algebra, implicitization of triangular and tensor- product Bézier surfaces, Cremona transformations and birationality criterion.