The implicitization of rational algebraic curves and surfaces, that is the computation of the image of a curve or surface parametrization, is an old and classical problem in elimination theory that has seen a renewed interest during the last thirty years because of its applications in geometric modeling: rational curves and surfaces are widely used for defining 3D shapes and implicitization methods are used to solve efficiently intersection problems between them. In this talk, I will report on methods based on syzygies and blowup algebras that have been developed in order to not only describe implicit equations of the image of a rational map, but also to analyze and determine the fibers of these rational maps, especially the finite fibers. Conceptually, the main idea is to use elimination matrices, mainly built from syzygies as representations of rational maps and to extract geometric informations from them [BC21].
[BC21] L. Busé and M. Chardin. Fibers of rational maps and elimination matrices: an application oriented approach. Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of his 75th Birthday, Springer, Irena Peeva editor, p. 189–217, 2021.
