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Parameterized algebraic curves and surfaces are widely used in geometricmodeling and their manipulation is an important task in the processing ofgeometric models. In particular, the determination of the intersection locibetween points, pieces of parameterized algebraic curves and pieces ofalgebraic surfaces is a key problem in this context. In this paper, we surveyrecent methods based on syzygies and blowup algebras for computing the imageand the finite fibers of a curve or surface parameterization, more generally ofa rational map. Conceptually, the main idea is to use elimination matrices,mainly built from syzygies, as representations of rational maps and to extractgeometric informations from them. The construction and main properties of thesematrices are first reviewed and then illustrated through several settings, eachof them highlighting a particular feature of this approach that combines toolsfrom commutative algebra, algebraic geometric and elimination theory.

Fibers of rational maps and elimination matrices: an application oriented approach