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abstract: 'Given a birational parameterization $\phi$ of an algebraic surface ${\mathscr S}$ in the projective space ${\mathbb P}^3$, the purpose of this paper is to investigate the sets of points on ${\mathscr S}$ whose preimage consists in $k$ or more points, counting multiplicities. They are described explicitly in terms of Fitting ideals of some graded parts of the symmetric algebra associated to the parameterization $\phi$.'
address:
- 'Departamento de Matemática, FCEN, Universidad de Buenos Aires, Argentina.'
- 'INRIA Sophia Antipolis - Méditerranée, Galaad team, 2004 route des Lucioles, B.P. 93, F-06902, Sophia Antipolis France.'
- 'Institut Mathématique de Jussieu et Université Pierre et Marie Curie, Boîte 247, 4 place Jussieu, F-75252 Paris CEDEX 05, France.'
author:
- Nicolás Botbol
- Laurent Busé
- Marc Chardin
title: 'Fitting ideals and multiple-points of surface parameterizations'
---

Introduction
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Parameterized algebraic surfaces are ubiquitous in geometric modeling because they are used to describe the boundary of 3-dimensional shapes. To manipulate them, it is very useful to have an implicit representation in addition to their given parametric representation. Indeed, a parametric representation is for instance well adapted for drawing or sampling whereas an implicit representation allows significant improvements in intersection problems that are 