Description:

The subject of this mini-course is the implicitization of rational algebraic curves and surfaces, that is the computation of the image of a curve or surface parametrization. This is an old and classical problem in elimination theory that has seen a renewed interest during the last twenty five years because of its usefulness in geometric modeling: rational curves and surfaces are widely used for defining 3D shapes and implicitization provides methods to solve efficiently intersection problems between them. In this mini-course we will explore how techniques from algebraic geometry and commutative algebra have been used to solve this problem, from the classical Sylvester resultant and Hilbert-Burch theorem, to the more recent approaches that are based on the study of certain blowup algebras, more specifically syzygy modules, associated to parameterizations. It will be divided into the following four lectures.

L1: Plane rational curves. The first lecture will deal with the case of rational plane curves. After a brief review of the classical Sylvester resultant, we will show how the first syzygy module of the ideal generated by the defining polynomials of a curve parameterization can be used to solve the implicitization problem, but also to get informations on the singularities of the curve and to provide a computational tool to deal with intersection problems.

L2: Matrix representations in Elimination Theory. The second lecture will be dedicated to elimination theory with a particular focus on methods that allows to obtain matrix-based formulas, such as resultants. For that purpose, we will review several tools and concepts from commutative algebra, including saturation of homogeneous ideals, Fittings ideals, Koszul and Cech complexes.

L3: Implicit matrix representations of rational curves and surfaces. In the third lecture we will introduce implicit matrix representations of parameterized space curves and surfaces. These representations slightly change the target of computing the image of a parameterization to computing a determinantal representation of this image via Fitting ideals.

L4: Computing intersection between curves and surfaces. Finally, in the last lecture applications of implicit matrix representations for solving intersection problems will be discussed. We will first focus on the intersection between a point and a curve or surface and then give an algorithm for solving the curve/surface intersection problem. This lecture will be the occasion to introduce methods that are at the interface of algebraic geometry and numerical linear algebra.

These four lectures will be illustrated with some computations using the software Macaulay2. The illustrative picture shows a visualization of an algebraic surface by means of ray tracing for which the intersection computations have been computed with the techniques that will be presented in this mini-course.