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

TL;DR

This paper surveys methods using elimination matrices derived from syzygies to analyze rational maps, focusing on their application in geometric modeling for computing intersections and images of algebraic curves and surfaces.

Contribution

It introduces a unified approach combining syzygies and blowup algebras to effectively compute geometric features of rational maps, with detailed construction and properties of elimination matrices.

Findings

Elimination matrices effectively represent rational maps.

Syzygies facilitate computation of images and fibers.

The approach integrates commutative algebra and algebraic geometry.

Abstract

Parameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points, pieces of parameterized algebraic curves and pieces of algebraic surfaces is a key problem in this context. In this paper, we survey recent methods based on syzygies and blowup algebras for computing the image and the finite fibers of a curve or surface parameterization, more generally of a rational map. 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. The construction and main properties of these matrices are first reviewed and then illustrated through several settings, each of them highlighting a particular feature of this approach that combines tools from commutative algebra, algebraic geometric and elimination theory.