Seminar

A central problem in geometric modeling is to find the implicit
equations for a curve or surface defined parametrically. From the
standpoint of commutative algebra, there is a strong connection between
the implicit equation and the syzygies of the base locus of the
parametrization. This relation is made explicit by the theory of
approximation complexes, as developed by Herzog-Simis-Vasconcelos,
Buse-Jouanolou, Cox, Chardin. I will give an overview of the use of
syzygies in implicitization and illustrate with the case of tensor
product surfaces. For tensor product surfaces of small degree, we can
determine explicitly all bigraded minimal free resolutions of the base
ideal. We study the singularities of these surfaces in relation to the
syzygies and obtain a simplified implicit equation. The work on tensor
product surfaces is joint with H. Schenck and J. Validashti.