Equations of parametric surfaces with base points via syzygies

Abstract

Suppose $S$ is a parametrized surface in complex projective 3-space $mathbf{P}^3$ given as the image of $phi: mathbf{P}^1 imes mathbf{P}^1 o mathbf{P}^3$. The implicitization problem is to compute an implicit equation $F=0$ of $S$ using the parametrization $phi$. An algorithm using syzygies exists for computing $F$ if $phi$ has no base points, i.e. $phi$ is everywhere defined. This work is an extension of this algorithm to the case of a surface with multiple base points of total multiplicity k. We accomplish this in three chapters. In Chapter 2, we develop the concept and properties of Castelnuovo-Mumford regularity in biprojective spaces. In Chapter 3, we give a criterion for regularity in biprojective spaces. These results are applied to the implicitization problem in Chapter 4