Equations of Parametric Surfaces with Base Points via Syzygies

Abstract

Let $S$ be a parametric surface in $\proj{3}$ given as the image of $ϕ: \proj{1} \times \proj{1} \to \proj{3}$. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of $S$ when certain base points are present. This work extends the algorithm provided by Cox for when $\varphi$ has no base points, and it is an analogous to some of the results of Busé, Cox and D'Andrea for the case when $ϕ: \proj{2} \to \proj{3}$ has base points.

22 pages. Revised version adds proofs that were originally quoted from Hoffman and Wang (arxiv.,org/abs/math.AG/0305125). The paper of Hoffman and Wang has now been withdrawn