Equations of Parametric Surfaces with Base Points via Syzygies

Let $S$ be a parametric surface in $\text{proj}3$ given as the image of $\phi: \proj{1} \times \proj{1} \to \proj{3}$. 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 Bus'{e}, Cox and D’Andrea for the case when $\varphi :\text{proj}2\to \text{proj}3$ has base points.