Implicitization of tensor product surfaces in the presence of a generic set of basepoints

Abstract

Given a $4$-dimensional vector subspace $U=\{f_0,\dots ,f_3\}$ of $H^0({\mathbb{P}}^1\times {\mathbb{P}}^1,\mathcal{O}(a,b))$, a tensor product surface, denoted by $X_U$, is the closure of the image of the rational map ${\lambda }_U:{\mathbb{P}}^1\times {\mathbb{P}}^1-\to {\mathbb{P}}^3$ determined by $U$. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of $X_U$ in ${\mathbb{P}}^3$. In this paper we show that if $U\subseteq H^0({\mathbb{P}}^1\times {\mathbb{P}}^1,\mathcal{O}(a,1))$ has a finite set of $r$ basepoints in generic position, then the implicit equation of $X_U$ is determined by two syzygies of $I_U=⟨f_0,\dots ,f_3⟩$ in bidegrees $\left(a-\lceil \frac{r}{2}\rceil ,0\right)$ and $\left(a-\lfloor \frac{r}{2}\rfloor ,0\right)$. This result is proved by understanding the geometry of the basepoints of $U$ in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The proof techniques for the main theorem also apply when $U$ is basepoint free.

19 pages, 3 figures