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

Abstract

Given a $4$-dimensional vector subspace $U=\{ f_{0},\ldots,f_{3}\}$ 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 $I_{U}=\langle f_{0},\ldots,f_{3} \rangle$ 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.