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

TL;DR

This paper provides a method to find the implicit equation of tensor product surfaces in projective space when there are a finite set of generic basepoints, using syzygies of the defining ideal.

Contribution

It introduces a novel approach to implicitization of tensor product surfaces with basepoints by leveraging syzygies in specific bidegrees, extending previous methods.

Findings

Implicit equation determined by two specific syzygies

Method applies to basepoint-free cases as well

Provides geometric understanding of basepoints in the process

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.