Tensor product surfaces and graded syzygies

TL;DR

This paper investigates the implicit equations of tensor product surfaces in projective space, focusing on cases where the associated ideal has a singly graded syzygy, with implications for geometric modeling.

Contribution

It extends previous work to explicitly determine the implicit equations of tensor product surfaces under new algebraic conditions involving syzygies.

Findings

Solved the implicitization problem for tensor product surfaces with singly graded syzygies.

Connected algebraic properties of the ideal to geometric features of the surface.

Enhanced understanding of the algebraic structure underlying tensor product surfaces.

Abstract

Let $U\subseteq H^0(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(a,b))$ be a four-dimensional vector space and consider the rational map $\phi_U:\,\mathbb{P}^1\times \mathbb{P}^1 \dashrightarrow \mathbb{P}^3$ defined by its basis of bihomogeneous polynomials. The tensor product surface $X_U\subseteq \mathbb{P}^3$ is the closed image of $\phi_U$, and a fundamental problem in this setting is to determine its implicit equation. As these surfaces are ubiquitous within the field of geometric modeling and design, knowledge of their implicit equations is particularly advantageous, allowing for more effective and efficient computations. In this article, we expand upon work of Duarte-Schenck and work of the present author to solve this implicitization problem when the bigraded ideal $I_U$ admits a singly graded syzygy.