Tensor product surfaces and quadratic syzygies

TL;DR

This paper investigates tensor product surfaces in algebraic geometry, focusing on their implicit equations through the lens of quadratic syzygies of associated ideals, extending previous work on linear syzygies.

Contribution

It advances understanding of the syzygies of tensor product surfaces by analyzing cases with quadratic syzygies, building upon prior work on linear syzygies.

Findings

Characterization of tensor product surfaces with quadratic syzygies

Extension of syzygy analysis from linear to quadratic cases

Implications for implicit equation computation in geometric modeling

Abstract

For $U\subseteq H^0(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(a,b))$ a four-dimensional vector space, a basis $\{p_0,p_1,p_2,p_3\}$ of $U$ defines a rational map $\phi_U:\,\mathbb{P}^1\times \mathbb{P}^1 \dashrightarrow \mathbb{P}^3$. The tensor product surface associated to $U$ is the closed image $X_U$ of the map $\phi_U$. These surfaces arise within the field of geometric modelling, in which case it is particularly desirable to obtain the implicit equation of $X_U$. In this paper, we study $X_U$ via the syzygies of the associated bigraded ideal $I_U=(p_0,p_1,p_2,p_3)$ when $U$ is free of basepoints, i.e. $\phi_U$ is regular. Expanding upon work of Duarte and Schenck for such ideals with a linear syzygy, we address the case that $I_U$ has a quadratic syzygy.