Tensor product surfaces and quadratic syzygies

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 ${\phi }_U:{\mathbb{P}}^1\times {\mathbb{P}}^1⇢{\mathbb{P}}^3$. 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.