Tensor product surfaces and linear syzygies

Abstract:

Let $U \subseteq H^0({\mathcal {O}_{\mathbb {P}^1 \times \mathbb {P}^1}}(a,b))$ be a basepoint free four-dimensional vector space, with $a,b \ge 2$. The sections corresponding to $U$ determine a regular map $\phi _U: {\mathbb {P}^1 \times \mathbb {P}^1} \longrightarrow \mathbb {P}^3$. We show that there can be at most one linear syzygy on the associated bigraded ideal $I_U \subseteq k[s,t;u,v]$. Existence of a linear syzygy, coupled with the assumption that $U$ is basepoint free, implies the existence of an additional “special pair” of minimal first syzygies. Using results of Botbol, we show that these three syzygies are sufficient to determine the implicit equation of $\phi _U(\mathbb {P}^1 \times \mathbb {P}^1)$, and that $\mathrm {Sing}(\phi _U(\mathbb {P}^1 \times \mathbb {P}^1))$ contains a line.