Tensor product surfaces and linear syzygies

Let U ⊆ H⁰(O_ℙ¹×ℙ¹ (a, b)) be a basepoint free four-dimensional vector space, with a, b ≥ 2. The sections corresponding to U determine a regular map ϕ_U : ℙ¹ × ℙ¹ → ℙ³. We show that there can be at most one linear syzygy on the associated bigraded ideal I_U ⊆ 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 ϕ_U (ℙ¹ × ℙ¹), and that Sing(ϕ_U (ℙ¹ × ℙ¹)) contains a line.