Syzygies and singularities of tensor product surfaces of bidegree (2, 1)

Let U ⊆ H⁰(O _ℙ1×ℙ1 (2, 1)) be a basepoint free four-dimensional vector space. The sections corresponding to U determine a regular map φU : ℙ¹ × ℙ¹ -→ ℙ³. We study the associated bigraded ideal I_U ⊆ k[s, t; u, v] from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation for φU(ℙ¹ × ℙ¹), via work of Busé-Jouanolou, Busé-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex Z. In four of the six cases I_U has a linear first syzygy; remarkably from this we obtain all differentials in the minimal free resolution. In particular, this allows us to explicitly describe the implicit equation and singular locus of the image.