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

Let U ⊆ H(OP1×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 IU ⊆ 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 [5], Busé-Chardin [6], Botbol [2] and Botbol-DickensteinDohm [3] on the approximation complex Z. In four of the six cases IU 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.