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

Abstract

Let U be a basepoint free four-dimensional subspace of the space of sections of O(2,1) on P^1 x P^1. The sections corresponding to U determine a regular map p_U: P^1 x P^1 --> P^3. We study the associated bigraded ideal I_U in 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 of the image p_U(P^1 x P^1), via work of Buse-Jouanolou, Buse-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex. 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 describe the implicit equation and singular locus of the image.