Syzygies and singularities of tensor product surfaces of bidegree (2,1)
  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. 
Hal Schenck, Alexandra Seceleanu, Javid Validashti
math.AC