  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.
