  We study the minimal bigraded free resolution of an ideal with three
generators of the same bidegree, contained in the bihomogeneous maximal ideal $
\langle s,t\rangle \cap \langle u,v \rangle$ of the bigraded ring K[s,t;u,v].
Our analysis involves tools from algebraic geometry (Segre-Veronese varieties),
classical commutative algebra (Buchsbaum-Eisenbud criteria for exactness,
Hilbert-Burch theorem), and homological algebra (Koszul homology, spectral
sequences). We treat in detail the case in which the bidegree is (1,n). We
connect our work to a conjecture of Fr\"oberg-Lundqvist on bigraded Hilbert
functions, and close with a number of open problems.
