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We study the minimal bigraded free resolution of an ideal with threegenerators 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, spectralsequences). We treat in detail the case in which the bidegree is (1,n). Weconnect our work to a conjecture of Fr\"oberg-Lundqvist on bigraded Hilbertfunctions, and close with a number of open problems.

The simplest minimal free resolutions in ${\mathbb{P}^1 \times \mathbb{P}^1}$