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

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.