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

Abstract

We study the minimal bigraded free resolution of an ideal with three generators of the same bidegree, contained in the bihomogeneous maximal ideal $⟨s,t⟩\cap ⟨u,v⟩$ 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 Fröberg-Lundqvist on bigraded Hilbert functions, and close with a number of open problems.

26 pages, 2 figures