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

TL;DR

This paper investigates the minimal free resolutions of certain ideals in a bigraded ring, using algebraic geometry and homological algebra tools, focusing on the case of bidegree (1,n) and relating to a conjecture on Hilbert functions.

Contribution

It provides a detailed analysis of minimal free resolutions for ideals with three generators of the same bidegree in a bigraded setting, especially for bidegree (1,n), connecting to existing conjectures.

Findings

Explicit description of minimal free resolutions for bidegree (1,n).

Connections established between resolutions and Hilbert function conjectures.

Open problems proposed for further research.

Abstract

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.