Weakened Minimal Resolution Conjecture in P × P

Establish the Weakened Minimal Resolution Conjecture for finite sets of points in the product of projective spaces P × P: Prove that for every integer N ≥ 2, there exists a dense open subset U of the parameter space (P × P)^N such that any N-tuple of points (P1, P2, ..., PN) in U determines a set X = {P1, ..., PN} with a generic Hilbert matrix, and for every bidegree (i, j) > (0, 0), the first bigraded Betti numbers of the Cox ring quotient S/IX satisfy β1,(i,j) ≠ 0 if and only if the difference-matrix entry DHX(i, j) is negative and DHX(i′, j′) ≤ 0 for all (i′, j′) ≤ (i, j) with (i′, j′) ≠ (0, 0); moreover, in that case β1,(i,j) = −DHX(i, j).

Background

Section 4 introduces a notion of sufficiently general points in P × P and defines the matrix DH derived from the bigraded Hilbert function to relate to Betti numbers. The authors require control over the first Betti numbers to construct short virtual resolutions via the virtual-of-a-pair method.

Conjecture 4.4 provides a precise prediction for the nonvanishing and values of the first Betti numbers β1,(i,j) based solely on the sign pattern of entries in DHX. This conjectural control is used to prove Theorem 1.3 and Theorem 5.4, yielding explicit length-3 virtual resolutions for sufficiently general sets of points in P × P.