Failure of WLP for non-square grids with d ≥ a

Prove that for any a × b grid X ⊂ P^3 with b > a ≥ 2 and any integer d ≥ a, the Artinian algebra R/A_{X,d}, where A_{X,d} is the ideal generated by the d-th powers of the linear forms dual to the points of X, fails to have the Weak Lefschetz Property.

Background

The paper studies the Weak Lefschetz Property (WLP) for quotients R/A_{X,d} where X is a grid of points on a smooth quadric in P^3 and A_{X,d} is generated by the d-th powers of linear forms dual to the points of X. The authors completely settle the square case (a × a grids), showing WLP holds precisely when 1 ≤ d ≤ a − 1 or d is a multiple of a − 1, and they determine the non-Lefschetz locus in the cases with WLP.

For non-square grids (a × b with b > a), they prove WLP holds when 1 ≤ d ≤ a − 1, but leave open the behavior for larger d. Conjecture 8.1 posits that all remaining non-square cases fail WLP, supported by computational evidence and cokernel-dimension estimates derived from Lemma 4.1 (see Remarks 8.2–8.4).