Betti tables forcing failure of the Weak Lefschetz Property

TL;DR

This paper demonstrates that specific Betti tables of Artinian algebras derived from point configurations in projective space can cause the failure of the Weak Lefschetz Property, revealing new geometric-algebraic obstructions beyond Hilbert functions.

Contribution

It establishes a novel link between Betti tables and the failure of WLP, focusing on cases where Hilbert functions do not indicate such failure.

Findings

Certain Betti tables force WLP failure

Failure occurs in non-level Artinian algebras

Hilbert functions do not detect these failures

Abstract

We study the Artinian reduction $A$ of a configuration of points $X \subset {\mathbb P}^n $, and the relation of the geometry of $X$ to Lefschetz properties of $A$. Migliore initiated the study of this connection, with a particular focus on the Hilbert function of $A$, and further results appear in work of Migliore--Mir\'o-Roig--Nagel. Our specific focus is on Betti tables rather than Hilbert functions, and we prove that a certain type of Betti table forces the failure of the Weak Lefschetz Property (WLP). The corresponding Artinian algebras are typically not level, and the failure of WLP in these cases is not detected in terms of the Hilbert function.