On the Weak Lefschetz Property for certain ideals generated by powers of linear forms

TL;DR

This paper investigates the Weak Lefschetz Property for ideals generated by powers of linear forms associated with grid configurations of points in projective space, providing complete results for square grids and conjectures for nonsquare grids.

Contribution

It offers a complete characterization of WLP for ideals from square grid point configurations and proposes a conjecture and approach for nonsquare grids.

Findings

Complete WLP characterization for square grid ideals

Identification of non-Lefschetz locus when WLP holds

Conjecture and approach for nonsquare grid cases

Abstract

Ideals $I\subseteq R=k[\mathbb P^n]$ generated by powers of linear forms arise, via Macaulay duality, from sets of fat points $X\subseteq \mathbb P^n$. Properties of $R/I$ are connected to the geometry of the corresponding fat points. When the linear forms are general, many authors have studied the question of whether or not $R/I$ has the Weak Lefschetz Property (WLP). We study this question instead for ideals coming from a family of sets of points called grids. We give a complete answer in the case of uniform powers of linear forms coming from square grids, and we give a conjecture and approach for the case of nonsquare grids. In the cases where WLP holds, we also describe the non-Lefschetz locus.