On the weak Lefschetz property for ideals generated by powers of general linear forms

TL;DR

This paper investigates the weak Lefschetz property (WLP) for ideals generated by powers of general linear forms, establishing bounds on the number of variables for which WLP holds or fails, with explicit constructions and sharp bounds.

Contribution

It provides a description of initial ideals for almost complete intersections and determines sharp bounds on the number of variables for WLP to hold or fail.

Findings

WLP holds for squares when n ≥ 3d-2

WLP holds for cubes when n ≥ (3d-3)/2

WLP fails for squares when n < 3d-2

Abstract

We provide a description of initial ideals for almost complete intersections generated by powers of general linear forms and prove that WLP in a fixed degree $d$ holds when the number of variables $n$ is sufficiently large compared to $d$. In particular, we show that if $n\geq 3d-2$ then WLP holds for the ideal generated by squares at the degree $d$ spot and for $n\ge \frac{3d-3}{2}$ WLP holds for ideal generated by cubes at the degree $d$ spot. Finally, we prove that WLP fails for the ideal generated by squares when $n< 3d -2$ at the $d$th spot by finding an explicit element in the kernel of the multiplication by a general linear form. This shows that our bound on $n$ is sharp in the case of the squares.