Hyperplane sections and the subtlety of the Lefschetz properties

TL;DR

This paper investigates the delicate nature of Lefschetz properties in Artinian algebras, demonstrating how small deformations can induce or destroy these properties, especially through hyperplane sections.

Contribution

It introduces a systematic deformation method for monomial ideals to achieve the weak Lefschetz property while preserving the Hilbert function.

Findings

General hyperplane sections often have the weak Lefschetz property in almost all characteristics.

Special hyperplane sections lack the weak Lefschetz property in all characteristics.

Deformations can control the presence or absence of Lefschetz properties in hyperplane sections.

Abstract

The weak and strong Lefschetz properties are two basic properties that Artinian algebras may have. Both Lefschetz properties may vary under small perturbations or changes of the characteristic. We study these subtleties by proposing a systematic way of deforming a monomial ideal failing the weak Lefschetz property to an ideal with the same Hilbert function and the weak Lefschetz property. In particular, we lift a family of Artinian monomial ideals to finite level sets of points in projective space with the property that a general hyperplane section has the weak Lefschetz property in almost all characteristics, whereas a special hyperplane section does not have this property in any characteristic.