Singular hypersurfaces characterizing the Lefschetz properties

TL;DR

This paper extends the link between singular hypersurfaces and Lefschetz properties to characterize when artinian ideals fail the Strong Lefschetz Property, providing new examples and connections to line arrangements and Terao's conjecture.

Contribution

It introduces a new geometric characterization of SLP failure via singular hypersurfaces and applies this to solve open problems and relate to line arrangements and Terao's conjecture.

Findings

Characterization of SLP failure through singular hypersurfaces.

New examples of ideals failing the SLP.

Connection between line arrangement stability and SLP failure.

Abstract

In the paper untitled "Laplace equations and the Weak Lefschetz Property" the authors highlight the link between rational varieties satisfying a Laplace equation and artinian ideals that fail the Weak Lefschetz property. Continuing their work we extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property. We characterize the failure of SLP (that includes WLP) by the existence of special singular hypersurfaces (cones for WLP). This characterization allows us to solve three problems posed by Migliore and Nagel and to give new examples of ideals failing the SLP. Finally, line arrangements are related to artinian ideals and the unstability of the associated derivation bundle is linked with the failure of SLP. Moreover we reformulate the so-called Terao's conjecture for free line arrangements in terms of artinian ideals failing the SLP.