Singular hypersurfaces characterizing the Lefschetz properties

In a recent paper, Mezzetti, Miró-Roig and Ottaviani [Mezzetti et al., "Laplace equations and the weak Lefschetz property", Canad. J. Math. 65 (2013) 634-654] highlight the link between rational varieties satisfying a Laplace equation and artinian ideals failing 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 the SLP (which includes WLP) by the existence of special singular hypersurfaces (cones for WLP). This characterization allows us to solve three problems posed in J. C. Migliore and U. Nagel ["A tour of the weak and strong Lefschetz properties", Preprint, 2011, arXiv:1109.5718, September 2011. J Commutative Algebra, to appear] 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 to the failure of the SLP. Moreover, we reformulate the so-called Terao's conjecture for free line arrangements in terms of artinian ideals failing the SLP