  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.
