Singular hypersurfaces characterizing the Lefschetz properties

Roberta Di Gennaro (1) , Giovanna Ilardi (2) , Jean Vallès (3)

1 Dip. Matematica e Applicazioni di Napoli
2 Dip. Matematica e Applicazioni di Napoli
3 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

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.

Mots clés: Weak and Strong Lefschetz Property; Laplace equations; Line arrangements

Domaines: Géométrie algébrique [math.AG]

Roberta Di Gennaro, Giovanna Ilardi, Jean Vallès. Singular hypersurfaces characterizing the Lefschetz properties. Journal of the London Mathematical Society, 2014, 89 (2), pp.194-212. ⟨10.1112/jlms/jdt053⟩. ⟨hal-00739676v4⟩