Laplace equations and the Strong Lefschetz Property

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]

We explore the link between artinian ideals that fail the Weak Lefschetz Property (WLP) in degree d-1 and projections of the Veronese varieties satisfying a Laplace equation of order d-1. We extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property (SLP) at the range k>0 in degree d-k. This generalization is not artificial; indeed for k=2 it is related to the so-called Terao's conjecture about free arrangements. We reformulate the Terao's conjecture for line arrangements in terms of artinian ideals failing the SLP at the range 2. Using this new link we propose non toric examples of ideals failing the SLP at the range 2. Moreover we add a new characterization of WLP or SLP in terms of singular hypersurfaces. Thanks to this characterization we produce many toric examples that fail the WLP and also the SLP at the range 2.

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

Domaines: Géométrie algébrique [math.AG]

Roberta Di Gennaro, Giovanna Ilardi, Jean Vallès. Laplace equations and the Strong Lefschetz Property. 2012. ⟨hal-00739676v1⟩