Lefschetz properties, Laplace equations, and Togliatti systems

In an article in collaboration with Rosa M. Mirò-Roig and Giorgio Ottaviani (Canad. J. Math. 65, 2013), we established a relation, due to apolarity, between Artinian homogeneous ideals of a polynomial ring not satisfying the Weak Lefschetz Property - WLP - and projective varieties that verify a Laplace equation of a certain order s, i.e. such that all the s-osculating spaces have dimension less than expected. Thanks to this relation, it has been possible to obtain various classification results of toric varieties verifying Laplace equations, extending some classical results of Eugenio Togliatti.
In the seminar, I will introduce these notions and I will speak of some recent results relating them to Galois cyclic coverings and circulant matrices.