On the Lefschetz properties

The Lefschetz properties of strong type (SLP) and of weak type (WLP) are algebraic abstractions associated to the Hard Lefschetz Theorem for the cohomology ring of complex smooth projective varieties. These properties have a deep link with questions of commutative algebra, combinatorics and projective geometry. I deal with the SLP and WLP for artinian graded Gorenstein K-algebras. I study the class of the GNP- polynomials of type (m, n, k, e), introduced by R. Gondim; in particular I study the Hilbert vector of a GNP- algebra and the annihilator of the GNP-polynomial of type (m, n, k, k + 1) via the use of an associated simplicial complex. In the end I examine the geometric structure of the GNP- hypersurfaces of type (m, n, k, e).