Jacobian ideals, arrangements and the Lefschetz properties

We consider in this paper the following questions: does the Jacobian ideal of a smooth hypersurface have the Weak Lefschetz Property? Does the Jacobian ideal of a smooth hypersurface have the Strong Lefschetz Property? We prove that if X is a hypersurface in Pn of degree d>2, such that its singular locus has dimension at most n−3, then the ideal J(X) has the WLP in degree d−2. Moreover we show that if X is a hypersurface in Pn of degree d>2, such that its singular locus has dimension at most n−3, then for every positive integer k