LAUSR

Abstract 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 P n 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 P n of degree d > 2 , such that its singular locus has dimension at most n − 3 , then for every positive integer k d − 1 the ideal J ( X ) has the SLP in degree d − k − 1 at range k. Finally we prove that if X ⊂ P n is a general hypersurface and J ( X ) its Jacobian ideal, then J ( X ) has the SLP. We present four famous line arrangements that give new examples of ideals failing SLP; they all arise from complex reflection groups. We conclude giving an infinite family of line arrangements that produce ideals failing SLP.