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 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<d−1 the ideal J(X) has the SLP in degree d−k−1 at range k. Finally we prove that if X⊂Pn 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.
Published Version
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