Perazzo n-folds and the weak Lefschetz property

In this paper, we determine the maximum h_max and the minimum h_min of the Hilbert vectors of Perazzo algebras A_F, where F is a Perazzo polynomial of degree d in n+m+1 variables. These algebras always fail the Strong Lefschetz Property. We determine the integers n, m, d such that h_max (resp. h_min) is unimodal, and we prove that A_F always fails the Weak Lefschetz Property if its Hilbert vector is maximum, while it satisfies the Weak Lefschetz Property if it is minimum, unimodal, and satisfies an additional mild condition. We determine the minimal free resolution of Perazzo algebras associated to Perazzo threefolds in P^4 with minimum Hilbert vectors. Finally we pose some open problems in this context.