Perazzo n-folds and the weak Lefschetz property

In this paper, we determine the maximum hₘₐₓ and the minimum hₘᵢₙ of the Hilbert vectors of Perazzo algebras AF, 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 hₘₐₓ (resp. hₘᵢₙ) is unimodal, and we prove that AF 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 P⁴ with minimum Hilbert vectors. Finally we pose some open problems in this context. Dedicated to Enrique Arrondo on the occasion of his 60ᵗʰ birthday.