Perazzo 3-folds and the weak Lefschetz property

We deal with Perazzo 3-folds in P4, i.e. hypersurfaces X = V(f) subset of P4 of degree d defined by a homogeneous polynomial f(x0, x1, x2, u, v) = p0(u, v)x0 +p1(u, v)x1 + p2(u, v)x2 + g(u, v), where p0, p1, p2 are algebraically dependent but linearly independent forms of degree d - 1 in u, v, and g is a form in u, v of degree d. Perazzo 3-folds have vanishing hessian and, hence, the associated graded Artinian Gorenstein algebra Af fails the strong Lefschetz Property. In this paper, we determine the maximum and minimum Hilbert function of Af and we prove that if Af has maximal Hilbert function it fails the weak Lefschetz Property while it satisfies the weak Lefschetz Property when it has minimum Hilbert function. In addition, we classify all Perazzo 3-folds in P4 such that Af has minimum Hilbert function.