Perazzo hypersurfaces and the weak Lefschetz property

We deal with Perazzo hypersurfaces X = V(f) in Pn+2 defined by a homogeneous polynomial f(x0, x1, ..., xn, u, v) = p0(u, v)x0 + p1(u, v)x1 + center dot center dot center dot + pn(u, v)xn + g(u, v), where p0, p1, . . . , pn are algebraically dependent but linearly independent forms of degree d - 1 in K[u, v] and g is a form in K[u, v] of degree d. Perazzo hypersurfaces have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra Af fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of Af , we prove that the Hilbert function of Af is always unimo dal and we determine when Af satisfies the weak Lefschetz property. We illustrate our results with many examples and we show that our results do not generalize to Perazzo hypersurfaces X = V(f) in Pn+3 defined by a homogeneous polynomial f(x0, x1, . . . , xn, u, v, w) = p0(u, v, w)x0 +p1(u, v, w)x1 + center dot center dot center dot + pn(u, v, w)xn + g(u, v, w), where p0, p1, . . . , pn are algebraically dependent but linearly independent forms of degree d - 1 in K[u, v, w] and g is a form in K[u, v, w] of degree d. (c) 2024 Elsevier Inc. All rights reserved.