Perazzo 3-folds and the weak Lefschetz property

Abstract

We deal with Perazzo 3 -folds in $\mathbb{P}^4$, i.e. hypersurfaces $X=$ $V(f) \subset \mathbb{P}^4$ of degree $d$ defined by a homogeneous polynomial $f\left(x_0, x_1, x_2, u, v\right)=p_0(u, v) x_0+p_1(u, v) x_1+p_2(u, v) x_2+$ $g(u, v)$, where $p_0, p_1, p_2$ 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 $A_f$ fails the strong Lefschetz Property. In this paper, we determine the maximum and minimum Hilbert function of $A_f$ and we prove that if $A_f$ 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 $\mathbb{P}^4$ such that $A_f$ has minimum Hilbert function.