Perazzo hypersurfaces and the weak Lefschetz property

We deal with Perazzo hypersurfaces $X=V(f)$ in ${\text{PP}}^{n+2}$ defined by a homogeneous polynomial $f(x_0,x_1,\dots ,x_n,u,v)=p_0(u,v)x_0+p_1(u,v)x_1+\cdots +p_n(u,v)x_n+g(u,v)$, where $p_0,p_1,\dots ,p_n$ 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 $A_f$ fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of $A_f$, we prove that the Hilbert function of $A_f$ is always unimodal and we determine when $A_f$ 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 ${\text{PP}}^{n+3}$ defined by a homogeneous polynomial $f(x_0,x_1,\dots ,x_n,u,v,w)=p_0(u,v,w)x_0+p_1(u,v,w)x_1+\cdots +p_n(u,v,w)x_n+g(u,v,w)$, where $p_0,p_1,\dots ,p_n$ 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 $d$.