Polynomials with vanishing Hessian and Lefschetz properties

The aim is to study Perazzo hypersurfaces $X=V(F)\subseteq\mathbb{P}(K^5)$, defined by $F(x_0,x_1,x_2,u,v) = p_0(u,v)x_0+p_1(u,v)x_1+p_2(u,v)x_2+g(u,v)$, 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 $d$. These hypersurfaces are the "building blocks" for all possible hypersuface in $\mathbb{P}^4$ with vanishing Hessian. A minimal and a maximal Hilbert vector is found for the associated Artinian Gorenstein $K$-algebras $A_F$: in the minimal case they satisfy the Weak Lefschetz property, but in the maximal case they don't. Furthermore, we classify all Perazzo $3$-folds with minimal $h$-vector. We also summarise basic knowledge and already known results about hypersurfaces with vanishing Hessian and their geometry in low dimension, and also about Artinian Gorenstein $K$-algebras.