Fiorindo, LucaMezzetti, EmiliaMiró-Roig, Rosa M. (Rosa Maria)2025-04-032025-04-032023-07-150021-8693https://hdl.handle.net/2445/220222We 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.26 p.application/pdfengcc-by-nc-nd (c) Luca Fiorindo et al., 2023http://creativecommons.org/licenses/by-nc-nd/4.0/HipersuperfíciesGeometria algebraicaHypersurfacesAlgebraic geometryinfo:eu-repo/semantics/article7324852025-04-03info:eu-repo/semantics/openAccess