Perazzo 3-folds and the weak Lefschetz property

TL;DR

This paper investigates Perazzo 3-folds in projective 4-space, analyzing their associated algebra's Hilbert functions and Lefschetz properties, revealing conditions under which these properties hold or fail.

Contribution

It determines the extremal Hilbert functions of the associated algebra and classifies Perazzo 3-folds with minimal Hilbert function.

Findings

Maximal Hilbert function algebra fails the weak Lefschetz property.

Minimal Hilbert function algebra satisfies the weak Lefschetz property.

Complete classification of Perazzo 3-folds with minimal Hilbert function.

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 $f(x_0,x_1,x_2,u,v)=p_0(u,v)x_0+p_1(u,v)x_1+p_2(u,v)x_2+g(u,v)$, 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 $d$. 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.