Perazzo $n$-folds and the weak Lefschetz property

TL;DR

This paper investigates the Hilbert vectors of Perazzo algebras, identifying conditions for unimodality and Weak Lefschetz Property satisfaction, and provides minimal free resolutions for specific cases, advancing understanding of algebraic properties related to Perazzo polynomials.

Contribution

It characterizes the extremal Hilbert vectors of Perazzo algebras and links their unimodality and Lefschetz properties to algebraic and geometric conditions, including resolutions for threefolds.

Findings

Maximum Hilbert vectors always fail the Weak Lefschetz Property.

Minimum Hilbert vectors are unimodal and satisfy the Weak Lefschetz Property under mild conditions.

Minimal free resolutions are determined for certain Perazzo threefolds.

Abstract

In this paper, we determine the maximum $h_{max}$ and the minimum $h_{min}$ of the Hilbert vectors of Perazzo algebras $A_F$, where $F$ is a Perazzo polynomial of degree $d$ in $n+m+1$ variables. These algebras always fail the Strong Lefschetz Property. We determine the integers $n,m,d$ such that $h_{max}$ (resp. $h_{min}$) is unimodal, and we prove that $A_F$ always fails the Weak Lefschetz Property if its Hilbert vector is maximum, while it satisfies the Weak Lefschetz Property if it is minimum, unimodal, and satisfies an additional mild condition. We determine the minimal free resolution of Perazzo algebras associated to Perazzo threefolds in $\mathbb P^4$ with minimum Hilbert vectors. Finally we pose some open problems in this context. Dedicated to Enrique Arrondo on the occasion of his $60^{th}$ birthday.