Polynomials with vanishing Hessian and Lefschetz properties

TL;DR

This paper investigates special hypersurfaces with vanishing Hessian, focusing on Perazzo hypersurfaces in projective 4-space, and explores their algebraic properties, Hilbert vectors, and Lefschetz properties.

Contribution

It classifies Perazzo 3-folds with minimal Hilbert vectors and analyzes the Lefschetz properties of associated Gorenstein algebras, advancing understanding of hypersurfaces with vanishing Hessian.

Findings

Identified minimal and maximal Hilbert vectors for associated Gorenstein algebras.

Proved minimal cases satisfy the Weak Lefschetz property.

Classified all Perazzo 3-folds with minimal h-vector.

Abstract

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.