Lefschetz properties of Jacobian algebras and Jacobian modules

TL;DR

This paper investigates the Lefschetz properties of Jacobian algebras and modules associated with hypersurfaces with isolated singularities, relating algebraic properties to the geometry of hyperplane sections and their singularities.

Contribution

It establishes new connections between Lefschetz properties of Jacobian algebras/modules and the singularities of hyperplane sections of hypersurfaces.

Findings

Lefschetz properties are linked to the singularities of hyperplane sections.

Results extend to Jacobian modules, not just algebras.

Provides criteria for Lefschetz properties based on singularity types.

Abstract

Let $V:f=0$ be a hypersurface of degree $d \geq 3$ in the complex projective space $\mathbb{P}^n$, $n \geq 3$, having only isolated singularities. Let $M(f)$ be the associated Jacobian algebra and $H: \ell=0$ be a hyperplane in $\mathbb{P}^n$ avoiding the singularities of $V$, but such that $V \cap H$ is singular. We related the Lefschetz type properties of the linear maps $\ell: M(f)_k \to M(f)_{k+1}$ induced by the multiplication by linear form $\ell$ to the singularities of the hyperplane section $V \cap H$. Similar results are obtained for the Jacobian module $N(f)$.