Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

TL;DR

This paper investigates the Lefschetz properties of certain Artinian algebras, using geometric techniques to establish strong Lefschetz properties in specific cases and exploring the geometric structure of loci related to these properties.

Contribution

It proves the strong Lefschetz property in degree 1 for codimension 6 complete intersection Artinian Gorenstein algebras presented by quadrics and explores geometric descriptions of special loci.

Findings

Proved strong Lefschetz property in degree 1 for codimension 6 algebras.

Established Lefschetz properties in higher codimensions.

Connected non-Lefschetz locus non-emptiness to lifting properties.

Abstract

In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of a weak Lefschetz property to an algebra from one of its quotients.