On mixed Hessians and the Lefschetz properties

TL;DR

This paper introduces the Mixed Hessian matrix to analyze Lefschetz properties in Artinian Gorenstein algebras, providing new criteria, proofs, and counterexamples for the Weak and Strong Lefschetz properties.

Contribution

It defines the Mixed Hessian, extends Lefschetz property criteria, and constructs new counterexamples of algebras not satisfying WLP, advancing understanding of algebraic Lefschetz properties.

Findings

Mixed Hessian effectively computes ranks of multiplication maps.

Recovered and generalized criteria for Strong and Weak Lefschetz elements.

Constructed minimal counterexamples of algebras without WLP.

Abstract

We introduce a new type of Hessian matrix, that we call Mixed Hessian. The mixed Hessian is used to compute the rank of a multiplication map by a power of a linear form in a standard graded Artinian Gorenstein algebra. In particular we recover the main result of \cite{MW} for identifying Strong Lefschetz elements, generalizing it also for Weak Lefschetz elements. This criterion is also used to give a new proof that Boolean algebras have the Strong Lefschetz Property (SLP). We also construct new examples of Artinian Gorenstein algebras presented by quadrics that does not satisfy the Weak Lefschetz Property (WLP); we construct minimal examples of such algebras and we give bounds, depending on the degree, for their existence. Artinian Gorenstein algebras presented by quadrics were conjectured to satisfy WLP in \cite{MN1,MN2}, and in a previous paper we construct the first counter-examples (see \cite{GZ}).