Lefschetz properties for Artinian Gorenstein algebras presented by quadrics

TL;DR

This paper introduces a family of Artinian Gorenstein algebras characterized by binomial structures, providing counterexamples to existing conjectures and generalizing criteria for Lefschetz properties.

Contribution

It constructs new binomial Gorenstein algebras presented by quadrics and disproves a conjecture about their Lefschetz properties, extending Hessian criteria for analysis.

Findings

Counterexamples to Migliore-Nagel conjectures in certain cases

Generalization of Hessian criterion for Lefschetz properties

Algebras presented by quadrics do not always satisfy weak Lefschetz property

Abstract

We introduce a family of standard bigraded binomial Artinian Gorenstein algebras, whose combinatoric structure characterizes the ones presented by quadrics. These algebras provide, for all socle degree grater than two and in sufficiently large codimension with respect to the socle degree, counter-examples to Migliore-Nagel conjectures, see \cite{MN1} and \cite{MN2}. One of them predicted that Artinian Gorenstein algebras presented by quadratics should satisfy the weak Lefschetz property. We also prove a generalization of a Hessian criterion for the Lefschetz properties given by Watanabe, see \cite{Wa1} and \cite{MW}, which is our main tool to control the Weak Lefschetz property.