Artinian Gorenstein algebras with binomial Macaulay dual generator

TL;DR

This paper systematically studies Artinian Gorenstein algebras with binomial Macaulay dual generators, revealing their strong Lefschetz property in codimension 3, their construction as doublings, and conditions for the weak Lefschetz property in higher codimensions.

Contribution

It provides a comprehensive analysis of binomial Macaulay dual generator algebras, including their Lefschetz properties, construction methods, and criteria for complete intersections.

Findings

All codimension 3 algebras satisfy the strong Lefschetz property.

These algebras can be constructed as doublings of 0-dimensional schemes in 2.

Sufficient and optimal conditions for the weak Lefschetz property in higher codimensions.

Abstract

This paper initiates a systematic study for key properties of Artinian Gorenstein \(K\)-algebras having binomial Macaulay dual generators. In codimension 3, we demonstrate that all such algebras satisfy the strong Lefschetz property, can be constructed as a doubling of an appropriate 0-dimensional scheme in \(\mathbb{P}^2\), and we provide an explicit characterization of when they form a complete intersection. For arbitrary codimension, we establish sufficient conditions under which the weak Lefschetz property holds and show that these conditions are optimal.