Artinian Gorenstein algebras with binomial Macaulay dual generator
Nasrin Altafi, Rodica Dinu, Sara Faridi, Shreedevi K. Masuti, Rosa M. Miró-Roig, Alexandra Seceleanu, Nelly Villamizar
TL;DR
This work analyzes Artinian Gorenstein algebras with binomial Macaulay dual generators, establishing that in codimension 3 these algebras satisfy the strong Lefschetz property, admit a doubling interpretation of a 0-dimensional subscheme in P^2, and allow explicit complete-intersection criteria. For arbitrary codimension, it provides a general sufficient condition for the weak Lefschetz property based on the binomial factorization F = g(m_1 - m_2), and proves optimality in this regime. The authors execute a detailed structural study in codimension 3: they compute minimal generators and resolutions for Ann(F), prove SLP in characteristic zero, and show that all such algebras arise as doublings of 0-dimensional schemes via skew-symmetric BE matrices. The results connect homological properties to geometric constructions (doubling), offering a clear pathway for understanding Lefschetz phenomena in this targeted binomial setting and informing broader questions about WLP/SLP in AG algebras. Overall, the paper advances the classification and Lefschetz theory for binomial AG algebras and links algebraic properties to geometric doublings in projective space.
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.