Complete intersection algebras with binomial Macaulay dual generator

TL;DR

We address identifying homogeneous forms $F$ in $K[X_1, obreak extotscrunch, obreak X_n]$ whose annihilator $Ann_R(F)$ is a complete intersection, with a focused treatment of the binomial case. The main contribution provides a precise coordinate-change classification: after a possible change of variables, $F$ has the form $F = X_1^{a_1} obreak obreak obreak o X_n^{a_n}(X_1^{b_1} obreak obreak obreak obreak o X_{n-1}^{b_{n-1}}-X_n^{b_n})$ with $ ext{sum}_{i=1}^{n-1} b_i = b_n>0$ and an inequality $a_i < q b_i$ for some $i$, where $q=loor{(a_n+1)/b_n}$; in this case $Ann_R(F)$ is generated by the $n-1$ monomial powers $x_i^{a_i+b_i+1}$ together with a specific form $G$, yielding a complete intersection. Conversely, if the binomial structure is not of this form (e.g., $r<n-1$ or $a_i less q b_i$ for all $i$), then $Ann_R(F)$ is not a CI. Moreover, these algebras satisfy the Strong Lefschetz property in characteristic zero, established via tensor-product stability and a reduction to monomial complete intersections through a monic relation in $x_n$. This work advances understanding of when Macaulay dual generators yield complete intersections and strengthens the link between CI structure and Lefschetz properties.

Abstract

In this paper, we characterize all Artinian complete intersection $K$-algebras $A_F$ whose Macaulay dual generator $F$ is a binomial. In addition, we prove that such complete intersection Artinian $K$-algebras $A_F$ satisfy the Strong Lefschetz property.

Theorems & Definitions (17)

• Lemma 2.1
• proof
• Remark 3.2
• Theorem 3.3
• proof
• Remark 3.4
• Example 3.5
• Definition 4.1
• Proposition 4.3
• proof