[论文解读] On Zero-Dimensional Glicci Monomial Ideals

Benjamin Mudrak|arXiv (Cornell University)|Feb 3, 2026

Commutative Algebra and Its Applications被引用 0

一句话总结

该论文证明在 k[x,y,z] 中最多有八个生成元的所有 m-主单项理想在同质 glicci 中，并在任意 n 中构建广义类别，属于 glicci 但非 licci，且给出明确的 Gorenstein 链接。

ABSTRACT

Consider the polynomial ring $R_n = k[x_1,...,x_n]$, where $k$ is a field. Let $m = (x_1,...,x_n)$ and $I$ be an $m$-primary monomial ideal in $R$. We consider the problem of determining whether such ideals are in the Gorenstein liasion class of a complete intersection (glicci). We prove that all $m$-primary monomial ideals in $k[x,y,z]$ with at most eight generators are homogeneously glicci. We also construct a large class of $m$-primary monomial ideals in $R_n$ for any $n$ with any number of minimal generators that are homogeneously glicci but not in the complete intersection liaison class of a complete intersection (licci). All Gorenstein links used are constructed explicitly and every second step links to another $m$-primary monomial ideal.

研究动机与目标

• Motivate the study of which m-primary monomial ideals lie in the Gorenstein liaison class of a complete intersection (glicci).
• Show that every m-primary monomial ideal in k[x,y,z] with at most eight generators is homogeneously glicci.
• Provide explicit Gorenstein linkage constructions that link to other m-primary monomial ideals in a stepwise manner.
• Describe a general construction to produce glicci m-primary monomial ideals in arbitrary polynomial rings R=k[x1,...,xn].

提出的方法

• Use Macaulay inverse systems to analyze homogeneous m-primary ideals and construct Gorenstein ideals explicitly.
• Apply double G-linkage (two successive G-links) to move from a given ideal to an easier one, while preserving monomial structure in each second step.
• Use Corollary 2.3 to perform double G-links and reduce the number of generators when possible.
• Employ a case-by-case analysis based on the T(I) invariant to control the linkage steps for ideals in k[x,y,z].
• Generalize to higher dimensions by iteratively building ideals I_i from an initial I_0 and a prescribed monomial framework, ensuring glicci property.
• Leverage Huneke-Ulrich licci criteria to distinguish glicci from licci families.]
• research_questions: ["Can every m-primary monomial ideal be connected to a complete intersection via homogeneous G-links (i.e., be glicci)?","Do all m-primary monomial ideals in k[x,y,z] with small generator counts (≤8) admit explicit glicci sequences with monomial intermediates?","How far can the glicci phenomenon be extended beyond three variables and small generator counts while preserving monomial Gorenstein links?","What structural patterns (e.g., T(I) types) determine the required linkage steps to reach a complete intersection?"]
• key_findings:[

实验结果

研究问题

• RQ1 Can every m-primary monomial ideal be connected to a complete intersection via homogeneous G-links (i.e., be glicci)?
• RQ2 Do all m-primary monomial ideals in k[x,y,z] with small generator counts (≤8) admit explicit glicci sequences with monomial intermediates?
• RQ3 How far can the glicci phenomenon be extended beyond three variables and small generator counts while preserving monomial Gorenstein links?
• RQ4 What structural patterns (e.g., T(I) types) determine the required linkage steps to reach a complete intersection?

主要发现

• All m-primary monomial ideals in k[x,y,z] with at most eight generators are homogeneously glicci via sequences of G-links with monomial intermediate ideals.
• There exist large classes of m-primary monomial ideals in any n with any number of generators that are glicci but not licci, with explicitly constructed Gorenstein links.
• Every second step in the constructed linkage remains within the realm of m-primary monomial ideals, enabling explicit tracking of the linkage path.
• A detailed case analysis based on the invariant T(I) guides the choice of G-links and ensures the generation count μ(I) stays within bounds to apply Corollary 2.3.
• The work provides an explicit constructive method for generating new glicci m-primary monomial ideals in arbitrary polynomial rings, and shows these ideals are not licci when gluings originate from high-height #I##.
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