On Zero-Dimensional Glicci Monomial Ideals

TL;DR

We address the glicci problem for $m$-primary monomial ideals by constructing explicit Gorenstein links. The paper proves that all such ideals in $k[x,y,z]$ with at most eight generators are homogeneously glicci and extends the construction to arbitrary $n$ producing glicci but non-licci examples, using a double G-link framework and Macaulay inverse systems. This yields a concrete, scalable method to obtain glicci monomial ideals and a broad, nontrivial class not in the complete intersection liaison class. The results advance understanding of Gorenstein linkage for monomial ideals and provide long, explicit chains of links that remain in the class of $m$-primary monomial ideals at each step.

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.