On Zero-Dimensional Glicci Monomial Ideals

TL;DR

This paper investigates the Gorenstein liaison class of zero-dimensional monomial ideals, proving that all such ideals with up to eight generators in three variables are homogeneously glicci, and constructing larger classes with specific properties.

Contribution

It establishes that all m-primary monomial ideals with up to eight generators in three variables are homogeneously glicci and constructs larger classes with explicit Gorenstein links.

Findings

All m-primary monomial ideals in k[x,y,z] with at most eight generators are homogeneously glicci.

Constructs a large class of m-primary monomial ideals in R_n with any number of generators that are homogeneously glicci but not licci.

Explicit Gorenstein links are constructed, linking each to another m-primary monomial ideal.

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.