Normality of monomial ideals in three variables

TL;DR

This paper investigates the conditions under which integrally closed monomial ideals in three-variable polynomial rings are normal, establishing that such ideals with up to seven generators are always normal, using combinatorial and valuation techniques.

Contribution

It provides a sharp bound of seven generators for the normality of integrally closed monomial ideals in three variables, extending previous results in higher dimensions.

Findings

Ideals with up to seven generators are always normal.

The proof combines valuation theory and combinatorial methods.

Provides a classification for normality in three-variable monomial ideals.

Abstract

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three and higher, this is no longer true in general, including monomial ideals in polynomial rings. In this paper, we study the normality of integrally closed monomial ideals in the polynomial ring $k[x,y,z]$ over a field $k$. We prove that every such ideal with at most seven minimal monomial generators is normal, thereby giving a sharp bound for normality in this setting. The proof is based on a detailed case-by-case analysis, combined with valuation-theoretic and combinatorial methods via Newton polyhedra.