Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2602.01782

Normality of monomial ideals in three variables

Maki Ataka·Naoyuki Matsuoka
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.AC

Keep this discovery

BibTeXRIS

Maki Ataka, Naoyuki Matsuoka. 2026-02-02. Normality of monomial ideals in three variables. https://arxiv.org/abs/2602.01782

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Categories of Multigraded Local Cohomology Modules: Serre Filtrations and Nakayama Duality

Let $\Bbbk$ be a field, let $S=\Bbbk[x_1,\ldots,x_n]$ with its standard $\mathbb N^n$-grading, and let $\mathfrak m=(x_1,\ldots,x_n)$. For $0\le i<n$ and $q=n-i$, we identify the category $\mathcal H_i(\mathbf t)$ of shifted multigraded local cohomology modules with \[ \Rep(U_q(\mathbf t)),\qquad U_q(\mathbf t)=\{\mathbf a\in[\mathbf0,\mathbf t]\mid |\operatorname{supp}(\mathbf a)|\ge q\}. \] This gives the finite and global Serre filtrations and their pure support-rank quotients. We organize the resulting torsion and quotient structures through abelian recollement: an order-ideal decomposition produces a canonical TTF triple, hereditary support torsion pairs, and Gabriel quotients. For finite posets both complementary recollement orientations exist, whereas for the global finite-support categories only the inward-finite orientation is automatic. These recollements admit bounded derived lifts. Under an additional finite-resolution condition the derived finite-support categories have right Serre functors, and derived Kan extensions satisfy a right-Serre exchange. In finite boxes we further construct a functorial rank-layer resolution comparing the left and right Kan sections; Nakayama--Serre duality transforms it into an explicit costandard rank complex. The exceptional top category $\mathcal H_n(\mathbf t)$ is treated separately via second cosyzygies.

math.AC↗
arXiv2026

Associated primes, witnesses, and omega invariants of monomial ideals

We introduce and study the omega invariant of a proper ideal in a Noetherian commutative ring, defined as the number of associated primes of the ideal. Our main objective is to investigate this invariant for monomial ideals and their powers. We characterize associated primes through monomial witnesses and provide an algorithmic procedure for constructing such witnesses from the exponent vectors of the minimal generators. These results lead to explicit formulas and bounds for the omega invariant without requiring the computation of a primary decomposition. We further establish alternative descriptions using irreducible decompositions and Alexander duality. A matrix-based approach is developed to detect associated primes of powers of monomial ideals directly from the exponent matrix of the original ideal. We also investigate the behavior of witnesses under passage from $I^n$ to $I^{n+1}$ and derive corresponding results for edge ideals of graphs.

math.AC↗
arXiv2026

Quadratic Gr\"obner bases for cut ideals of cycles and ring graphs

Let $C_n$ be the cycle of length $n\ge3$ and let $I_{C_n}$ be its cut ideal. We show that $I_{C_n}$ has a quadratic Gr\"obner basis with respect to an explicit weight order. Since the defining configuration consists of $(0,1)$-vectors, the initial monomials of such a basis are automatically squarefree. As the cut polytope of a cycle is the parity polytope, the result gives a regular unimodular flag triangulation of this classical polytope. Together with the known tree case and the clique-sum theorem for cut ideals, the cycle result also yields a quadratic Gr\"obner basis for the cut ideal of every connected ring graph with at least one edge, thereby supplying the missing cycle input and establishing the result for connected ring graphs.

math.AC↗