{
  "abstract": "Let $k$ be an arbitrary field and let $S=k[x,y,z]$ be the polynomial ring with three variables $x,y,z$. We study integrally closed $(x,y,z)$-primary ideals of $S$ that are homogeneous for a positive weighted grading but need not be homogeneous for the standard grading. From a numerical semigroup $H$ of embedding dimension three, we obtain such ideals as inverse images $I_h=φ_H^{-1}(t^hk[t]\\cap k[H])$. If $\\ell$ is the least degree of a defining relation of $k[H]$, then $I_\\ell$ is monomial, whereas $I_{\\ell+1}$ has a binomial generator in the cases considered here. We determine $I_{\\ell+1}$ for numerical semigroups of embedding dimension three and multiplicity three or four. The six-generated cases arising in multiplicity three and in the symmetric multiplicity-four case form two explicit families. For every ideal $I$ in these families, we prove that its Rees algebra is a Cohen--Macaulay normal domain. In the non-symmetric multiplicity-four case, $I_{\\ell+1}$ is seven-generated; for $H=\\langle4,9,15\\rangle$, we prove that its Rees algebra is again a Cohen--Macaulay normal domain.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18087",
  "author_total": 1,
  "authors": [
    {
      "name": "Naoyuki Matsuoka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4957xexd9v9cz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18087",
  "fulltext_url": null,
  "id": "4957xexd9v9cz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18087",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7213503015",
  "pdf_url": "https://arxiv.org/pdf/2609.18087",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4957xexd9v9cz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18087",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18087"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Normality of ideals beyond the standard graded setting: families from numerical semigroup rings",
  "title_text": "Normality of ideals beyond the standard graded setting: families from numerical semigroup rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/4957xexd9v9cz",
  "venue": "arXiv",
  "versions": []
}