{
  "abstract": "Let $A=\\{0=a_0<a_1<\\dots<a_{k-1}=d\\}$ be integers with greatest common divisor $1$, let $C_A$ be the projective monomial curve they define, and let $\\mathrm{reg}(A)$ be the Castelnuovo-Mumford regularity of its coordinate ring. Gimenez and González-Sánchez proved $\\mathrm{reg}(A)\\leσ(A)+1$ for the sumsets regularity $σ(A)$; the reverse inequality $σ(A)\\le\\mathrm{reg}(A)$ was proposed as an open conjecture in a 2024 talk. We show that it fails and measure by how much. Let $c_1,c_2$ be the conductors of the two numerical semigroups attached to $A$ and $N=\\lceil(c_1+c_2)/d\\rceil$. For curves in $\\mathbb{P}^3$ that are not arithmetically Cohen-Macaulay, $\\mathrm{reg}(A)\\ge(c_1+c_2)/d+2$, with equality exactly for the arithmetically Buchsbaum curves, and $σ(A)=\\mathrm{reg}(A)$. For arithmetically Cohen-Macaulay curves in $\\mathbb{P}^3$ we prove $9N\\le10\\,\\mathrm{reg}(A)-2$, with $9/10$ best possible, and give an explicit family with $σ(A)-\\mathrm{reg}(A)$ unbounded. In $\\mathbb{P}^4$ a family with a band-shaped Apéry set has $\\mathrm{reg}(A)/σ(A)\\to(2+\\sqrt2)/4$ along the solutions of a Pell equation, and gluing carries this bound to every $\\mathbb{P}^r$ with $r\\ge5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24121",
  "author_total": 2,
  "authors": [
    {
      "name": "Prashant Chaudhary"
    },
    {
      "name": "Kapil Khandelwal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5h4g6869d7gk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24121",
  "fulltext_url": null,
  "id": "c5h4g6869d7gk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24121",
  "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"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7223011896",
  "pdf_url": "https://arxiv.org/pdf/2610.24121",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5h4g6869d7gk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24121",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24121"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sumsets regularity versus Castelnuovo-Mumford regularity for monomial curves",
  "title_text": "Sumsets regularity versus Castelnuovo-Mumford regularity for monomial curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/c5h4g6869d7gk",
  "venue": "arXiv",
  "versions": []
}