{
  "abstract": "Let $X$ be a smooth projective Calabi-Yau variety of dimension $3$, with Picard number one. Let $n$ be the self-intersection of the ample generator of the Néron-Severi group of $X$. Let $\\mathsf{GV}_{g,d}$ be the Gopakumar-Vafa invariant of $X$ of genus $g$ and degree $d$. It is conjectured that {\\it if $g>\\frac{d^{2}}{2n}+\\frac{d}{2}+1$, then $\\mathsf{GV}_{g,d}=0$}. In this paper, we prove the implication when $d\\gg n$ and $X\\subset \\mathbb P^r$ is a complete intersection in projective space. The vanishing follows by proving that if $C$ is a closed subscheme of $X$ of dimension $1$, arithmetic genus $p_a(C)$, and degree $d\\gg n$, then $p_a(C) \\leq \\frac{d^2}{2n}+\\frac{d}{2}+1$. Actually, we prove a somewhat more general result. In particular, we prove that $C$ is an extremal curve if and only if $C$ is the complete intersection on $X$ with a hyperplane $H$ and a hypersurface $F$, namely $C=X\\cap H\\cap F$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25065",
  "author_total": 3,
  "authors": [
    {
      "name": "Vincenzo Di Gennaro"
    },
    {
      "name": "Antonio Rapagnetta"
    },
    {
      "name": "Pietro Sabatino"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cgwhs79ztny8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25065",
  "fulltext_url": null,
  "id": "1cgwhs79ztny8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25065",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223060501",
  "pdf_url": "https://arxiv.org/pdf/2610.25065",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cgwhs79ztny8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25065",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25065"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The genus of a curve on Calabi-Yau threefolds",
  "title_text": "The genus of a curve on Calabi-Yau threefolds",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/1cgwhs79ztny8",
  "venue": "arXiv",
  "versions": []
}