{
  "abstract": "For a smooth projective $3$-fold $V$ of intermediate Kodaira dimension, we establish Noether-type inequalities relating the Iitaka volume $\\operatorname{Ivol}(V)$ and the geometric genus $p_g(V)$. Specifically, for a smooth projective $3$-fold $V$ of Kodaira dimension $2$, we show that $\\operatorname{Ivol}(V)\\geq \\frac{3}{4}p_g(V)-\\frac{5}{4};$ while if the Kodaira dimension of $V$ is $1$, then $\\operatorname{Ivol}(V)\\geq p_g(V)-1.$ Both inequalities are sharp. In the Kodaira dimension $2$ case, we prove that the equality holds only if the bicanonical system of $V$ induces the Iitaka fibration and the canonical system of $V$ is composed with a pencil of surfaces whose general member has Iitaka volume $\\frac12$ provided that $p_g(V)\\geq 4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02689",
  "author_total": 4,
  "authors": [
    {
      "name": "Meng Chen"
    },
    {
      "name": "Yong Hu"
    },
    {
      "name": "Chen Jiang"
    },
    {
      "name": "Yang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7dzq6r4c6hb75.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02689",
  "fulltext_url": null,
  "id": "7dzq6r4c6hb75",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02689",
  "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": "W7220459997",
  "pdf_url": "https://arxiv.org/pdf/2610.02689",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7dzq6r4c6hb75.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02689",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02689"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sharp Noether-type inequalities for $3$-folds of intermediate Kodaira dimensions",
  "title_text": "Sharp Noether-type inequalities for $3$-folds of intermediate Kodaira dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/7dzq6r4c6hb75",
  "venue": "arXiv",
  "versions": []
}