{
  "abstract": "For a complex $\\varepsilon$-log canonical Fano surface of Picard number one, we prove that the anticanonical volume is at least $\\varepsilon^2/100$ and that every integral nef and big Weil divisor has volume at least $\\varepsilon^7/1728$. Both exponents are optimal. For $\\varepsilon$-log canonical log Calabi--Yau surface pairs of arbitrary Picard number, we prove a lower bound of order $\\varepsilon^9/(1+\\log_2(1/\\varepsilon))$, without a bigness assumption or further restrictions on the boundary coefficients. We also prove $τ(X,H)\\le18\\varepsilon^{-3}$ for the pseudo-effective threshold on every projective $\\varepsilon$-log canonical surface, with optimal exponent three. A volume-to-birationality estimate gives a sufficient bound of order $\\varepsilon^{-11/2}\\sqrt{1+\\log_2(1/\\varepsilon)}$ in the log Calabi--Yau setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23763",
  "author_total": 1,
  "authors": [
    {
      "name": "Pinxian Bie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/efpg0kg03smer.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23763",
  "fulltext_url": null,
  "id": "efpg0kg03smer",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23763",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214007751",
  "pdf_url": "https://arxiv.org/pdf/2609.23763",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/efpg0kg03smer.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.23763",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23763"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces",
  "title_text": "Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/efpg0kg03smer",
  "venue": "arXiv",
  "versions": []
}