{
  "abstract": "Let $\\mathcal{F}$ be a canonical foliation of general type on a smooth projective surface $X$, and write $\\mathrm{vol}(\\mathcal{F}):=\\mathrm{vol}(K_{\\mathcal{F}})$. Let $G\\subseteq\\operatorname{Bir}(X,\\mathcal{F})$ be a finite subgroup, and let $\\mathcal{G}:=\\mathcal{F}/G$ be the quotient foliation in the birational sense. For a canonical foliation $\\mathcal{H}$, define its $r$-th pluricanonical index by \\[ δ_r(\\mathcal{H}) := \\min \\bigl\\{ m\\in\\mathbb{Z}_{>0} \\mid h^0(mK_{\\mathcal{H}})\\geq r \\bigr\\}, \\] where $\\min\\varnothing:=\\infty$. For an arbitrary foliation, these indices are computed on any canonical birational model. If $κ(\\mathcal{G})\\geq0$, we prove \\[ |G| \\leq \\begin{cases} 4δ_1(\\mathcal{G})\\,\\mathrm{vol}(\\mathcal{F}), &κ(\\mathcal{G})=0,\\\\[1mm] \\displaystyle \\frac{4}{3}δ_2(\\mathcal{G})\\,\\mathrm{vol}(\\mathcal{F}), &κ(\\mathcal{G})=1,\\\\[3mm] δ_2(\\mathcal{G})^2 \\bigl(1+δ_2(\\mathcal{G})\\bigr)\\, \\mathrm{vol}(\\mathcal{F}), &κ(\\mathcal{G})=2. \\end{cases} \\] Since $\\operatorname{Bir}(X,\\mathcal{F})$ is finite, one may in particular take $G=\\operatorname{Bir}(X,\\mathcal{F})$. When $κ(\\mathcal{G})=0$ or $1$, the effective bounds $δ_1(\\mathcal{G})\\leq12$ and $δ_2(\\mathcal{G})\\leq42$ give \\[ |G|\\leq48\\,\\mathrm{vol}(\\mathcal{F}) \\qquad\\text{and}\\qquad |G|\\leq56\\,\\mathrm{vol}(\\mathcal{F}), \\] respectively. The main new ingredient is a cluster formula for adjoint volumes, which yields index-dependent lower bounds for tangency-free foliated surface pairs whose underlying foliation has Kodaira dimension zero or one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29900",
  "author_total": 1,
  "authors": [
    {
      "name": "Shi Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9t4s05ac2e11y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29900",
  "fulltext_url": null,
  "id": "9t4s05ac2e11y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29900",
  "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": "W7204877394",
  "pdf_url": "https://arxiv.org/pdf/2608.29900",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9t4s05ac2e11y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29900",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29900"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Birational Automorphism Bounds for General-Type Foliations on Surfaces via Pluricanonical Indices",
  "title_text": "Birational Automorphism Bounds for General-Type Foliations on Surfaces via Pluricanonical Indices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/9t4s05ac2e11y",
  "venue": "arXiv",
  "versions": []
}