{
  "abstract": "We define and study the $F$-signature of big divisors on projective varieties in positive characteristics. For a normal projective variety $X$, we show that the $F$-signature function defines a continuous function on the big cone of $X$, extending our previous results on the ample cone. By studying the the Frobenius-alpha invariant on the big cone, we prove that the positivity of the $F$-signature of any big $\\mathbb{R}$-divisor on $X$ is equivalent to the global $F$-regularity of $X$. As applications, we prove a birational transformation rule for the $F$-signature in terms of the $F$-signature of certain graded ideal sequences. We also prove that in a reduction modulo $p$ situation, a characteristic independent lower bound for the $F$-signature of $X$ with respect to any ample divisor $L$ spreads out to a similar bound for arbitrary big divisors on $X$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36321",
  "author_total": 2,
  "authors": [
    {
      "name": "Seungsu Lee"
    },
    {
      "name": "Suchitra Pande"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ed5espgxsnjym.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36321",
  "fulltext_url": null,
  "id": "ed5espgxsnjym",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36321",
  "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.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214953095",
  "pdf_url": "https://arxiv.org/pdf/2609.36321",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ed5espgxsnjym.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36321",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36321"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The $F$-signature Function on the Big Cone",
  "title_text": "The F-signature Function on the Big Cone",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/ed5espgxsnjym",
  "venue": "arXiv",
  "versions": []
}