{
  "abstract": "We study the Seshadri function on the ample cone of a smooth complex projective surface at a fixed point. At a submaximal class, we show that the function is differentiable if and only if there is a unique Seshadri curve. At a maximal class $L$, differentiability is equivalent to every curve $C$ computing the Seshadri constant satisfying $\\frac{[C]}{{\\rm mult}_x C}=\\frac{[L]}{\\sqrt{L^{2}}}$. We also prove that the non-differentiability locus is nowhere dense. For the blow-up of $\\mathbb P^{2}$ at $r\\ge 9$ very general points, assuming the $(-1)$-curves conjecture for the blow-up at $r+1$ very general points, we construct a non-empty open subset of the ample cone on which the Seshadri function at a very general point equals $\\sqrt{L^{2}}$. Consequently, this function is not piecewise linear. Under the assumption of the Strong SHGH conjecture, we obtain a common non-empty open subset on which maximality holds at every point when $r\\ge36$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01138",
  "author_total": 1,
  "authors": [
    {
      "name": "Cyril J. Jacob"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51pepr95fhcme.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01138",
  "fulltext_url": null,
  "id": "51pepr95fhcme",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01138",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01138",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51pepr95fhcme.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01138",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01138"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Seshadri functions on surfaces",
  "title_text": "Seshadri functions on surfaces",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/51pepr95fhcme",
  "venue": "arXiv",
  "versions": []
}