{
  "abstract": "We construct an irrational one-point Seshadri constant on the blow-up $X_9$ of $\\mathbb{P}^2$ at nine very general points. The divisor $L=9H-3(E_1+\\cdots+E_5)-2(E_6+\\cdots+E_9)$ is ample and satisfies $\\varepsilon(L;x)=2\\sqrt{5}$ at a very general point $x\\in X_9$. To prove this, we establish that the Seshadri constant of $\\mathcal{O}_{\\mathbb{P}^1\\times\\mathbb{P}^1}(1,1)$ at ten very general points is $1/\\sqrt{5}$, completing the reflection approach proposed by Dionne and Roth for the ten-point case. We also prove that the same equality holds at a very general free orbit of a fixed dihedral group of order ten. This yields an irrational one-point Seshadri constant on the singular quotient surface. A plane model of its minimal resolution and a deformation of the blow-up centers then give the result on $X_9$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26521",
  "author_total": 2,
  "authors": [
    {
      "name": "Antonio Laface"
    },
    {
      "name": "Luca Ugaglia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/exn8b9j2dhn0q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26521",
  "fulltext_url": null,
  "id": "exn8b9j2dhn0q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26521",
  "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": "W7214094483",
  "pdf_url": "https://arxiv.org/pdf/2609.26521",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/exn8b9j2dhn0q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.26521",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26521"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Irrational Seshadri constants from dihedral orbits",
  "title_text": "Irrational Seshadri constants from dihedral orbits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/exn8b9j2dhn0q",
  "venue": "arXiv",
  "versions": []
}