{
  "abstract": "Laface and Ugaglia recently constructed an irrational one-point Seshadri constant on the blow-up of $\\mathbb{P}^2$ at nine very general points by combining a dihedral orbit on $\\mathbb{P}^1\\times\\mathbb{P}^1$, a sequence of de Jonquières transformations, and a reflection argument along a $(-4)$-curve with balanced normal bundle. We show that the same mechanism extends uniformly to every odd integer $n\\geq 5$. For $n=2k+1$ we prove $$ \\varepsilon\\bigl(\\mathcal{O}_{\\mathbb{P}^1\\times\\mathbb{P}^1}(n-4,1);p_1,\\ldots,p_{2n}\\bigr)=\\sqrt{\\frac{n-4}{n}} $$ at $2n$ very general points, and already at a very general free orbit of a fixed dihedral group of order $2n$. For every odd $n\\geq 7$ this produces an explicit ample line bundle on the blow-up of $\\mathbb{P}^2$ at $k+7=(n+13)/2$ very general points whose one-point Seshadri constant at a very general point equals $$ 2\\sqrt{n(n-4)}. $$ More precisely, after one quadratic transformation we obtain the ample divisor $$ L_n=(3n-4)H-nE_1-(n-2)(E_2+\\cdots+E_5)-4(E_6+\\cdots+E_{k+5})-2(E_{k+6}+E_{k+7}), $$ with $L_n^2=4n(n-4)$ and $\\varepsilon(L_n;x)=\\sqrt{L_n^2}$. We also isolate an abstract balanced-reflection principle underlying the construction: a nef class on a special fiber can be reflected across a rational curve of square $-2a$ whenever the curve has normal bundle $\\mathcal{O}_{\\mathbb{P}^1}(-a)^{\\oplus 2}$ in the total space, and the reflected class is nef on very general fibers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01783",
  "author_total": 4,
  "authors": [
    {
      "name": "Grzegorz Malara"
    },
    {
      "name": "Łukasz Merta"
    },
    {
      "name": "Justyna Szpond"
    },
    {
      "name": "Marcin Zieliński"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2v30xxatrn30v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01783",
  "fulltext_url": null,
  "id": "2v30xxatrn30v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01783",
  "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": "W7215744641",
  "pdf_url": "https://arxiv.org/pdf/2610.01783",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2v30xxatrn30v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01783",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01783"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Dihedral reflections and an infinite series of irrational Seshadri constants",
  "title_text": "Dihedral reflections and an infinite series of irrational Seshadri constants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/2v30xxatrn30v",
  "venue": "arXiv",
  "versions": []
}