{
  "abstract": "Let $S_E$ be the surface in Xiao's degree-four family identified by Polizzi as a smooth divisor in $E(3)$. Starting from Polizzi's branch model and Xiao's classification of the thirteen singular fibers, we derive the local spherical braids and their Picard--Lefschetz lifts, including the colored $3+3$ partitions at the seven reducible fibers. We identify the normalization of each bisection with $E/\\{\\pm1\\}$ and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove $π_1(S_E)\\cong\\mathbb Z^2$, compute the elementary-divisor-$4$ Albanese kernel of a regular genus-two fiber, and show that the six nonseparating Picard--Lefschetz transformations represent the six cusps of $Γ(4)$. We also prove that the natural product tori near an elliptic section are nullhomologous and cannot lower $b_1$ below $2$ by torus surgery. As a separate application of the degree-three monodromy, we give a twisted-double construction of an exotic $\\mathbb CP^2\\#7\\overline{\\mathbb CP}^{\\,2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08655",
  "author_total": 2,
  "authors": [
    {
      "name": "Anar Akhmedov"
    },
    {
      "name": "Sümeyra Sakallı"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qv83p7t7hgvg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08655",
  "fulltext_url": null,
  "id": "4qv83p7t7hgvg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08655",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220968880",
  "pdf_url": "https://arxiv.org/pdf/2610.08655",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qv83p7t7hgvg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08655",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08655"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Xiao's Degree-Four Fibration and the Polizzi Model",
  "title_text": "Xiao's Degree-Four Fibration and the Polizzi Model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/4qv83p7t7hgvg",
  "venue": "arXiv",
  "versions": []
}