{
  "abstract": "We determine the vanishing cycles of Xiao's genus-two fibrations $X_d\\to P^1$, $d=3,4,5$, whose fibers admit degree-$d$ maps to a fixed elliptic curve $E$. We tile the base $P^1$ by triangles and realize the surface $X_d$ as a degree-$d$ branched cover of $P^1\\times E$. The covering description gives an explicit algorithm for the vanishing cycles. A vanishing path is recorded by the sides it crosses in the tiling; each crossing changes a labelled picture of the fiber by a local hexagon move, and the vanishing cycle is read from the terminal picture. We obtain the $7$, $13$, and $31$ vanishing cycles as explicit curves in a marked reference fiber. For $d=4,5$, these give new positive factorizations of the identity in the genus-two mapping class group, of types $(6,7)$ and $(12,19)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34042",
  "author_total": 1,
  "authors": [
    {
      "name": "Zoltán Szabó"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdvpp36yrds5f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34042",
  "fulltext_url": null,
  "id": "fdvpp36yrds5f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34042",
  "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"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214789667",
  "pdf_url": "https://arxiv.org/pdf/2609.34042",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdvpp36yrds5f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34042",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34042"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5",
  "title_text": "The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/fdvpp36yrds5f",
  "venue": "arXiv",
  "versions": []
}