{
  "abstract": "Let $p$ be a prime. We prove that every compatible branch datum of degree $p$ over the sphere is realizable by a connected branched cover. The three-point case is constructed in residue characteristic $p$. Henrio's moment theorem supplies the distinct-point moment solutions from which we construct a special primitive tail for each prescribed partition; a second application underlies the new tail required by a positive source genus. These tails are joined by a logarithmic deformation datum and embedded in one subgroup of $S_p$ containing a common regular subgroup of order $p$. Wewers's lifting theorem produces a three-point Galois cover in characteristic zero. The quotient by a point stabilizer has degree $p$ and the prescribed three ramification profiles. The fusion and realization results of Edmonds--Kulkarni--Stong then give the assertion for an arbitrary number of branch values. As consequences, the connected prime-degree Hurwitz potential has full support on the Riemann--Hurwitz locus, every corresponding connected relative Gromov--Witten invariant of $\\mathbf P^1$ is nonzero, the two-relative-point disconnected sector with even completed-cycle orders at most $p$ is strictly positive subject to the dimension constraint, and the connected transposition sector is strictly positive.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20572",
  "author_total": 3,
  "authors": [
    {
      "name": "Jijian Song"
    },
    {
      "name": "Hailin Wen"
    },
    {
      "name": "Zebao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eamhvg097kyj4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20572",
  "fulltext_url": null,
  "id": "eamhvg097kyj4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20572",
  "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"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213600416",
  "pdf_url": "https://arxiv.org/pdf/2609.20572",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eamhvg097kyj4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20572",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20572"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Hurwitz existence problem in prime degree",
  "title_text": "The Hurwitz existence problem in prime degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/eamhvg097kyj4",
  "venue": "arXiv",
  "versions": []
}