{
  "abstract": "Let $S$ be a minimal surface of general type with $K_S^2=1$ and $p_g(S)=2$, called a $(1,2)$-surface. We give a geometric recovery and refinement of D. Wen's bound $|\\mathrm{Aut}(S)|\\leq 200$ using the canonical genus-two fibration and the double-cover model over the Hirzebruch surface $Σ_2$. Computing $\\mathrm{Aut}(Σ_2)$ from its Cox ring separates the action on the base $\\mathbb{P}^1$ from the vertical kernel. If $\\overline K_0$ denotes the vertical image on $Σ_2$, then $\\overline K_0$ is cyclic of order $5$, $4$, $2$, or $1$, and we prove the respective sharp bounds $|\\mathrm{Aut}(S)|\\leq 200, 192, 48, 80$. The proof combines singularity indices of genus-two fibrations with invariant divisors arising from the branch equation. We also construct admissible branch divisors attaining all four bounds and show that their full automorphism groups are, respectively, $(C_{10}\\times C_{10})\\rtimes C_2$, $(BO_{48}\\times C_8)/\\langle(-I,J^4)\\rangle$, $(BT_{24}\\times C_4)/\\langle(-I,J^2)\\rangle$, and $C_{40}\\times C_2$. Here $BO_{48}$ and $BT_{24}$ denote the binary octahedral and binary tetrahedral groups, respectively, and $J$ denotes a generator of the corresponding cyclic factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.25554",
  "author_total": 1,
  "authors": [
    {
      "name": "Hang Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8fy5hgxwrbbqs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.25554",
  "fulltext_url": null,
  "id": "8fy5hgxwrbbqs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.25554",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7165907812",
  "pdf_url": "https://arxiv.org/pdf/2606.25554",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8fy5hgxwrbbqs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2606.25554",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.25554"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Automorphism Groups of (1,2)-Surfaces",
  "title_text": "On Automorphism Groups of (1,2)-Surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/8fy5hgxwrbbqs",
  "venue": "arXiv",
  "versions": []
}