{
  "abstract": "Let $S$ be a minimal complex surface of general type with $K_S^2=p_g(S)=1$. By a theorem of Catanese, the canonical model of $S$ is a complete intersection of two sextics in $\\mathbf{P}(1,2,2,3,3)$ whose equations admit a normal form, and the global Torelli theorem fails for these surfaces. We describe $\\operatorname{Aut}(S)$ in terms of the coefficients of the normal form, prove that $|\\operatorname{Aut}(S)|\\leq 72$, and classify the surfaces attaining the bound: two surfaces with singular canonical model and a one-parameter family $X_r$ of smooth surfaces. For the surfaces $X_r$ we compute a basis of the primitive cohomology consisting of eigenvectors of a subgroup $\\mathbf{Z}/6\\mathbf{Z} \\times \\mathbf{Z}/6\\mathbf{Z}$ of $\\operatorname{Aut}(X_r)$, given by residues of explicit polynomials, together with the intersection form in this basis. We then show that the natural map from $\\operatorname{Aut}(X_r)$ to the group of Hodge isometries of the primitive lattice modulo $\\pm1$ is injective but has image of index at least $331\\,776$, so that the Torelli principle for automorphisms fails for these surfaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33364",
  "author_total": 3,
  "authors": [
    {
      "name": "Jorge Duque Franco"
    },
    {
      "name": "Alvaro Liendo"
    },
    {
      "name": "Pedro Montero"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2h5rvyp6c9g2z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33364",
  "fulltext_url": null,
  "id": "2h5rvyp6c9g2z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33364",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223998861",
  "pdf_url": "https://arxiv.org/pdf/2610.33364",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2h5rvyp6c9g2z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33364",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33364"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Automorphisms and Hodge isometries of surfaces with $K^2=p_g=1$",
  "title_text": "Automorphisms and Hodge isometries of surfaces with K^2=p_g=1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/2h5rvyp6c9g2z",
  "venue": "arXiv",
  "versions": []
}