{
  "abstract": "We classify abelian canonical covers of degree at most $8$ of smooth surfaces of minimal degree, and of $\\mathbb P^2$ and Hirzebruch surfaces not embedded as surfaces of minimal degree. Let $π: X \\longrightarrow W$ be an abelian canonical cover of a smooth surface of minimal degree $W$ with group $G$ and degree $n$. Degrees $n \\le 4$ were classified earlier; we complete the picture for $5 \\le n \\le 8$: - No such covers exist if $n$ is prime and $n > 3$, nor if $G=\\mathbb Z_8$. - For $G=\\mathbb Z_6$, $\\mathbb Z_2 \\times \\mathbb Z_2 \\times \\mathbb Z_2$ and $\\mathbb Z_2 \\times \\mathbb Z_4$ we give the complete list of covers. Only $W =\\mathbb P^2$, $\\mathbb F_0$ and $\\mathbb F_1$ arise. The list contains several families with unbounded $p_g$, most of them new; $X$ is smooth for general covers with $G=\\mathbb Z_2 \\times \\mathbb Z_2 \\times \\mathbb Z_2$, but singular if $G=\\mathbb Z_6$ and, with one exception, if $G=\\mathbb Z_2 \\times \\mathbb Z_4$. When $W$ is $\\mathbb P^2$ or a Hirzebruch surface not embedded as a surface of minimal degree, we show that abelian canonical covers exist only for $n = 2$ or for $G =\\mathbb Z_2 \\times \\mathbb Z_2$, and we classify the latter completely: they form infinitely many families, most new, with unbounded $p_g$ and, most of them, with unbounded irregularity. The slopes $K^2/χ$ of all these $X$ accumulate at $6$, at 8, and at $8α/(α+1)$ for every integer $α\\ge 2$. These results rest on more general ones, valid for $X$ normal, locally Gorenstein and of arbitrary dimension, and $W$ smooth with $p_g(W) = 0$: we determine which branch divisors of an abelian canonical cover must vanish, and, for canonical covers whose $π_*\\mathcal O_X$ splits as a sum of line bundles, we describe the $\\mathcal O_W$-module structure of $π_*\\mathcal O_X$, computing it completely when $W$ is $\\mathbb P^2$ or a Hirzebruch surface.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35724",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexandre Dorothée"
    },
    {
      "name": "Francisco Javier Gallego"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6tnjjy64w9wjw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35724",
  "fulltext_url": null,
  "id": "6tnjjy64w9wjw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35724",
  "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": "W7214743362",
  "pdf_url": "https://arxiv.org/pdf/2609.35724",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6tnjjy64w9wjw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35724",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35724"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Abelian canonical covers of minimal rational surfaces and $\\mathbb F_1$ up to degree $8$",
  "title_text": "Abelian canonical covers of minimal rational surfaces and 𝔽_1 up to degree $8$",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/6tnjjy64w9wjw",
  "venue": "arXiv",
  "versions": []
}