{
  "abstract": "Bryan, Oberdieck, Pandharipande and Yin predicted the number of hyperelliptic curves on general abelian surfaces under a transversality condition. Some of the numbers have already been confirmed. In this paper we construct singular hyperelliptic curves on $(1,d)$ polarised abelian surfaces for $d\\leq 15$ confirming a few predicted numbers, namely of (geometric) genus $g=d$ and of geometric genus $k+3$ on $(1,2k)$ and $(1,2k+1)$ polarised surfaces. Moreover, we have found that all genus 7 hyperelliptic curves on general $(1,8)$ polarised surfaces are double coverings of curves locally embedded into $(1,4)$ polarised surfaces. More interestingly, there is a unique genus 9 hyperelliptic curve (up to translation) locally embedded into a general $(1,12)$ polarised surface and it is a Klein covering of a curve locally embedded into a $(1,3)$ polarised surface.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40989",
  "author_total": 3,
  "authors": [
    {
      "name": "Paweł Borówka"
    },
    {
      "name": "Paolo Grossi"
    },
    {
      "name": "Anatoli Shatsila"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7e9s9dqcdrpx4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40989",
  "fulltext_url": null,
  "id": "7e9s9dqcdrpx4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40989",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40989",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7e9s9dqcdrpx4.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40989",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40989"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Hyperelliptic curves on $(1,d)$ abelian surfaces for $d\\leq 15$",
  "title_text": "Hyperelliptic curves on (1,d) abelian surfaces for d≤ 15",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/7e9s9dqcdrpx4",
  "venue": "arXiv",
  "versions": []
}