{
  "abstract": "We study singularities of height functions on singular surfaces in $\\mathbb{R}^3$ parameterized by smooth map-germs $\\mathcal{A}$-equivalent to $H_k$ in Mond's classification, and the versality of the family of the height functions. We also study relations of the singularities of the height functions with the parabolic locus of these singular surfaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22712",
  "author_total": 1,
  "authors": [
    {
      "name": "Masaru Hasegawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/67e4b0yvr5nh2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22712",
  "fulltext_url": null,
  "id": "67e4b0yvr5nh2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22712",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204210265",
  "pdf_url": "https://arxiv.org/pdf/2608.22712",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/67e4b0yvr5nh2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.22712",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22712"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Height functions on singular surfaces parameterized by smooth maps $\\mathcal{A}$-equivalent to $H_k$",
  "title_text": "Height functions on singular surfaces parameterized by smooth maps 𝒜-equivalent to H_k",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/67e4b0yvr5nh2",
  "venue": "arXiv",
  "versions": []
}