{
  "abstract": "We construct explicit families of maximal singular curves over finite fields. A maximal singular curve is a curve that attains the Aubry-Perret bound, which is the natural extension of the Hasse-Weil-Serre bound on the maximum number of rational points on a smooth curve over a finite field. Starting from a smooth elliptic or hyperelliptic curve $\\tilde{C}$ over $\\mathbb{F}_{q}$ ($q$ odd), we generate singular curves $C$ with non-split nodes or cusps. In our approach we use linear projections of geometric embeddings, and we apply Stöhr's embedding of hyperelliptic Gorenstein curves and the Rosa-Stöhr theory of trigonal Gorenstein curves. The construction is applied to the Tafazolian and Tafazolian-Torres smooth maximal curves. Finally, we also determine the gonality of all obtained curves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22557",
  "author_total": 3,
  "authors": [
    {
      "name": "Cesar Hilario"
    },
    {
      "name": "Rodrigo Salomão"
    },
    {
      "name": "Renato Vidal Martins"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5kw42026ajd4p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22557",
  "fulltext_url": null,
  "id": "5kw42026ajd4p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22557",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214009223",
  "pdf_url": "https://arxiv.org/pdf/2609.22557",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5kw42026ajd4p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.22557",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22557"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Maximal singular curves over finite fields from elliptic and hyperelliptic curves",
  "title_text": "Maximal singular curves over finite fields from elliptic and hyperelliptic curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/5kw42026ajd4p",
  "venue": "arXiv",
  "versions": []
}