{
  "abstract": "For a smooth plane quartic $C$, the $3$-Weierstrass points are the $300$ points (counted with weights) at which some plane cubic has contact order at least ten with $C$. Revisiting the work of Kamel and Elshareef, we write the third Veronese Wronskian as an $\\text{SL}_3$-covariant form $Ω_3$ of degree $75$, given by a determinant of iterated Hamiltonian derivatives, and explain the special $3$-Weierstrass points through contact cubics and homologies preserving $C$. We determine $Ω_3$ for the Fermat, Komiya--Kuribayashi, Klein and Picard quartics; for the Klein quartic $Ω_3\\equivΦ_6^2\\,Φ_{21}\\,(3Φ_{14}^3+21962\\,Φ_6^7)$ modulo the curve, up to a non-zero scalar. Finally, the dectactic points of the Fermat quartic are not $4$-Weierstrass points, which disproves a conjecture of Kamel and Elshareef. All computations are supported by \\texttt{SINGULAR} scripts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30992",
  "author_total": 1,
  "authors": [
    {
      "name": "Ewelina Nawara"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/epcxbnqj1y756.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30992",
  "fulltext_url": null,
  "id": "epcxbnqj1y756",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30992",
  "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": "W7223274144",
  "pdf_url": "https://arxiv.org/pdf/2610.30992",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/epcxbnqj1y756.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30992",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30992"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Third-Order Weierstrass Points on Plane Quartics",
  "title_text": "Third-Order Weierstrass Points on Plane Quartics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/epcxbnqj1y756",
  "venue": "arXiv",
  "versions": []
}