{
  "abstract": "We study the local geometry, singularity classification, and wavefront dynamics of complex evolutoids for regular holomorphic curves in $\\mathbb{C}^2$ equipped with the standard complex bilinear metric. By integrating the generating family of rotated complex affine lines, we derive an explicit parametrization for the 1-parameter family of complex wavefronts $Γ^{θ, w}$ and establish that their singular cusp locus sweeps out the complex evolutoid $E^θ$. Using a square-root-free formulation, we classify the local behavior of $E^θ$, proving that evolutoids remain regular and tangent to the curve at isotropic points ($q=0$) while escaping to infinity at complex inflections ($κ=0$). Finally, we analyze the three-dimensional complex discriminant surface $\\mathcal{D}_F \\subset \\mathbb{C}^3$ formed by the evolving family $E^θ$. We prove that $A_2$ cusps are versally unfolded by spatial parameters, whereas $A_3$ swallowtail singularities are holomorphically versally unfolded by $(x_1, x_2, θ)$ if and only if the complex non-degeneracy condition $κ^4 + κ_s^2 \\neq 0$ holds. Crucially, we demonstrate that simple isotropic points drive the breakdown of this versality condition, inducing degenerate, non-transverse sections of the complex swallowtail surface.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38546",
  "author_total": 2,
  "authors": [
    {
      "name": "Jorge Luiz Deolindo-Silva"
    },
    {
      "name": "Mostafa Salarinoghabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8cxw1vmaqwr0f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38546",
  "fulltext_url": null,
  "id": "8cxw1vmaqwr0f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38546",
  "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.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7215124832",
  "pdf_url": "https://arxiv.org/pdf/2609.38546",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8cxw1vmaqwr0f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38546",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38546"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Complex evolutoids of holomorphic plane curves",
  "title_text": "Complex evolutoids of holomorphic plane curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/8cxw1vmaqwr0f",
  "venue": "arXiv",
  "versions": []
}