{
  "abstract": "We develop a theory of nonlinear neighborhoods of smooth projective curves of genus at least two, constructed by gluing products of open subsets of a curve with a disc using nonlinear changes of the disc coordinate. Their zero sections have normal bundles of negative degree and contract to normal surface singularities. For the minimal good resolution $(X,E) \\to (V,0)$ of a normal complex surface singularity other than a rational double point, the Wahl defect is defined by $Δ(V,0) := h^1(\\mathcal{O}_X)-h^1(S_X)+h^1(\\bigwedge^2S_X)$, where $E$ is reduced and $S_X = Θ_X(-\\log E)$. We give a finite formula for this defect in terms of extension obstructions when $E$ is a single smooth curve. For our constructed neighborhoods, the formula involves extensions of functions alone, includes obstructions at every order, and gives a sharp lower bound depending only on the genus. We apply the theory to hyperelliptic curves and smooth plane curves. Both families consist of singularities that are neither $\\mathbb{Q}$-Gorenstein nor quasi-homogeneous. The hyperelliptic examples realize every nonpositive integer as a Wahl defect and attain the genus bound in every genus at least two. They disprove Wahl's Main Conjecture, which asserts nonnegativity for non-Gorenstein singularities, with equality precisely in the quasi-homogeneous case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21093",
  "author_total": 2,
  "authors": [
    {
      "name": "Heesang Park"
    },
    {
      "name": "Dongsoo Shin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cjhdmkaavve7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21093",
  "fulltext_url": null,
  "id": "1cjhdmkaavve7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21093",
  "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"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7222703217",
  "pdf_url": "https://arxiv.org/pdf/2610.21093",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cjhdmkaavve7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.21093",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21093"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Nonlinear neighborhoods of smooth curves and arbitrarily negative Wahl defects",
  "title_text": "Nonlinear neighborhoods of smooth curves and arbitrarily negative Wahl defects",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/1cjhdmkaavve7",
  "venue": "arXiv",
  "versions": []
}