{
  "schema_version": 1,
  "problem_number": "OWR-12697689-012",
  "title": "An Arc Hilbert-Poincaré Criterion for Rational Double Points",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a normal complex algebraic surface germ, we prove that its focused scheme-theoretic arc Hilbert-Poincare series equals (1-q)^(-3) times the product of (1-q^j)^(-2) for j >= 2 if and only if it is an ADE rational double point. The known forward formula is complemented by a converse: low-weight coefficients recover a hypersurface of multiplicity two, and a graded annihilator argument forces every finite prefix of its jet equations to be a regular sequence. The resulting fiber dimension bound gives irreducibility of all jet schemes, so Mustata's criterion applies. The same argument gives a converse for isolated normal hypersurface singularities of maximal canonical multiplicity. Normality is essential for the surface statement: the known normal-crossing example xy=0 already has the same series. The forward formulas and that example are attributed to Bruschek, Mourtada and Schepers. This four-page English preprint addresses the explicitly normal category of UnsolvedMath OWR-12697689-012; it does not claim a first historical or unrestricted source-question resolution. Exact rational computations through weight ten supplement the written proof. AI-assisted and self-audited; not independently peer reviewed or formally verified, and no exhaustive priority certification is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "keywords": [
    "OWR-12697689-012",
    "arc Hilbert-Poincare series",
    "rational double points",
    "ADE singularities",
    "jet schemes",
    "regular sequences",
    "canonical singularities"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12697689-012/",
  "pdf_url": "https://eulersolve.org/papers/owr-12697689-012/paper.pdf?v=3e2a3da445bf",
  "doi": "10.5281/zenodo.23662448",
  "zenodo_record_url": "https://zenodo.org/records/23662448",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A normal complex algebraic surface germ has focused scheme-theoretic arc Hilbert-Poincare series (1-q)^(-3) product_{j>=2}(1-q^j)^(-2) if and only if it is an ADE rational double point. More generally, for an isolated normal d-dimensional hypersurface of multiplicity d, the maximal-canonical-multiplicity product is equivalent to canonical singularities. Complete characterization for normal complex algebraic surfaces and an isolated hypersurface extension. The forward formulas and the nonnormal example xy=0 are already in Bruschek-Mourtada-Schepers. No unrestricted or first historical source-question resolution is claimed. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "3e2a3da445bfaebafd5b569e5a7d06348b55b2d19f1b7f4c4c3544754e4828b4"
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    "source.zip": {
      "sha256": "14b5ed9f3c41f1a7b7b3b5840a0f024d1f78e9a30dc7288f0583b232b94b9f94"
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    "verification_report.md": {
      "sha256": "11b881ee9a72c6bea0ab7d86b5fcc03b056a48202fdcdce7e2fb47121bbb203c"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}