{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0318",
  "title": "A Nodal Decic Counterexample to the Generic Projection Incidence Bound",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct a degree-ten hypersurface in complex projective four-space with exactly sixty ordinary double points whose general plane projection lies on a sextic. Thus the projected nodes violate the proposed bound k(n-1), since 60 > 6(10-1) = 54, even though 60 < (10-1)^2. This answers negatively the projection question in Cheltsov's Problem 18 in an AIM workshop list and gives a counterexample to Cheltsov-Park Conjecture 1.6. The node set satisfies the stronger ambient incidence bound 7k for curves of degree k. We use the known sextic-projection property of the projectivized H4 roots, realize those points as nodes of an explicit symmetric decic surface, and remove every unwanted singularity after extending to a threefold. Exact Hessian, Groebner-basis and interpolation certificates accompany the construction. The threefold is factorial; the separate sharp factoriality theorem is not contradicted. This is an AI-assisted, self-audited, unrefereed preprint; it makes no independent-review, formal-certification or absolute-priority claim.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AG"
  ],
  "keywords": [
    "AIM-ALGEBRAIC_GEOMETRY-0318",
    "nodal hypersurfaces",
    "generic projection",
    "H4 root configuration",
    "ordinary double points",
    "counterexample"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0318/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0318/paper.pdf?v=22286ed43968",
  "doi": "10.5281/zenodo.23453999",
  "zenodo_record_url": "https://zenodo.org/records/23453999",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the projection subquestion of AIM Problem 18 and Cheltsov-Park Conjecture 1.6, including the small-node and ambient-incidence hypotheses. The broader factoriality classification is not resolved. The H4 sextic projection and sharp factoriality theorem are known and credited. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "22286ed439687766cd3d9d1a298163271191632fa05c967efaf658b5deb90957"
    },
    "source.zip": {
      "sha256": "9e1e1924268196d80c0780c9dabe79d206bc53f5259f6283a31da035fb1c84b2"
    },
    "verification_report.md": {
      "sha256": "88e057ea4c1e209e9cce31dc9e60a821df0aac382f865c66c93bdca818e86be9"
    }
  },
  "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."
}