{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0111",
  "title": "Low-Degree Curves and Smooth Hilbert Loci on the Fermat Fourfold",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For the degree-d Fermat fourfold in complex projective 5-space, and for delta>=2 with d>10delta-6, we describe the integral degree-delta Hilbert locus as a disjoint union of 15d^3 smooth open plane-form spaces. The point classification applies Salberger's diagonal-curve inequality after a minimal-support descent already present in the frozen research aid. The scheme-level extension follows from a three-term Wronskian lemma: the normal bundle of a standard plane has no sections on any integral degree-delta plane curve when d>delta+3, including singular curves. Reznick's published nonstandard conic on X_14 makes the conic cutoff d>=15 sharp. Combining the published degree and genus inequalities also gives a stronger necessary genus bound for nonstandard curves.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "keywords": [
    "Fermat hypersurface",
    "Hilbert scheme",
    "normal bundle",
    "Wronskian",
    "AIM-ALGEBRAIC_NUMBER_THEORY-0111",
    "math.AG",
    "math.NT"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-number-theory-0111/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-number-theory-0111/paper.pdf?v=5202a6a0b897",
  "doi": "10.5281/zenodo.23031474",
  "zenodo_record_url": "https://zenodo.org/records/23031474",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Not full AIM programme closure; inherited point classification and known conic credited, possible folklore, no absolute priority. Self-audited and unrefereed, with AI assistance; no independent peer review or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "5202a6a0b897c9e872554880af65b95272b8572d1933b950f7654b535fffb57d"
    },
    "source.zip": {
      "sha256": "89ef32adec4ea99a39e2238f4e7e91f91ccb1aa0c6357caae443367cbc328372"
    },
    "verification_report.md": {
      "sha256": "a71aed2f4fcb9f462df0a326f484940e19b27e3eee10ace88d50886e3a0c7f7f"
    }
  },
  "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."
}