{
  "schema_version": 1,
  "problem_number": "AIM-ARITHMETIC_GEOMETRY-0078",
  "title": "A Generically Nonreduced Component for Hilbert Function (1,4,10,10)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give a computer-assisted proof that the very-compressed locus with Hilbert function (1,4,10,10) is the reduced support of a generically nonreduced irreducible component of the Hilbert scheme of 25 points on affine four-space over an algebraically closed field of characteristic zero. The support is A^4 times Gr(10,20) and has dimension 104. This supplies the case a=10 omitted from Jelisiejew's published theorem for (1,4,10,a), a=6,7,8,9, and identifies the generic component throughout the interval in AIM Problem 31. The new computation gives 244 primary-obstruction quadrics in 46 normal variables over F_3. Exact F4 rounds produce a positive pure leading power of every variable, proving that the normal obstruction algebra is zero-dimensional. A properness argument transfers this certificate to characteristic zero, where the published Bialynicki-Birula criterion applies. The general obstruction framework and the four earlier cases are established prior work. We do not compute the generic nilpotent local algebra or claim absolute priority.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "keywords": [
    "Hilbert scheme of points",
    "generically nonreduced component",
    "very-compressed algebra",
    "primary obstruction",
    "Groebner basis",
    "computer-assisted proof",
    "Hilbert function (1,4,10,10)",
    "AIM-ARITHMETIC_GEOMETRY-0078",
    "open mathematics",
    "mathematical proof",
    "math.AG",
    "math.AC"
  ],
  "manuscript_version_date": "2026-09-05",
  "publication_date": "2026-09-05",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-05",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-arithmetic-geometry-0078/",
  "pdf_url": "https://eulersolve.org/papers/aim-arithmetic-geometry-0078/paper.pdf?v=e70bc6b6abdf",
  "doi": "10.5281/zenodo.22328644",
  "zenodo_record_url": "https://zenodo.org/records/22328644",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Computer-assisted a=10 extension of Jelisiejew's published a=6,7,8,9 cases. Two decisive runs use the same patched solver; no independent ideal-membership kernel or computation of the generic nilpotent local algebra is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "e70bc6b6abdf072fd503cb8fc7555e9ba0b286637df5fd0b04c87a2ae5f7491d"
    },
    "source.zip": {
      "sha256": "f88414aa82a6a27a12b9e2c873ffaedf69ec72a16ba257a588503bba7e3e5d60"
    },
    "verification_report.md": {
      "sha256": "a0d6d764632f4864c6d8b364e1798965df81950f614a051bf3d32518394639e0"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}