{
  "schema_version": 1,
  "problem_number": "AIM-ARITHMETIC_GEOMETRY-0079",
  "title": "Sharp Interpolation for Equidegree Hilbert–Burch Curves",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let H(t,a) be the Hilbert component of curves in projective three-space defined by the maximal minors of a (t+1)-by-t matrix of degree-a forms, over an algebraically closed field of characteristic zero. We prove that its interpolation number is t(t+1)(binom(a+3,3)-2)/2, exactly half its dimension. Smooth integral curves attain the bound, while the upper bound includes the whole component and its boundary. A bidiagonal specialization yields two triangular normal-evaluation systems indexed by intervals. Normal-section evaluation has maximal rank at every number of general markings on a general smooth member. The complete-intersection and linear cases, and a small quadratic example, are credited to prior work. This is a complete result for a specified infinite equidegree family, not a solution of arbitrary ACM degree matrices or the full AIM Hilbert-component question. Seven-page unrefereed AI-assisted preprint with exact supplementary checks; no independent review or absolute-priority claim.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "keywords": [
    "Hilbert schemes",
    "Hilbert–Burch curves",
    "interpolation",
    "normal bundles",
    "ACM curves",
    "AIM-ARITHMETIC_GEOMETRY-0079"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-arithmetic_geometry-0079/",
  "pdf_url": "https://eulersolve.org/papers/aim-arithmetic_geometry-0079/paper.pdf?v=fea09d52eeee",
  "doi": "10.5281/zenodo.23435412",
  "zenodo_record_url": "https://zenodo.org/records/23435412",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "This theorem determines the interpolation number for all positive t,a in the equidegree Hilbert–Burch family over an algebraically closed field of characteristic zero. The full AIM arbitrary-component problem and arbitrary ACM degree matrices remain unresolved here. Complete-intersection, linear and small quadratic cases are credited to prior work. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed. For a >= 4, this is maximal-rank normal-section evaluation, not nonspecial normal-bundle interpolation.",
  "files": {
    "paper.pdf": {
      "sha256": "fea09d52eeeef27a8cf1d70635b1ec961dc30fb9699932e7e7e7ae76acfe4108"
    },
    "source.zip": {
      "sha256": "880823932eb9560d7e922a92f0d04ec82c8d12af1b64705473166cee41a68326"
    },
    "verification_report.md": {
      "sha256": "1840df98ad658f54e2f1cfa37e3cb73e9ce08d3a9f5774e7a25276d0c1b38c0d"
    }
  },
  "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."
}