{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0078",
  "title": "Generic Multiplicities of Hilbert Schemes on Thickened Lines",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let T_(n,beta) be the Hilbert scheme of n points on the thickened line defined by y^beta=0 over C. For the component indexed by a partition lambda of n with largest part at most beta, we compute its ambient generic multiplicity as the product of binomial(beta,lambda_i). The proof uses disjoint-support factorization and a cyclic chart in which a monic polynomial divides t^beta; the remainder equations form a weighted complete intersection. In particular, a chart of T_(2,3) has generic local ring K[u]/(u^3), whereas imposing a repeated-root discriminant produces K[u]/(u^2). This distinguishes ambient multiplicity from the multiplicity of a support stratum. We compare with the known multinomial multiplicity of the characteristic-polynomial nilpotent fiber, and describe reduced Jordan strata and their closure obstruction. The stated geometric theorem is complete; the broad AIM categorical cell question remains unresolved. This preprint is unrefereed and makes no absolute-priority or formal-verification claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "keywords": [
    "Hilbert scheme",
    "nonreduced curve",
    "thickened line",
    "generic multiplicity",
    "Jordan stratum",
    "nilpotent fiber",
    "monic divisor",
    "complete intersection",
    "AIM-ALGEBRAIC_GEOMETRY-0078"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0078/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0078/paper.pdf?v=ec24da5986f2",
  "doi": "10.5281/zenodo.23377436",
  "zenodo_record_url": "https://zenodo.org/records/23377436",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The formula computes ambient generic lengths for the actual thickened-line Hilbert scheme, not lengths after imposing a support-discriminant slice. The known component classification and characteristic-polynomial multinomial formula are credited. The broad AIM categorical cell question remains unresolved. Self-audited, unrefereed and AI-assisted; no independent review, formal verification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "ec24da5986f235adb1726ffab9f897c3cab60a5a41a99e78eb91872e4f66ea23"
    },
    "source.zip": {
      "sha256": "e613aa6d911fdb5d2f4190b8f9e24786fcb8f7fea3656bfa73ab22d6676da711"
    },
    "verification_report.md": {
      "sha256": "ec3fc0da2521cd8c56a47a5a7c12254c75e5eee5cce50ee6270c95eaeab8db2d"
    }
  },
  "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."
}