{
  "schema_version": 1,
  "problem_number": "AIM-ARITHMETIC_GEOMETRY-0050",
  "title": "Frobenius Factorization Loci for Gauss Maps of Smooth Plane Curves",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct canonical closed subschemes representing relative Frobenius factorization of a morphism from a smooth projective curve family, including over nonreduced bases. For Gauss maps of smooth plane curves of degree d in characteristic p, we identify these subschemes with explicit linear coefficient loci. For q = p^e > 2, the eth locus is empty unless q divides d - 1; otherwise it is the smooth open in the space of equations sum_i X_i R_i(X_0^q, X_1^q, X_2^q), of dimension 3 binomial((d-1)/q + 2, 2) - 1. The classification on geometric points is due to Pardini and Homma. Our scheme-level argument uses the pulled-back Euler sequence, Serre duality and a regular-sequence calculation to rule out infinitesimal thickenings. We obtain dimensions and geometric irreducibility of exact-height strata, handle the characteristic-two exception, and give explicit smooth families with height jumps.\n\nThese are complete results for the specified relative factorization and smooth-plane scopes associated with AIM-ARITHMETIC_GEOMETRY-0050. They do not construct characteristic-only components of space-curve Hilbert schemes or close the full AIM source question. The preprint is AI-assisted, self-audited and unrefereed. Independent review and proof-assistant formalization are not claimed; novelty and absolute priority remain undetermined.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG"
  ],
  "keywords": [
    "Gauss maps",
    "Frobenius factorization",
    "smooth plane curves",
    "inseparability",
    "Hilbert schemes",
    "AIM-ARITHMETIC_GEOMETRY-0050"
  ],
  "manuscript_version_date": "2026-10-05",
  "publication_date": "2026-10-05",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-05",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-arithmetic_geometry-0050/",
  "pdf_url": "https://eulersolve.org/papers/aim-arithmetic_geometry-0050/paper.pdf?v=af108ba1284d",
  "doi": "10.5281/zenodo.23168593",
  "zenodo_record_url": "https://zenodo.org/records/23168593",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete results for the relative factorization and smooth-plane scopes associated with AIM-ARITHMETIC_GEOMETRY-0050 and Problem 3 of the 2010 AIM Hilbert Schemes workshop. The geometric plane classification is credited to Pardini and Homma. No characteristic-only component of a space-curve Hilbert scheme is constructed, and the full source question is not closed. Written general proofs are supplemented by 136 portable exact mathematical checks; the private dossier additionally verifies eight source files. AI-assisted, self-audited and unrefereed; no independent review or proof-assistant formalization is claimed. Novelty and absolute priority remain undetermined.",
  "files": {
    "paper.pdf": {
      "sha256": "af108ba1284d437475ac6a8f2363eefb65cca7c738648e1212948b34396195ab"
    },
    "source.zip": {
      "sha256": "1e22d3497799b8b4d24d6eb102fb64addb160c69af48682319322a189fabd3ae"
    },
    "verification_report.md": {
      "sha256": "8057ffcaa337da7b8560364c198c621ac0837e0af2576fc407533d2df092cd44"
    }
  },
  "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."
}