{
  "schema_version": 1,
  "problem_number": "OWR-1452-024",
  "title": "Counterexamples over F̄_p to a Generic Equivalence Problem of Kraft and Russell",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In Oberwolfach Report 01/2007, Kraft and Russell stated that two morphisms of varieties over an algebraically closed field of infinite transcendence degree, whose fibres over all closed points are isomorphic, become isomorphic after a dominant étale base change, and asked whether this holds over every algebraically closed field, or for counterexamples over F̄_p or Q̄. In 2014 they proved the statement for affine morphisms, with a dominant base change of finite degree. The étale form already fails over every algebraically closed field of positive characteristic, because of Russell's classical purely inseparable forms of the affine line; we record this, and note that it immediately answers a question on positive characteristic raised in a remark of Kaliman. We show that over F̄_p even the finite-degree form fails. For every prime p we give a pair of smooth affine families of threefolds over an open subset Y of the affine line, defined over F_p, whose fibres over each closed point of Y are isomorphic as F̄_p-varieties, but which do not become isomorphic after any base change U → Y whose image contains the generic point, whether it is étale, of finite degree, or neither. The fibres are affine modifications of G_m² × A¹ at two points, and their isomorphism classes are governed by GL₂(Z)-orbits. Over F̄_p every point of G_m² has finite order, and the Frobenius twist stays in the orbit at every closed point but not at the generic point. For p ≡ 1 (mod 4) we give smooth projective families with the same properties: blow-ups of E × E at two points, where E is the curve y² = x³ − x with complex multiplication by Z[i], parametrised by E × E minus the origin or by a curve in it. The case of Q̄ remains open. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "keywords": [
    "Kraft–Russell generic equivalence theorem",
    "isotriviality",
    "families with isomorphic fibres",
    "algebraic closure of a finite field",
    "affine modification",
    "abelian surface",
    "complex multiplication",
    "forms of the affine line",
    "counterexample",
    "Oberwolfach Reports",
    "OWR-1452-024",
    "math.AG",
    "math.NT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1452-024/",
  "pdf_url": "https://eulersolve.org/papers/owr-1452-024/paper.pdf?v=6635bae30e74",
  "doi": "10.5281/zenodo.23062534",
  "zenodo_record_url": "https://zenodo.org/records/23062534",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the F̄_p part of Kraft and Russell's Problem 2 (OWR 01/2007) negatively, for the finite-degree and arbitrary-base-change forms. The failure of the étale form in positive characteristic is classical (Russell's forms of the affine line) and is not claimed as new; the Q̄ case remains open.",
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    "source.zip": {
      "sha256": "96734d2baa51a4bc5374531a667ad53fd0bbe55299562204d2e9f7ca75dd154a"
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    "verification_report.md": {
      "sha256": "f64123d87abc5c008e80b317c50579a88d2333789db14c8a50fe3e94f086b49f"
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  "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."
}