{
  "schema_version": 1,
  "problem_number": "OWR-14601-005",
  "title": "The Affine Cone over the Second Secant Variety of (P^1)^5 Is Gorenstein: On Three Questions of Michałek",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In his abstract in Oberwolfach Report 19/2016, Michałek asked whether the affine cone over the second secant variety σ₂ of the Segre product P¹ × P¹ × P¹ × P¹ × P¹ is Gorenstein. We prove that it is. The homogeneous coordinate ring R of σ₂((P¹)⁵) ⊂ P³¹, the variety of binary tensors of border rank at most two, is Cohen–Macaulay and Gorenstein, with canonical module R(−4), Hilbert series (1 + 20t + 210t² + 772t³ + 1250t⁴ + 772t⁵ + 210t⁶ + 20t⁷ + t⁸)/(1 − t)¹², degree 3256 and Castelnuovo–Mumford regularity 8. The proof combines the toric chart of Michałek, Oeding and Zwiernik, on which the variety is a product of an affine space and a Gorenstein toric variety; the decomposition of the coordinate ring due to Raicu, from which we derive the Hilbert series in all degrees; and standard tools, namely Kawamata–Viehweg vanishing for varieties with rational singularities and Serre duality. The identity ω = O(−4) for the canonical sheaf is also proved directly, by gluing the toric charts. Michałek, Oeding and Zwiernik had shown that the variety is Gorenstein locally; Oeding (2016) conjectured the arithmetic property. To our knowledge the theorem and its proof are new; no proof that this cone is Gorenstein was found in the literature accessible to us, and a search that finds nothing is not a proof of novelty. The same abstract asks two further questions. Whether the convex hull of the lattice points in any ball is a normal polytope: the answer is no, by a companion note (a ball in R⁶, and examples in every dimension at least six). Whether, for a homogeneous polynomial f and a generic linear subspace L, the closure of ∇f(L) misses L^⊥: the answer is yes over C, as the source itself records, attributing the observation to B. Teissier; we add a short self-contained proof and an example showing that it fails in characteristic p dividing deg f, and claim nothing new for this part. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "keywords": [
    "secant varieties",
    "Segre varieties",
    "binary tensors",
    "border rank",
    "arithmetically Gorenstein",
    "Cohen–Macaulay rings",
    "Hilbert series",
    "secant cumulants",
    "toric geometry",
    "canonical sheaf",
    "lattice polytopes",
    "gradient map",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-14601-005",
    "math.AG",
    "math.AC",
    "open mathematics"
  ],
  "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/owr-14601-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-14601-005/paper.pdf?v=73335ec9b137",
  "doi": "10.5281/zenodo.23377176",
  "zenodo_record_url": "https://zenodo.org/records/23377176",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the three questions of M. Michałek (Oberwolfach Report 19/2016): (a) the affine cone over the second secant variety of (P^1)^5 is Gorenstein, with canonical module R(-4), Hilbert series and degree 3256 (proved here, new to our knowledge; Oeding conjectured it in 2016); (b) the lattice hull of a ball need not be normal, by the companion note on record OWR-15587-006; (c) the gradient question has the answer yes over C, as the source itself records (Teissier); nothing new is claimed for (c). Novelty is not certified.",
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    "source.zip": {
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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."
}