The Affine Cone over the Second Secant Variety of (P^1)^5 Is Gorenstein: On Three Questions of Michałek

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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14601-005 (Michałek, Toric structures in nontoric varieties, Oberwolfach Report 19/2016, pp. 999–1001).