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 \(\sigma_2\) of the Segre product \(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\) is Gorenstein. We prove that it is. The homogeneous coordinate ring \(R\) of \(\sigma_2((\mathbb{P}^1)^5)\subset\mathbb{P}^{31}\), the variety of binary tensors of border rank at most two, is Cohen–Macaulay and Gorenstein, with canonical module \(R(-4)\), Hilbert series \((1+20t+210t^2+772t^3+1250t^4+772t^5+210t^6+20t^7+t^8)/(1-t)^{12}\), 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 \(\omega=\mathcal{O}(-4)\) 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 \(\mathbb{R}^6\), and examples in every dimension at least six). Whether, for a homogeneous polynomial \(f\) and a generic linear subspace \(L\), the closure of \(\nabla f(L)\) misses \(L^\perp\): the answer is yes over \(\mathbb{C}\), 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.