On the quadratic equations for odeco tensors

Elina Robeva discovered quadratic equations satisfied by orthogonally decomposable ("odeco") tensors. Boralevi-Draisma-Horobe\c{t}-Robeva then proved that, over the real numbers, these equations characterise odeco tensors. This raises the question to what extent they also characterise the Zariski-closure of the set of odeco tensors over the complex numbers. In the current paper we restrict ourselves to symmetric tensors of order three, i.e., of format $n \times n \times n$. By providing an explicit counterexample to one of Robeva's conjectures, we show that for $n \geq 12$, these equations do not suffice. Furthermore, in the open subset where the linear span of the slices of the tensor contains an invertible matrix, we show that Robeva's equations cut out the limits of odeco tensors for dimension $n \leq 13$, and not for $n \geq 14$ on. To this end, we show that Robeva's equations essentially capture the Gorenstein locus in the Hilbert scheme of $n$ points and we use work by Casnati-Jelisiejew-Notari on the (ir)reducibility of this locus.