On the quadratic equations for odeco tensors

Biaggi, Benjamin; Draisma, Jan; Seynnaeve, Tim (2022). On the quadratic equations for odeco tensors. Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 54 Università di Trieste 10.13137/2464-8728/34260

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Elina Robeva discovered quadratic equations satisfied by orthogonally decomposable (“odeco”) tensors. Boralevi-DraismaHorobet¸-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×n×n. By providing an explicit counterexample to one of Robeva’s conjectures, we show that for n ≥ 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 ≤ 13, and not for n ≥ 14. 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-JelisiejewNotari on the (ir)reducibility of this locus.

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