Conjecture on non-defectivity for degrees at least (3,3)(3,3)

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Let n=(m,n)∈N2\mathbf{n}=(m,n)\in\mathbb{N}^2 and a=(a,b)∈N2\mathbf{a}=(a,b)\in\mathbb{N}^2, and let Xn,aX_{\mathbf{n},\mathbf{a}} be the two-factor Segre-Veronese variety associated with these parameters. Assume that a≥(3,3)\mathbf{a}\geq(3,3) coordinatewise.

Non-defectivity conjecture for degrees at least (3,3)(3,3). There are no defective two-factor Segre-Veronese varieties Xn,aX_{\mathbf{n},\mathbf{a}} for any n∈N2\mathbf{n}\in\mathbb{N}^2.

This is a weaker consequence suggested by the paper's broader classification conjecture. The paper gives computational evidence and known results supporting it, but does not establish the claim in full.

References

Primary source

Hirotachi Abo and Maria Chiara Brambilla, “On the dimensions of secant varieties of Segre-Veronese varieties”, arXiv:0912.4342 (2011).

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