Secant varieties of Segre-Veronese varieties ${\mathbb{P}}^m\times {\mathbb{P}}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\ge 3$

Abstract

We prove that for any $m\ge 3$, $n\gg m^3$, all secant varieties of the Segre-Veronese variety ${\mathbb{P}}^m\times {\mathbb{P}}^n$ have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we focus on the superabundant case. We generalize an approach due to Brambilla and Ottaviani into a construction we call the inductant. With this, the proof of non-defectivity reduces to checking a finite collection of base cases, which we verify using a computer-assisted proof.

arXiv comment34 pages, 6 figures. Source code and computational data available at https://github.com/tatarkajenej/SegVer-O-1-2-

References in corpus (6)

• Higher secant varieties of ${\mathbb{P}}^n\times {\mathbb{P}}^1$ embedded in bi-degree $(a,b)$
• Higher secant varieties of ${\mathbb{P}}^n\times {\mathbb{P}}^m$ embedded in bi-degree $(1,d)$
• Most secant varieties of tangential varieties to Veronese varieties are nondefective
• Non-defectivity of Segre-Veronese varieties
• Secant varieties of P^2 x P^n embedded by O(1,2)
• All secant varieties of the Chow variety are nondefective for cubics and quaternary forms