Higher secant varieties of ${\mathbb{P}}^n\times {\mathbb{P}}^1$ embedded in bi-degree $(a,b)$

Abstract

In this paper we compute the dimension of all the higher secant varieties to the Segre-Veronese embedding of ${\mathbb{P}}^n\times {\mathbb{P}}^1$ via the section of the sheaf $\mathcal{O}(a,b)$ for any $n,a,b\in {\mathbb{Z}}^{+}$. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann $(1,s-1)$-defective Veronese varieties.

19 pages

References in corpus (3)

• Ideals of varieties parameterized by certain symmetric tensors
• On the dimensions of secant varieties of Segre-Veronese varieties
• Higher secant varieties of ${\mathbb{P}}^n\times {\mathbb{P}}^m$ embedded in bi-degree $(1,d)$

Cited by in corpus (5)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• Four lectures on secant varieties
• Secant non-defectivity via collisions of fat points
• Generic identifiability of pairs of ternary forms
• Secant varieties of Segre-Veronese varieties ${\mathbb{P}}^m\times {\mathbb{P}}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\ge 3$