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

Abstract

Let $X_{(1,d)}^{(n,m)}$ denote the Segre-Veronese embedding of ${\mathbb{P}}^n\times {\mathbb{P}}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study the dimensions of higher secant varieties of $X_{(1,d)}^{(n,m)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when $\left(\genfrac{}{}{0ex}{}{m+d}{d}\right)$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X_{(1,d)}^{(n,m)}$ has the expected dimension for every $s$.

8 pages

References in corpus (1)

• Higher secant varieties of Segre-Veronese varieties

Cited by in corpus (6)

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