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

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

2010-04-15

arxiv.org/abs/1004.2614v1

J. Pure Appl. Algebra. 215, (2011), pp. 2853-2858

Mathematics

Algebraic Geometry

8 pages

Scientific paper

10.1016/j.jpaa.2011.04.005

Let $X^{(n,m)}_{(1,d)}$ 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 $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.

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