Higher secant varieties of Segre-Veronese varieties

Abstract

We study the dimension of the higher secant varieties $X^s$ of $\mathbb{X}={\mathbb{P}}^{n_1}\times ...\times {\mathbb{P}}^{n_t}$ embedded the morphism given by ${\mathcal{O}}_{\mathbb{X}}(a_1,...,a_t)$. We call it a {\it Segre-Veronese variety} and the embedding a {\it Segre-Veronese embedding}. In several cases (e.g. for 3 factors {\Bbb P}^{1}, or when $s$ is small) we show that $X^s$ has the expected dimension except for a few cases. A list of examples of defective $X^s$'s is given.

12 pages, plain TEX