Higher secant varieties of Segre-Veronese varieties

We study the dimension of the higher secant varieties $X^s$ of ${\Bbb X} = {\Bbb P}^{n_1}\times …\times {\Bbb P}^{n_t}$embeddedthemorphismgivenby${\cal O}_{\Bbb X}({a_1,…,a_t})$. 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.