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

Let $X_{(1,d)}^{(n,m)}$ denote the Segre-Veronese embedding of $$\mathbb{P}$^n \times $\mathbb{P}$^m$viathesectionsofthesheaf$$\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 $\left(\genfrac{}{}{0px}{}{m+d}{d}\right)$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X_{(1,d)}^{(n,m)}$ has the expected dimension for every $s$.