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

In this paper we compute the dimension of all the higher secant varieties to the Segre-Veronese embedding of ${\mathbb{P}}^n\times {\mathbb{P}}^1$ via the section of the sheaf $\mathcal{O}(a,b)$ for any $n,a,b\in {\mathbb{Z}}^{+}$. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann $(1,s-1)$-defective Veronese varieties.