Higher secant varieties of Pn×Pm embedded in bi-degree (1,d)

Let denote the Segre–Veronese embedding of Pn×Pm via the sections of the sheaf O(1,d). We study the dimensions of higher secant varieties of and we prove that there is no defective sth secant variety, except possibly for n values of s. Moreover, when is a multiple of (m+n+1), the sth secant variety of has the expected dimension for every s.