Higher secant varieties of P^n X P^m embedded in bi-degree (1,d)

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