On non-defectivity of certain Segre-Veronese varieties

Let Xm,n be the Segre-Veronese variety Pm×Pn embedded by the morphism given by O(1,2) and let σs(Xm,n) denote the sth secant variety of Xm,n. In this paper, we prove that if m=n or m=n+1, then σs(Xm,n) has the expected dimension except for σ6(X4,3). As an immediate consequence, we will give functions s1(m,n)≤s2(m,n) such that if s≤s1(m,n) or if s≥s2(m,n), then σs(Xm,n) has the expected dimension for all positive integers m and n.