On non-defectivity of certain Segre-Veronese varieties

Let X(m,n) be the Segre-Veronese variety P(m) x P(n) embedded by the morphism given by O(1,2) and let sigma(s)(X(m,n)) denote the sth secant variety of X(m,n). In this paper, we prove that if m = n or m = n + 1, then sigma(s)(X(m,n)) has the expected dimension except for sigma(6)(X(4,3)). As an immediate consequence, we will give functions s(1)(m, n) <= s(2)(m, n) such that if s <= s(1) (m, n) or if s >= s(2)(m, n), then sigma(s)(X(m,n)) has the expected dimension for all positive integers m and n. (C) 2010 Elsevier Ltd. All rights reserved.