On the concept of k-secant order of a variety

For a variety X of dimension n in P-r, r >= n(k + 1) + k, the kth secant order of X is the number mu(k)(X) of (k + 1)-secant k-spaces passing through a general point of the kth secant variety. We show that, if r > n(k + 1) + k, then mu k(X) = 1 unless X is k-weakly defective. Furthermore we give a complete classification of surfaces X subset of P-r, r > 3k + 2, for which mu(k) (X) > 1.