ON THE CONCEPT OF k-SECANT ORDER OF A VARIETY

For a variety X of dimension n in Pr, r n(k + 1)+k, the kth secant order of X is the number μk(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 μk(X) = 1 unless X is k-weakly defective. Furthermore we give a complete classification of surfaces X ⊂ Pr, r > 3k + 2, for which μk(X) > 1.