The homogeneous ideals of higher secant varieties

Let image be a projective space over an algebraically closed field of characteristic zero. Let image be a closed, irreducible subvariety, not lying on a hyperplane. The kth higher secant variety of X, denoted Xk, is the closure of the union of all linear spaces spanned by k points of X. We prove that I(Xk), the homogeneous ideal of Xk, is contained in the kth symbolic power of I(X). As a consequence, Xk lies on no hypersurface of degree less than k+1. Furthermore, if X is a curve, and deg Xk=k+1, we prove that X is a rational normal curve.