The homogeneous ideals of higher secant varieties

Abstract

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