Numerical degeneracy of two families of rational surfaces

Let be a closed irreducible n-dimensional subvariety. The kth higher secant variety of X, denoted Xk, is the Zariski closure of the union (in ) of the linear spaces spanned by k points of X. A simple dimension count shows that dim Xk k(n+1)−1, and that when equality holds, there is a non-empty (Zariski) open subset U Xk and a positive integer seck(X), such that for all z U, there are exactly seck(X) k-secant (k−1)-planes to X through z. Assume that dim Xk=k(n+1)−1, so that seck(X) is defined. For Xk non-linear we expect seck(X)=1, otherwise we say that Xk is numerically degenerate. In this paper, we consider the embeddingsX of and by their respective very ample line bundles and classify thosek for which Xk is numerically degenerate. In the classification we prove a result of independent interest, showing that a rational normal scroll X (of arbitrary dimension) never has seck(X)>1.