Numerical degeneracy of two families of rational surfaces

AbstractLet X⊂PN be a closed irreducible n-dimensional subvariety. The kth higher secant variety of X, denoted Xk, is the Zariski closure of the union (in PN) of the linear spaces spanned by k points of X. A simple dimension count shows that dimXk⩽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 dimXk=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 embeddings X of P2 and P1×P1 by their respective very ample line bundles and classify those k 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