Numerical degeneracy of two families of rational surfaces

✍ Scribed by Michael L. Catalano-Johnson

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 134 KB
• Volume: 176
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(02)00065-8

✦ Synopsis

Let X ⊂ P N be a closed irreducible n-dimensional subvariety. The kth higher secant variety of X , denoted X k , is the Zariski closure of the union (in P N ) of the linear spaces spanned by k points of X . A simple dimension count shows that dim X k 6 k(n + 1) -1, and that when equality holds, there is a non-empty (Zariski) open subset U ⊂ X k and a positive integer sec k (X ), such that for all z ∈ U , there are exactly sec k (X ) k-secant (k -1)-planes to X through z. Assume that dim X k = k(n + 1) -1, so that sec k (X ) is deÿned. For X k non-linear we expect sec k (X ) = 1, otherwise we say that X k is numerically degenerate. In this paper, we consider the embeddings X of P 2 and P 1 × P 1 by their respective very ample line bundles and classify those k for which X k is numerically degenerate. In the classiÿcation we prove a result of independent interest, showing that a rational normal scroll X (of arbitrary dimension) never has sec k (X ) ¿ 1.