Higher secant varieties of curves and the theorem of Nagata on ruled surfaces (Q1066224)

Let \(X\subset {\mathbb{P}}^ N\) be a nondegenerate algebraic curve over an algebraically closed field of arbitrary characteristic and let \(Sec_ n(X)\) be the closure of the union of all (n-1)-planes of \({\mathbb{P}}^ N\) contaning n smooth points of X (1\(\leq n\leq N)\). The author proves that dim Sec\({}_ n(X)=Min(2n-1,N)\). Using this result he gives a new proof of the following classical theorem of Nagata: the self-intersection number of all sections of a \({\mathbb{P}}^ 1\) bundle over a smooth projective curve Y does not exceed the genus of Y.