Höhere Sekantenvarietäten und Vektorbündel auf Kurven. (Higher secant varieties and vector bundles on curves) (Q1073126)

The main purpose of the article is to prove the following nice result: Let E be a locally free sheaf of rank 2 on a non-singular irreducible curve of genus g defined over an algebraically closed field of characteristic zero. An invertible (split) subsheaf M of E is called ''maximal'' if \(c_ 1(M)\) is maximal. Let \(s=c_ 1(E)-2c_ 1(M)\) for M maximal. Assume that the following three properties hold: \((1)\quad 2\leq s\leq g-1;\) \((2)\quad h^ 0(X,\det (E\otimes M^{-1}))\leq 1\) for a maximal M; \((3)\quad E\) has only a finite number of maximal subbundles. - Then we have that the number of maximal subbundles is at most equal to \(\sum^{s}_{k=0}\left( \begin{matrix} g-1-k\\ s-k\end{matrix} \right)\left( \begin{matrix} g\\ k\end{matrix} \right)+1.\) The proof uses results and techniques of the author and \textit{M. S. Narasimhan} [Math. Ann. 266, 55-72 (1983; Zbl 0507.14005)] to relate the above result to results about s-secant (s-1)-planes. Such secant planes are studied using Schwarzenberger's theory of secant planes to curves. The latter theory is reviewed in the article.