Secant varieties and Hirschowitz bound on vector bundles over a curve

Abstract

For a vector bundle V of rank n over a curve X and for each integer r in the range 1 ≤ r ≤ n − 1, the Segre invariant s r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we generalize Lange and Narasimhan’s results on rank 2 bundles which related the invariant s 1 to the secant varieties of the curve inside certain extension spaces. For any n and r, we find a way to get information on the invariant s r from the secant varieties of certain subvariety of a scroll over X. Using this geometric picture, we obtain a new proof of the Hirschowitz bound on s r .