  For a vector bundle V over a curve X of rank n and for each integer r in the
range 1 \le r \le 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.

Algebraic Geometry