Rational curves on moduli spaces of vector bundles

W describe the unobstructed components of the Hilbert Scheme of rational curves of fixed degree k in the moduli space U-C(r,L) of stable vector bundles of rank r and determinant L on a curve C. We show that for every k, there are gcd(r,deg L) such components. We construct obstructed components of the Hilbert Scheme. We also obtain an upper bound on the degree of rational connectedness of U-C(r,L) which is linear in the dimension.