Rational curves on moduli spaces of vector bundles

W describe the unobstructed components of the Hilbert Scheme of rational curves of fixed degree [Formula: see text] in the moduli space [Formula: see text] of stable vector bundles of rank [Formula: see text] and determinant [Formula: see text] on a curve [Formula: see text]. We show that for every [Formula: see text], there are [Formula: see text] such components. We construct obstructed components of the Hilbert Scheme. We also obtain an upper bound on the degree of rational connectedness of [Formula: see text] which is linear in the dimension.