Curves in grassmannians and spanned stable bundles

Fix integers g, r, d with g greater than or equal to 2, r greater than or e qual to 2 and d > rg. Let X be a smooth curve of genus g and E a general st able vector bundle on X with rank(E) = r and deg(E) = d. Assume E spanned a nd let h(E) :X --> G(r, H-0 (X, E)) be the induced map into a Grassmannian. Here we give lower bounds on d which assure that the natural map a(E): Lam bda (tau)(H-0 (X, E)) --> H-0(X, det(E)) is surjective and similar results for general subspaces, V, of H-0 (X, E). Use the Plucker embedding to see G (r, H-0 (X, E)) as a subvariety of P(Lambda (tau) (H-0 (X, E))). According to M. TEIXIDOR I BIGAS this type of results are related to the geometry of the curve h(E)(X) subset of P(Lambda (tau) (H-0 (X, E))) and give the dimen sion of the linear span of h(E)(X).