HOMOMORPHISMS OF VECTOR BUNDLES ON CURVES AND PARABOLIC VECTOR BUNDLES ON A SYMMETRIC PRODUCT

Let S-n(X) be the symmetric product of an irreducible smooth complex projective curve X. Given a vector bundle E on X, there is a corresponding parabolic vector bundle V-E* on S-n(X). If E is nontrivial, it is known that V-E* is stable if and only if E is stable. We prove that H-0 (S-n (X), Hom(par)(V-E*, V-F*)) = H-0(X, F circle plus E-v)circle plus(H-0(X, F)circle plus H-0 (X, E-v)). As a consequence, the map from a moduli space of vector bundles on X to the corresponding moduli space of parabolic vector bundles on S-n(X) is injective.