A NOTE ON RANK TWO STABLE BUNDLES OVER SURFACES

Let $\pi: X \longrightarrow C$ be a fibration with reduced fibers over curve $C$ and consider polarization $H$ on the surface $X$. $E$ stable vector bundle of rank $2$ $C$. We prove that pullback $\pi^*E$ is $H-$stable This result allows us to relate corresponding moduli spaces bundles $\mathcal{M}_C(2,d)$ $\mathcal{M}_{X,H}(2,df,0)$ through an injective morphism. study induced morphism at level Brill-Noether loci construct examples fibered surfaces. Results concerning emptiness follow as consequence generalization Clifford's Theorem for two