Moduli spaces of stable vector bundles of rank 2 on complex surfaces have been studied by several authors. The structures of the moduli spaces of stable bundles on surfaces such as rational surfaces([Ba],[Hu],[DP]), ruled surfaces([Br],[Q1]), K3 surfaces([Mu1,2],[Ty1,2]), elliptic surfaces([FM],[F],[OV]) and some surfaces of general type([Bh],[DK]) have been described. In this paper, we study the possible types of the moduli spaces of stable vector bundles of rank two on the surfaces with the big Picard group. Good examples with that property are Enriques surfaces. We classify the possible types of them on Enriques surfaces. The unversal covering space of an Enriques surface is a K3 surface. So, we apply our methods to these surfaces. Furthermore, Every Enriques surface is elliptic. In principle, these methods can also be applied to any other elliptic surface with a section.
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