User:Bencwbrown/sandbox

Moduli space of Vector Bundles
Let $$X$$ be a connected Riemann surface. The Picard group $$H^{1}(X,\mathcal{O})$$ parametrising the isomorphism classes of holomorphic vector bundles on $$X$$, where the group structure is given by the tensor product of line bundles. If we fix the first Chern class -- which we will call degree from now on -- to be d then $$Pic^{d}(X)$$ is the set of equivalence classes of degree d.