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Probably not a very deep question follows. I struggle to understand the idea of the Picard group. I want to understand when two line bundles are isomorphic.

Is there a nice illustrative example? For example what are the isomorphism classes of the the 1-forms living on the tangent space of a curve (i.e. $f \in \Omega^1(S)$ and $[f]\in H^1(S)$).

Also, are there some easy counter-examples? And what about objects other than line bundles, like vector bundles? For those one usually constructs moduli spaces. Thus can we consider Picard group as being the moduli space of line bundles?

Note: I have the impression that, in the example of 1-forms, we consider two of them isomorphic if they differ by an exact form (I think this makes sense in terms of cohomology). But still, I would like to see a concrete counter-example as well as an example that has different kind of line bundles)

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    $\begingroup$ Over a smooth variety, the Picard group is isomorphic to the divisor class group. This gives a classification of line bundles, and describes fairly explicitly what all line bundles look like. $\endgroup$ – hwong557 Nov 22 '16 at 0:18

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