The Picard Group and Picard Variety

Given a topological space, T, you can define the Picard Group, Pic(T). The Picard Variety is also sometimes Pic(T).

You can build a fiber bundle by attaching 1D vector spaces at every point on T. You would get a line bundle.

Some of the line bundles will be isomorphic to each other. The set of all isomorphism classes of these line bundles forms the Picard Group. The group operation is the tensor product.

If you parametrize these isomorphism classes (13 params, 18 param, or 389213912, doesn't matter) into, say, 13 parameters, and somehow plotted them, you would get a 13D object. This is the Picard Variety.