Construction of Picard group and set theoretic issue. I am currently learning Algebraic geometry and I came over something that is bothering me. 
I have an algebraic variety and the Picard group over it is defined as the set of classes of line bundles quotiented by the the relation of being isomorphic. 
First of all it seems to me that these classes are actually proper classes and not sets, since I can equip almost any set that has the right cardinal with a structure of line bundle over a given variety. How can I collect them then into a set if they are in fact proper classes?
And even if I do, how can I guarantee that I actually won't get a proper class?

This is a common issue you will have to get used to. It's similar to the observation that zero-dimensional $k$-vector spaces are a proper class, because for any set $A$ we can give $\lbrace A\rbrace$ the structure of a $k$-vector space.
However up to isomorphisms you of course obtain an actual set.
Ususally this is resolved by taking Grothendieck universes. This has the disadvantage that it assumes the existence of certain cardinals, whose existence is supposed to be independent of ZFC. So in order to do this properly you have to add one axiom to your set theory.
