TITLE: Is the Picard Group countable?
QUESTION [7 upvotes]: Is the Picard group of a (smooth, projective) variety always countable? 
This seems likely but I have no idea if it's true. 
If so, is the Picard group necessarily finitely generated?

REPLY [7 votes]: No. Even for curves there is an entire variety which parametrizes $Pic^0$(X). It is called the jacobian variety. The jacobian is g dimensional (where g is the genus of the curve), so in particular if g > 0 and you are working over an uncountable field the picard group will be uncountable.

REPLY [4 votes]: It is noted here and also in math.SE/56356 that the Picard group of a non-rational curve is not finitely generated.