Smooth projective variety $X$ with $Cl(X)$ not finitely generated

Question

I am trying to find an example of a smooth projective variety $X$ such that $Cl(X)$ is not finitely generated.

I'm actually a bit surprised such an example exists. For example if we can remove finitely many prime components such that we eventually get down to a $\operatorname{Spec}(A)$ for $A$ a UFD, then by exact sequences of the form $\mathbb{Z} \mapsto Cl(X) \rightarrow Cl(X - Z) \rightarrow 0$ we get $Cl(X)$ is finitely generated. In particular we can remove finitely many proper closed subschemes and get the something which also has non f.g. class group. I don't have any intuition for such things, aside from thinking of the class group as something like a homology group, but I have no ideas.

Any help is appreciated.

Answer 1

This is a community wiki post recording the discussion in the comments so that this post may be marked as answered (once this is upvoted or accepted).

If $X$ is regular, its local rings are indeed UFD, but in general, you can't remove closed subset until you get a $\operatorname{Spec} A$ with $A$ a UFD. For example, on an elliptic curve (which is regular), there are no open affine $\operatorname{Spec} A$ such that $A$ is a UFD. In fact, an elliptic curve over $\Bbb C$ (say) is an example of $X$ such that $Cl(X)$ is not finitely generated. – Roland