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Let $X$ be a smooth proper geometrically connected variety over a field $K$ of characteristic 0. Let $\overline{K}$ denote an algebraic closure of $K$.

What other conditions on $X$ are needed so to guarantee that $Pic(X_{\overline{K}})$ is of finite type?

Here Pic denotes the Picard group.

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  • $\begingroup$ This should almost never happen. Think about the fact that this group is like the $\bar{K}$-points of an abelian variety which have torsion of all orders. $\endgroup$ – Alex Youcis Aug 30 '15 at 7:51
  • $\begingroup$ In particular, it is 23234's suggestion that you need the abelian variety to be dimension $0$ which needs $H^1(X,\mathcal{O}_X)=0$. $\endgroup$ – Alex Youcis Aug 30 '15 at 8:29
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The vanishing of $H^1(X,\mathcal O_X)$ implies that $Pic^0$ is trivial (in your situation, as the base field is of characteristic zero).

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  • $\begingroup$ That's not quite true—in positive characteristic one can have non-reduced Picard scheme. $\endgroup$ – Alex Youcis Aug 30 '15 at 9:11

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