# When is the geometric Picard group $Pic(X_{\overline{K}})$ of finite type?

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.

• 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. – Alex Youcis Aug 30 '15 at 7:51
• In particular, it is 23234's suggestion that you need the abelian variety to be dimension $0$ which needs $H^1(X,\mathcal{O}_X)=0$. – Alex Youcis Aug 30 '15 at 8:29

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).