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Let $X$ be a smooth projective variety over a finite field $k$. Is the Picard group finitely generated? Equivalently, is $\text{Pic}^0(X)$ finitely generated?

(I am not assuming $k$ is separably closed, so I don't see an immediate relation to the Picard scheme and the Mordell-Weil Theorem).

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    $\begingroup$ $\mathrm{Pic}^0(X)$ is finite, since it is a subgroup of $\underline{\mathrm{Pic}}^0_{X/k}(k)$, and $\underline{\mathrm{Pic}}^0_{X/k}$ (the unit component of the Picard scheme of $X$) is a $k$-group scheme of finite type and thus has only finitely many $k$-points. $\endgroup$ – Laurent Moret-Bailly Feb 25 '18 at 8:35

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