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Let $X$ be a rationally connected variety over an algebraically closed field $k$. Then (1) is the Brauer group $\text{Br}(X)$ finite? (2) is the Picard group $\text{Pic}(X)$ torsion-free?

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  • $\begingroup$ Partial results can be found in arxiv.org/pdf/1703.05735.pdf . For instance, if X is separably rationally connected, then Pic(X) is torsion-free. Also, if X is rationally chain connected, then there is an integer $m>0$ such that Br(X) is killed by $m$. $\endgroup$ – Ariyan Javanpeykar Oct 20 '18 at 19:06
  • $\begingroup$ Also have a look at Remark 4.3 in that paper. $\endgroup$ – Ariyan Javanpeykar Oct 20 '18 at 22:01

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