# Finiteness of the Brauer group and torsion-freeness of Picard group of a rationally connected variety over an algebraically closed field

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?

• 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 $m$. – Ariyan Javanpeykar Oct 20 '18 at 19:06
• Also have a look at Remark 4.3 in that paper. – Ariyan Javanpeykar Oct 20 '18 at 22:01