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Is the Picard group of a Fano variety always finitely generted and torsion free?

This is well known over fields of characteristic 0, so the question is about the case of positive characteristic.

Note that there can only be $p$-torsion in characteristic $p$: Picard groups of Fano varieties in positive characteristic

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  • $\begingroup$ Are you primarily interested in smooth projective Fano varieties, or a more general notion? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 23, 2023 at 22:39
  • $\begingroup$ In my paper with Zhiyu Tian and Ruhong Zong, we also have some results if the Fano variety lifts to characteristic $0$. It seems like the hardest case is Fano varieties that do not lift to characteristic $0$. $\endgroup$
    – Jason Starr
    Commented Feb 24, 2023 at 0:30
  • $\begingroup$ @R.vanDobbendeBruyn: Just the smooth projective ones. It would be interesting to know however about related results or counter-examples for rationally (chain) connected varieties. $\endgroup$
    – Daniel Loughran
    Commented Feb 24, 2023 at 9:27
  • $\begingroup$ @JasonStarr: Is this the paper "Weak approximation for Fano complete intersections in positive characteristic"? Where is the result in there? Also what is your intuition? Do you expect the Picard group to be always torsion-free? $\endgroup$
    – Daniel Loughran
    Commented Feb 24, 2023 at 9:45
  • $\begingroup$ Yes, that is the paper. The result that I mean is Corollary 2.4: if the scheme deforms to characteristic $0$, and if $p$ is prime to the "torsion order" and the "uniruling index", then the Picard group is torsion-free. $\endgroup$
    – Jason Starr
    Commented Feb 24, 2023 at 11:48

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