Does there exist a connected scheme, smooth, proper, and positivedimensional over $\mathbb{C}$ with finite Picard group? Note that Picard group has cardinality$>1$. Also note that this can not happen for projective schemes.
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4$\begingroup$ No, as shown in this mathoverflow.net/a/122718/39304 answer any such variety admits a divisor $D$ which intersects some curve $C$ positively. The class of $D$ in the Picard group is cannot be torsion because if $nD$ was linearly equivalent to $0$, the intersection $nD\cdot C=n(D\cdot C)$ would vanish. $\endgroup$– SashaPApr 26, 2019 at 2:53

1$\begingroup$ Meta discussion here: meta.mathoverflow.net/questions/4200/floodofnewusers $\endgroup$– Steven LandsburgMay 2, 2019 at 15:00
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