# Finite Picard group

Does there exist a connected scheme, smooth, proper, and positive-dimensional over $$\mathbb{C}$$ with finite Picard group? Note that Picard group has cardinality$$>1$$. Also note that this can not happen for projective schemes.

• 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. Apr 26, 2019 at 2:53
• Meta discussion here: meta.mathoverflow.net/questions/4200/flood-of-new-users May 2, 2019 at 15:00