Is the Picard Group countable?

Question

Is the Picard group of a (smooth, projective) variety always countable?

This seems likely but I have no idea if it's true.

If so, is the Picard group necessarily finitely generated?

The Néron–Severi Theorem asserts that $\mathrm{Pic}(X) / \mathrm{Pic}^0(X)$ is finitely generated if $X$ is a proper smooth variety. So your second question is equivalent to: Is $\mathrm{Pic}^0(X)$ finitely generated? But I don't think that this is true, because otherwise this Theorem would not consider the quotient. Also, I have found several papers whose results have the assumption that the Picard group is finitely generated. So probably it's not automatic. Interesting question by the way. — Martin Brandenburg, Mar 17 '13 at 00:30

score 8 · Accepted Answer · answered Mar 17 '13 at 00:46

No. Even for curves there is an entire variety which parametrizes $Pic^0$(X). It is called the jacobian variety. The jacobian is g dimensional (where g is the genus of the curve), so in particular if g > 0 and you are working over an uncountable field the picard group will be uncountable.

score 5 · Answer 2 · edited Apr 13 '17 at 12:21

5

It is noted here and also in math.SE/56356 that the Picard group of a non-rational curve is not finitely generated.

edited Apr 13 '17 at 12:21

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answered Mar 17 '13 at 00:46

Martin Brandenburg

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Is the Picard Group countable?

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