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Aug 9 at 14:23 vote accept Jooh
Aug 5 at 10:28 comment added Lazzaro Campeotti I forgot to say something about the second question. The answer to that is "no"; if $Y$ is an elliptic curve and $X=Y \times Y$ then its Picard group has rank at least 3 (cf. Hartshorne Ex IV.4.10.)
Aug 5 at 9:19 history answered Lazzaro Campeotti CC BY-SA 4.0