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Jan 31, 2013 at 21:22 comment added Sándor Kovács There is also a recent preprint by Kollár, Grothendieck-Lefschetz type theorems for the local Picard group (arxiv.org/pdf/1211.0317v2.pdf). It does not address this question directly, but obtains some nice results on the local Picard group.
Jan 31, 2013 at 12:15 vote accept Francesco Polizzi
Jan 31, 2013 at 12:15 comment added Francesco Polizzi Dear Hailong, thank you again. I find your answer and the included references very satisfactory, so I will accept it. Best, Francesco
Jan 31, 2013 at 9:37 comment added Hailong Dao Dear Francesco, it looks fine to me. By the way, I forgot a more recent reference by Hartshorne-Polini, front.math.ucdavis.edu/1301.3222.
Jan 31, 2013 at 8:58 comment added Francesco Polizzi Dear Hailong, thank you very much for your nice answer! It was the best thing I was hoping for :-) Since I started to work on these questions quite recently, and so I'm always worried of making some trivial mistake, may I ask you if the argument I sketched giving the implication (Question 2 true => Question 1 true) seems to you correct? Best, Francesco
Jan 31, 2013 at 5:56 history answered Hailong Dao CC BY-SA 3.0