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Jul 23, 2010 at 13:09 comment added Francesco Polizzi I think that there must be a counterexample, too. But, as it is clear from your interesting answers, finding it could be rather difficult.
Jul 22, 2010 at 20:59 history edited Donu Arapura CC BY-SA 2.5
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Jul 22, 2010 at 12:21 history edited Donu Arapura CC BY-SA 2.5
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Jul 22, 2010 at 12:00 comment added Donu Arapura Good point. The example of a a cone shows that this nonsense as written.
Jul 22, 2010 at 11:57 history edited Donu Arapura CC BY-SA 2.5
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Jul 22, 2010 at 11:54 comment added damiano That's what I was also thinking, but maybe it would be safer to assume that both curves have positive genus, to rule out the possibility of cones over plane curves. In fact, just abelian surfaces might work.
Jul 22, 2010 at 11:50 history answered Donu Arapura CC BY-SA 2.5