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Oct 25, 2012 at 16:35 history edited Sándor Kovács CC BY-SA 3.0
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Oct 12, 2012 at 2:39 comment added Sándor Kovács Actually, I think that verifying that the pre-image of $E$ is smooth should not be too hard as its pre-image ought to be just the blow up of $E$ at those points, which are smooth points, so the blow-up remains smooth. Then its a smooth Cartier divisor, so the ambient space has to be smooth along it, but it is already smooth everywhere else.
Oct 11, 2012 at 19:11 comment added HNuer Yeah, in my head the only question that was bothering me was the smoothness. It seemed clear to me that the exceptional locus would just be $\mathbb P^1$'s which wouldn't contribute to the discrepancy. As usual you have explained things very clearly.
Oct 11, 2012 at 16:10 history answered Sándor Kovács CC BY-SA 3.0