Timeline for Is a flop on Calabi-Yau threefolds always Atiyah flop?
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| Feb 21, 2013 at 9:32 | history | edited | Kim | CC BY-SA 3.0 |
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| Feb 19, 2013 at 13:26 | comment | added | Kim | Thank you for the comment, Sasha. You could have posted your comment as an answer! I now see that my argument on $\mathcal{O}\oplus\mathcal{O}(-2)$ makes sense only up to first order. Is it then true that the exceptional curve is always fixed? (i.e. it is not in a member of non-trivial family of rational curves) Could you kindly explain why the exceptional loci must be swap by rational curves? | |
| Feb 19, 2013 at 3:07 | comment | added | Sasha | Your argument forbidding the curve to have $N = O \oplus O(-2)$ is wrong. There are examples of flops in such curves. The reason is that although such curve has a nontrivial tangent space $H^0(N)$ to the deformation, it also has an obstruction $H^1(N)$ which prevents the curve from deforming. On the other had, the positive genus is impossible, since exceptional loci of birational morphisms are always swept by rational.curves. | |
| Feb 18, 2013 at 20:19 | history | edited | Kim | CC BY-SA 3.0 |
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| Feb 18, 2013 at 20:04 | history | asked | Kim | CC BY-SA 3.0 |