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Is it true that any flop on a Calabi-Yau threefold is given by the Atiyah flop? That is, there always exists a rigid rational curve $\mathbb{P}^1$ with normal bundle $\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus2}$ and the flop is given by contracting the $\mathbb{P}^1$ and "inserting $\mathbb{P}^1$ in other direction".

Here is my thought. First of all, the exceptional locus is of dimension 2, so it must be curve. If it is rational curve, then the normal bundle is either $\mathcal{O}(-1)^{\oplus2}$ or $\mathcal{O}\oplus \mathcal{O}(-2)$, but the latter cannot happen as the curve can move. In the former case, I think, the only possibily is Atiyah flop. Another thing one has to prove is that the exceptional curve cannot have genus greater than 1, but I am stuck here.

Edit As Sasha pointed out below, my claim is not true in general. Let me now ask another question that came up from my initial question and Sasha's answer to it.

Is it true that the exceptional curve is always fixed? (i.e. it is not in a member of non-trivial family of rational curves)

Thank you for your help.

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    $\begingroup$ 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. $\endgroup$
    – Sasha
    Commented Feb 19, 2013 at 3:07
  • $\begingroup$ 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? $\endgroup$
    – Kim
    Commented Feb 19, 2013 at 13:26

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