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.

Thank you for your help.

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.

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".

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.

Thank you for your help.