# Recognizing a Mukai flop

Let us first review the usual construction of a Mukai flop. Suppose $M$ is a smooth $2m$-dimensional projective variety over $\mathbb C$ containing a closed $m$-dimensional subvariety $W$, and suppose we have a projective birational morphism $\nu:M\to\overline M$ such that $\nu$ is an isomorphism away from $W$, and such that the restriction of $\nu$ to $W$ is a projective bundle $W\cong \mathbb P(\mathcal V)\to Y$, where $\mathcal V$ is a vector bundle on a smooth subvariety $Y\subset \overline M$. Assume further that the normal bundle $N_{W/M}\to W$ restricts to the cotangent bundle on each fibre of this projective bundle.

We can then perform the Mukai flop of $M$ along $W$ as follows: let $\widetilde M\to M$ be the blowup of $M$ along $W$. This has exceptional divisor $E = \mathbb P(N_{W/M})$. By the assumption on the normal bundle made above, using the Euler sequence we have an embedding

$$E\subset \mathbb P(\mathcal V)\times \mathbb P(\mathcal V^*)$$ which is described on each fibre over $y\in Y$ as the incidence variety

$$\{(L,H)\in \mathbb P(\mathcal V_y)\times \mathbb P(\mathcal V_y^*): L\subset H\}.$$ It can then be shown that there exists a variety $M'$ and a birational morphism $\widetilde M\to M'$ which is an isomorphism away from $E$ and whose restriction to $E$ is the projection onto $\mathbb P(\mathcal V^*)$ with image $W'\cong\mathbb P(\mathcal V^*)$, and we have a projective birational morphism $\nu':M'\to \overline M$ which is an isomorphism away from $W'$ and restricts to $\mathbb P(\mathcal V^*)\to Y$ on $W'$. The birational morphism $M\dashrightarrow M'$ is the Mukai flop of $M$ along $W$.

My question is as follows. Suppose we had all the hypotheses in the first paragraph above, and suppose we are given a projective birational map $\nu':M'\to \overline M$ which is an isomorphism away from $Y$ and which restricts to the dual projective bundle $\mathbb P(\mathcal V^*)\to Y$ away from $Y$. Can we then conclude that $M\dashrightarrow M'$ is the Mukai flop along $W$? If not, are there some additional conditions that we can impose on $\nu'$ which make it true?

Thanks kindly.

• Imagine that the vector bundle $\mathcal{V}$ is self-dual. Then you can take $M' = M$, $\nu' = \nu$. But clearly this is NOT the flop. This shows that you definitely need an additional condition. – Sasha Mar 27 '14 at 16:10
• @Sasha When you say "self-dual", do you mean that there is an isomorphism $\mathcal V\cong\mathcal V^*$? Would we not need to somehow extend the isomorphism $\mathbb{P}(\mathcal V)\cong\mathbb{P}(\mathcal V^*)$ to an automorphism of $M$ and twist $\nu$ by it? – bradhd Apr 8 '14 at 17:46
• No, I want to say the following. Assume that $\mathcal{V} \cong \mathcal{V}^*$. Then the morphism $M \to \bar{M}$ (your initial data) satisfies all your assumptions for $M' \to \bar{M}$. But clearly it does not give you a flop. So, in a sense, the main problem is to distinguish between $M$ and $M'$. This is usually done (as Sandor explains) by considering additional divisor $D$ which is assumed to be positive for one morphism and negative for the other. – Sasha Apr 8 '14 at 18:38

If you consider your flop a $(K+D)$-flip, then you might find the additional condition you need. Any flop is actually a $(K+D)$-flip for a well chosen $D$. (For the definition of a $(K+D)$-flip see 3.33 of Birational Geometry of Algebraic Varieties  By Kollár-Mori).
In this case, $\det N_{W/M}$ (or rather an associated divisor) seems like a good candidate for $D$. An easy calculation shows that $(\det N_{W/M})^{-1}$ is $\nu$-ample. If you can verify that it's strict transform on the Mukai flop is $\nu'$-ample, then it would work as $D$. In any case, you want to find a divisor $D$ for which $-D$ is $\nu$-ample and its strict transform on the Mukai flop is $\nu'$-ample.
If you have such a $D$, then the condition that its strict transform on $M'$ is $\nu'$-ample implies that $\nu'$ is the flop you you want (in this case the Mukai flop), because with this additional condition the flip is unique by (for example) Lemma 6.2 of the above book by Kollár-Mori.