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.



Recognizing a Mukai flop
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](http://books.google.com/books?id=YrysxvPbBLwC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)).


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](http://books.google.com/books?id=YrysxvPbBLwC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false) of the above book by Kollár-Mori.


