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Let $f:Y \to X$ be a small contraction morphism of projective normal varieties. Then what is the usual way (if any) to define the flop $f^+: Y^+ \to X$ ?

I could find a clear and uniform way to define flips in the literature, however, for flops things become very confusing.

In the paper "Flops" by Kollár, it defines (Def 2.1) the $D$-flop of $f$, which is the small map $f^+ : Y^+ \to X$ such that $D^+$ (the proper transform of $D$) is $f^+$-ample (there are some other assumptions in the definition which I omit). However, something must be missing in that definition, because by that definition, $f$ itself is the flop of $f$. Moreover, this is the definition of $D$-flop, then what is the definition of flop? Is it the case $D = K_Y$?

In Debarre's "Higher-Dimensional Algebraic Geometry" (see footnote in page 173), the flop is "defined" by a small map which "removes the curve on which the canonical divisor has degree 0 and replaces it with another curve with the same property. However, there is a Cartier divisor that is negative on the first curve and positive on the second".

Last, in Bridgeland's paper "Flops and derived categories", flop is "defined" as "if $D$ is a divisor such that $-D$ is $f$-nef, then its proper transform $D'$ is $f^+$-nef" (See page 12 (4.6)).

Then what are the relations between these definitions? Are they all the same?

Besides, are the flops special case of flips? --- I always confused by this!

Li Yutong
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1 Answers1

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First note that the name of the author is Kollár, not Kollar. I don't speak Hungarian, but apparently there is a big difference!

In fact the definition in Kollár's paper has a typo! He writes

Let $D$ be a divisor on $Z$ such that $D$ is $f$-ample.

It should rather be

Let $D$ be a divisor on $Z$ such that $-D$ is $f$-ample.

Hopefully then you see how the above definitions agree. Maybe it's helpful to consider the dimension 3 case: here we are "cutting out" a curve $C$ that satisfies $K \cdot C =0, \, D\cdot C <0$ and replacing it with a curve $C'$ such that $K \cdot C'=0, \, D^+ \cdot C' >0$.

Next: "Moreover, this is the definition of $D$-flop, then what is the definition of flop? Is it the case $D=K_Y$"? No. A flop is a map which is a $D$-flop for some $D$. But by definition that $D$ can never be $K_Y$, because $K_Y$ is always trivial on the flopping locus (i.e. the subvariety contracted by $f$), whereas we want $-D$ to be $f$-ample on that locus.

Finally: "Besides, are the flops special case of flips?" With the "basic" defintion of flip, the answer is no: a flip should replace a locus on which $K_X$ is negative with one where $K_X$ is positive, whereas a flop changes a locus where $K_X$ is trivial. However, there is a more general notion of flip, called a log flip, where (roughly speaking) for some "good" divisor $D$ on $X$ we replace a locus where $K_X+D$ is negative with one where $K_X+D$ is positive. Then a $D$-flop is indeed a log flip in this sense.

  • Thank you so much!!! May I ask some further questions? (1) So in the definition, $D$ should really be $f$-ample rather than $f$-nef ? (2) It seems in Kollár's definition, he did not require canonical sheaf is trivial on the exceptional loci. (3) Should you write $K \cdot C =0$ in your example? (4) I heard several times that "flop is unique", so the uniqueness of flops really depends on the $D$, and different $D$ could give different flops, am I correct? – Li Yutong Jan 25 '14 at 23:54
  • I should be more precise on (2): it is required that $K_Y = f^*K_X$, this certianlly implies that $K_Y$ is trivial on exceptional loci. However, there is no symmetric requirement on the flop side $Y^+ \to X^+$. – Li Yutong Jan 26 '14 at 00:44
  • Dear Li Yutong: (1) Yes, $-D$ should be f-ample. Bridgeland's description is a little different; he is saying that for any divisor $D$ such that $-D$ is f-nef, the proper transform $D^+$ will be $f^+$-nef. It's not immediately obvious, but it is true, that these are equivalent. For (2), the equalities $K_Y=f^K_X$ and $K_{Y^+} = (f^+)^ K_X$ follow from the fact that $f$ and $f^+$ are small. (Note there is no such thing as $X^+$ in your notation.) –  Jan 26 '14 at 14:28
  • For (3), yes, sorry for my typo! I'll fix it now. For (4), uniqueness means that given $f: Y \rightarrow X$ there is a unique flop $f^+$. But even for a single $D$, there can be many possibilities for $f$. –  Jan 26 '14 at 14:30
  • Thank you very very much!!! Just want to make sure I understand correctly: if $f: Y \to X$ is fixed, no matter which $D$ we choose, the flops $f^+$ are all the same. --- This is really strange since I was guessing the flop (as in the case of flip) might be construct from some finitely generated sheaf of algebra, and it should involved with $D$ somewhere... – Li Yutong Jan 26 '14 at 14:52
  • @LiYutong: that's true, $Y^+$ can be described as $\mathrm{Proj}X(fD)$. The point is that for different choices of $D$ such that $-D$ is $f$-ample, all the varieties $\mathrm{Proj}_X(f_D)$ will be isomorphic (over $X$). –  Jan 26 '14 at 19:07