Let $X, Y$ be two birational projective varieties which are isomorphic in codimension 1. Suppose $H$ is an ample divisor on $Y$, and $H'$ be its strict transform on $X$, suppose we can run MMP with respect to $K_X + H'$, is it true that there are only flips in this process(i.e. no divisoral or fibre contractions).

I am not sure if I oversimplify the picture, the question comes from the Kawamata's result that flops connect minimal models, where he claimed that running MMP with scaling between minimal models, there are only flips appearing. I can understand the corresponding statement in [BCHM], where they run MMP over a common ample model, hence no divisoral or fibre contractions. But I was wondering if this is a phenomenon only relies on the codimension - i.e. MMP only changes the part where two varieties are non-isomorphism?


You need to be a little careful -- unless you put a coefficient on $H'$, the pair won't be lc in general. But even if it is, this probably won't be true unless $X$ is minimal: a $K_X$ divisorial contraction can be a $K_X+H'$ divisorial contraction also (this will work for any contraction, if you stick a small enough coefficient on $H'$), and so there's a run of the MMP with a divisorial contraction.

  • $\begingroup$ Yes, assume the singularity is good ($X,Y$ are lc, and make a small multiple of $H'$ etc.), then why there is no divisorial/fibre contraction? (I don't understand why $X$ is minimal is used here) $\endgroup$ – Li Yutong Nov 8 '14 at 20:17
  • $\begingroup$ If $X$ is minimal, then any $K_X+H'$ contraction is of an $H'$-negative ray (since $K_X$ nef). But $H'$ has codimension $2$ base locus since it's the strict transform of an ample under a small modification. So a $K_X+H'$-contraction must be flipping. (Again if $X$ isn't minimal this is false.) $\endgroup$ – user47305 Nov 8 '14 at 23:37
  • $\begingroup$ Beautiful!!! I understand it now. Thank you very very much!!! $\endgroup$ – Li Yutong Nov 9 '14 at 1:25

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