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In MMP, when we get a small extremal contraction $f:X\rightarrow Y$, we will flip it to $f^+:X^+\rightarrow Y$ such that $K_{X^+}+D^+$ is $f^+$-ample. Technically, I understand that is because we want to avoid small contractions which destroy $\mathbb Q$-factorial property. But I don't know if there is a more natural way to understand it.

My Questions:

  1. Geometrically, what does the birational map $\phi:X\dashrightarrow X^+$ look like?

  2. Is there any direct relation between the 'convex cone diagrams' of $X$ and $X^+$?

  3. Flip seems to be a very symmetric operation. So why could we expect things to become better after flip? For example, is that possible when we flip twice, we get back to the original one?

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