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Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre.

If I do (relative) log mmp over a klt pair $(X,D)$ what I obtain is a klt pair $(X',D')$ as output.

If I do (relative) mmp over a terminal $X$, I obtain a terminal $X'$ as output.

Now suppose $(X,D)$ is terminal. I would like to apply (relative) log mmp over it to obtain $(X',D')$ terminal as output. Of course this will not happen in general, but I was wondering if there are:

1) sufficient conditions on $(X,D)$ for this to happen, or

2) whether there is/may be some freedom when running MMP to make a choice in some/all the steps to guarantee this.

Probably the answer to both questions is NO, or even worse: 'probably not. I thought I would give it a try, anyway. It is kind of too specific to find it out of a book without learning all MMP (which I am actually trying to do at the same time).

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