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In many articles of the minimal model program the authors work with sub-lc pairs instead of lc-pairs. In other words, they consider non-necesarilly effective boundary divisors $B$.

Is it expected (conjectured) that any sequence of flips for sub-lc pairs terminates?

Is this known for 3-folds? If yes, what is the standard reference for this?

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    $\begingroup$ I doubt that. Assuming that, for simplicity, $X$ is smooth, then any divisor $D$ can be written as $K_X+H_1-H_2$ where $H_1, H_2$ are very ample divisors. This pair is of course a sub-lc pair. So your question is basically asking the termination of $D$-MMP for arbitraty divisor $D$ on $X$, which is definitely not true. $\endgroup$
    – Chen Jiang
    Commented Nov 29, 2017 at 14:17
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    $\begingroup$ To elaborate slightly: take your favorite infinite sequence of flops (if you don't already have a favorite, the classic example is due to Kawamata, and you can find it exposited in Reid's "minimal models of canonical 3-folds": it's a threefold elliptically fibered over a surface, and you keep alternately flopping the two components of a reducible fiber). It's likely that your sequence is an infinite sequence of $D$-flips for some suitable non-effective $D$, and so you can't expect termination. $D$ can even be pseff and exhibit this sort of behavior (this is the case in the Kawamata example). $\endgroup$
    – user47305
    Commented Nov 30, 2017 at 21:19

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