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Oct 21, 2014 at 15:54 vote accept Li Yutong
Oct 21, 2014 at 15:29 comment added Hacon That is a valid concern; the point is that since $K_X+\Delta +\lambda C$ is $f$-trivial, there is a chance that it is actually a pull-back from $Z$. To check this is not easy; it is essentially a consequence of the BPF thm; see 3.7(4) in Koll\'ar-Mori 1998
Oct 21, 2014 at 15:22 comment added Li Yutong Dear Prof. Hacon, thank you very much for your answer! I have a little concern about the formula $K_X + \Delta + \lambda C = f^*(K_Z + \Delta' + \lambda C')$. When $f: X \to Z$ is a small contraction, could $K_Z + \Delta' + \lambda C'$ no longer $\mathbb{Q}$-Cartier?
Oct 21, 2014 at 14:54 history answered Hacon CC BY-SA 3.0