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When $X$ is a smooth projective variety, one can use Mori's bend-and-break trick to establish the cone theorem. However, when $X$ has singularity (say klt. singularity), the cone theorem is obtained by a series of hard results: vanishing theorem -> non-vanishing theorem -> rationality theorem -> cone theorem.

I was wondering what would go wrong when we follow the argument in the smooth case with some simple minded modifications --- like using resolution or cyclic covering? Moreover, the bend-and-break lemma itself does not require any smoothness.

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up vote 6 down vote accepted

When we use the bend-and-break technique in the proof of the Cone Theorem, we not only need to know that under certain conditions there are rational curves through a point of our variety $X$, but we also require an upper bound on their degree (with respect to a given polarization $H$).

Such a bound is only available when $X$ is smooth. See [Debarre, Higher dimensional algebraic geometry, Theorem 3.6 p. 67] for the result that we need.

See also the introductions to Chapters 6 and 7 of the same book for a discussion about the different approach to the proof of the Cone Theorem in the smooth and in the singular case.

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I am not fully convinced by your answer. Do you mean that some estimates in the dimension of certain moduli space force people only to work in smooth case? On the other hand, it seems that some bounds can also be obtained without smooth assumption, for example, Theorem 7.46 in Debarre's book. – Li Yutong Jun 15 '14 at 21:33
It is ture that Debarre wrote the bend-and-break is "completely powerless" in singular case, unfortunately, he did not specify the reason! – Li Yutong Jun 15 '14 at 21:35
I'm saying that Theorem 3.6 p. 67 in Debarre's book (about the estimation of the $H$-degree of the rational curve $\Gamma$) only holds in the smooth case, and that that version of the theorem is used in the proof of the Cone Theorem in Chapter 6. The estimate used in Theorem 7.46 seems to me weaker. – Francesco Polizzi Jun 15 '14 at 23:12
The crucial estimate is actually $$-K_X \cdot \Gamma \leq \dim X +1,$$ and according to Debarre one needs smoothness for this (see the discussion that immediately follows the statement of Theorem 3.6). It is used at page 152, in First step of the proof of the Cone Theorem. – Francesco Polizzi Jun 15 '14 at 23:21

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