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.



What goes wrong to use "bend-and-break" trick for singular varieties?
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. 


