The purpose of this chapter is to give a proof of the cone theorem, providing the necessary key ingredients: the rationality theorem and boundedness of the denominator. We prove these two theorems by applying the same cohomological arguments developed for the proofs of the base point freeness theorem and the non-vanishing theorem of the previous chapter. We note that our point of view for discussing the behavior of divisors following Kawamata—Reid—Shokurov—Kollar is “dual” to the original approach of Mori, who discusses the behavior of curves directly via deformation theory. We will study Mori’s argument in Chapter 10.
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