In this chapter we explore the relationship between the canonical class Kx of a smooth projective variety X and rational curves on X.
The first section considers the case when –Kx is ample; these are called Fano varieties. The main result shows that X contains a rational curve C ⊂ X which has low degree with respect to – Kx. This result, due to [Mor79], is one of the starting points of the minimal model theory. It is quite interesting that even for varieties over ℂ, the proof proceeds through positive characteristic.
In section 2 we generalize these results to the case when – Kx is no longer ample, but it has positive intersection number with some curve. The proofs are very similar to the earlier ones, we just have to keep track of some additional information carefully.
This leads to the geometric proof of the Cone Theorem for smooth projective varieties in section 3, due to [Mor82]. Unfortunately, for most applications this is not strong enough, and we prove a more general Cone Theorem in Chapter 3 with very different methods.
In section 4 we illustrate the use of the Cone Theorem by using it to construct minimal models of surfaces. The rest of the book is essentially devoted to generalizing these results to higher dimensions.
The last section contains the proof of some of the basic ampleness criteria.
Unfortunately, the methods of this chapter are not sufficient to complete the minimal model program in higher dimensions. In fact, they are not used in subsequent chapters. Nonetheless, we feel that these results provide a very clear geometric picture, which guides the later, more technical works.