In Chapter 1 we proved the Cone Theorem for smooth projective varieties, and we noted that the proof given there did not work for singular varieties. For the minimal model program certain singularities are unavoidable and it is essential to have the Cone Theorem for pairs (X, Δ). Technically and historically this is a rather involved proof, developed by several authors. The main contributions are [Kaw84a, Rei83c, Sho85].

Section 1 states the four main steps of the proof and explains the basic ideas behind it. There is a common thread running through all four parts, called the basepoint-freeness method. This technique appears transparently in the proof of the Basepoint-free Theorem. For this reason in section 2 we present the proof of the Basepoint-free Theorem, though logically this should be the second step of the proof.

The basepoint-freeness method has found applications in many different contexts as well, some of which are explained in [Laz96] and [Kol97, Sec.5].

The remaining three steps are treated in the next three sections, the proof of the Rationality Theorem being the most involved.

In section 6 we state and explain the relative versions of the Basepoint-free Theorem and the Cone Theorem.

With these results at our disposal, we are ready to formulate in a precise way the log minimal model program. This is done in section 7. In dimension two the program does not involve flips, and so we are able to treat this case completely.

In section 7 we study minimal models of pairs. It turns out that this concept is not a straightforward generalization of the minimal models of smooth varieties (2.13).