The Cone Theorem and Minimal Models

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The most important application of the bend-and-break techniques of [Mori79] is his proof of the Cone Theorem and the applications to minimal models of smooth varieties. Later development of this theory strayed far from the methods considered in this book. The cohomological methods of Kawamata— Reid—Shokurov are described in detail in [CKM88]. Minimal models are not fully covered in any monograph,1 since the proof of the crucial flip theorem [Mori88] is rather long and complicated. The ideas of the Cone Theorem and of Mori’s Program (or Minimal Model Program) are very simple, powerful and influential. The aim of this short chapter is to illustrate these in rather special cases where the technical details do not overwhelm them.