Birational Geometry of Algebraic Varieties Cambridge Tracts in Mathematics Paperback

One Of The Major Discoveries Of The Last Two Decades Of The Twentieth Century In Algebraic Geometry Is The Realization That The Theory Of Minimal Models Of Surfaces Can Be Generalized To Higher Dimensional Varieties. This Generalization, Called The Minimal Model Program Or Mori's Program, Has Developed Into A Powerful Tool With Applications To Diverse Questions In Algebraic Geometry And Beyond. This Book Provides The A Comprehensive Introduction To The Circle Of Ideas Developed Around The Program, The Prerequisites Being Only A Basic Knowledge Of Algebraic Geometry. It Will Be Of Great Interest To Graduate Students And Researchers Working In Algebraic Geometry And Related Fields. 1. Rational Curves And The Canonical Class. 1.1. Finding Rational Curves When K[subscript X] Is Negative. 1.2. Finding Rational Curves When K[subscript X] Is Not Nef. 1.3. The Cone Of Curves Of Smooth Varieties. 1.4. Minimal Models Of Surfaces. 1.5. Ampleness Criteria -- 2. Introduction To The Minimal Model Program. 2.1. Introduction To Mori's Program. 2.2. Extensions Of The Minimal Model Program. 2.3. Singularities In The Minimal Model Program. 2.4. The Kodaira Vanishing Theorem. 2.5. Generalizations Of The Kodaira Vanishing Theorem -- 3. Cone Theorems. 3.1. Introduction To The Proof Of The Cone Theorem. 3.2. Basepoint-free Theorem. 3.3. The Cone Theorem. 3.4. The Rationality Theorem. 3.5. The Non-vanishing Theorem. 3.6. Relative Versions. 3.7. Running The Mmp. 3.8. Minimal And Canonical Models -- 4. Surface Singularities Of The Minimal Model Program. 4.1. Log Canonical Surface Singularities. 4.2. Du Val Singularities. 4.3. Simultaneous Resolution For Du Val Singularities. 4.4. Elliptic Surface Singularities. 4.5. Deformations Of Hypersurface Singularities -- 5. Singularities Of The Minimal Model Program. 5.1. Rational Singularities. 5.2. Log Terminal Singularities. 5.3. Canonical And Terminal Threefold Singularities. 5.4. Inversion Of Adjunction. 5.5. Duality Theory -- 6. Three-dimensional Flops. 6.1. Flips And Flops. 6.2. Terminal Flops. 6.3. Terminalization And Q-factorialization. 6.4. Canonical Flops -- 7. Semi-stable Minimal Models. 7.1. Semi-stable Mmp. 7.2. Semi-stable Reduction Theorem. 7.3. Special Semi-stable Flips. 7.4. Semi-stable Flips. 7.5. Applications To Families Of Surfaces. 7.6. A Survey Of Further Results. János Kollár, Shigefumi Mori, With The Collaboration Of C.h. Clemens And A. Corti. Includes Bibliographical References (p. 241-247) And Index. One of the major discoveries of the past two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program, or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the first comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.