Birational Geometry of Algebraic Varieties
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...
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This book provides the first comprehensive introduction to the circle of ideas developed around Mori's program.
PrefaceIntroduction11Rational Curves and the Canonical Class71.1Finding Rational Curves when K[subscript x] is Negative81.2Finding Rational Curves when K[subscript x] is not Nef161.3The Cone of Curves of Smooth Varieties181.4Minimal Models of Surfaces261.5Ampleness Criteria292Introduction to the Minimal Model Program362.1Introduction to Mori's Program372.2Extensions of the Minimal Model Program452.3Singularities in the Minimal Model Program502.4The Kodaira Vanishing Theorem622.5Generalizations of the Kodaira Vanishing Theorem673Cone Theorems743.1Introduction to the Proof of the Cone Theorem753.2Basepoint-free Theorem783.3The Cone Theorem813.4The Rationality Theorem863.5The Non-vanishing Theorem913.6Relative versions933.7Running the MMP963.8Minimal and Canonical Models1064Surface Singularities of the Minimal Model Program1114.1Log Canonical Surface Singularities1124.2Du Val Singularities1224.3Simultaneous Resolution for Du Val Singularities1284.4Elliptic Surface Singularities1364.5Deformations of Hypersurface Singularities1445Singularities of the Minimal Model Program1525.1Rational Singularities1535.2Log Terminal Singularities1585.3Canonical and Terminal Threefold Singularities1645.4Inversion of Adjunction1725.5Duality Theory1796Three-dimensional Flops1876.1Flips and Flops1886.2Terminal Flops1926.3Terminalization and Q-factorialization1956.4Canonical Flops2017Semi-stable Minimal Models2077.1Semi-stable MMP2087.2Semi-stable Reduction Theorem2147.3Special Semi-stable Flips2207.4Semi-stable Flips2247.5Applications to Families of Surfaces2297.6A Survey of Further Results236Bibliography241Index249
