## Higher-Dimensional Algebraic GeometryHigher-dimensional algebraic geometry studies the classification theory of algebraic varieties. This very active area of research is still developing, but an amazing quantity of knowledge has accumulated over the past twenty years. The author¿s goal is to provide an easily accessible introduction to the subject. The book covers preparatory and standard definitions and results, moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps towards Mori¿s minimal model program of classification of algebraic varieties by proving the cone and contraction theorems. The book is well organized and the author has kept the number of concepts that are used but not proved to a minimum to provide a mostly self-contained introduction to graduate students and researchers. |

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### Contents

III | 1 |

IV | 2 |

V | 4 |

VI | 10 |

VII | 13 |

VIII | 14 |

IX | 16 |

X | 19 |

XLII | 111 |

XLIII | 113 |

XLIV | 116 |

XLV | 120 |

XLVI | 122 |

XLVII | 127 |

XLVIII | 130 |

XLIX | 131 |

XI | 21 |

XII | 23 |

XIII | 27 |

XIV | 29 |

XV | 35 |

XVI | 37 |

XVII | 38 |

XVIII | 39 |

XIX | 45 |

XX | 47 |

XXI | 52 |

XXII | 55 |

XXIII | 56 |

XXIV | 60 |

XXV | 63 |

XXVI | 66 |

XXVII | 70 |

XXVIII | 73 |

XXIX | 76 |

XXX | 79 |

XXXI | 80 |

XXXII | 84 |

XXXIII | 85 |

XXXIV | 86 |

XXXV | 89 |

XXXVI | 96 |

XXXVII | 99 |

XXXIX | 100 |

XL | 104 |

XLI | 108 |

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### Common terms and phrases

abelian variety ample divisor Assume birational morphism blow-up canonical divisor canonical singularities Cartier divisor characteristic zero closure codimension codimension at least coefficients coherent sheaf cone of curves cone theorem construction contained Corollary curve f defined degree dense open subset desingularization dim(X dimension effective divisor equivalence class evaluation map exact sequence exceptional divisor exceptional locus exists extremal ray Fano variety fiber finite follows free rational curve geometric hence hyperplane section hypersurface implies induction intersection number invertible sheaf irreducible component irreducible curve isomorphic let f line bundle linear system minimal model Mori's multiple NE(X nef and big nef divisor nonnegative nonzero normal bundle open subset parametrized Picard number polynomial positive integer proof proper morphism Proposition prove Q-factorial quasi-projective rational map rationally chain-connected simple normal crossings smooth projective variety smooth variety subscheme subvariety sufficiently large surface surjective vanishing theorem variety and let