Algebraic Geometry

The aim of this book is to introduce the reader to the geometric theory of algebraic varieties, in particular to the birational geometry of algebraic varieties. This volume grew out of the author's book in Japanese published in 3 volumes by Iwanami, Tokyo, in 1977. While writing this English version, the author has tried to rearrange and rewrite the original material so that even beginners can read it easily without referring to other books, such as textbooks on commutative algebra. The reader is only expected to know the definition of Noetherin rings and the statement of the Hilbert basis theorem. The new chapters 1, 2, and 10 have been expanded. In particular, the exposition of D-dimension theory, although shorter, is more complete than in the old version. However, to keep the book of manageable size, the latter parts of Chapters 6, 9, and 11 have been removed. I thank Mr. A. Sevenster for encouraging me to write this new version, and Professors K. K. Kubota in Kentucky and P. M. H. Wilson in Cam­ bridge for their careful and critical reading of the English manuscripts and typescripts. I held seminars based on the material in this book at The University of Tokyo, where a large number of valuable comments and suggestions were given by students Iwamiya, Kawamata, Norimatsu, Tobita, Tsushima, Maeda, Sakamoto, Tsunoda, Chou, Fujiwara, Suzuki, and Matsuda.

1 Schemes.- 1.1 Spectra of Rings.- 1.2 Examples of Spectra as Topological Spaces.- 1.3 Rings of Fractions, the Case Af.- 1.4 Rings and Modules of Fractions.- 1.5 Nullstellensatz.- 1.6 Irreducible Spaces.- 1.7 Integral Extension of Rings.- 1.8 Hilbert Nullstellensatz.- 1.9 Dimension of Spec A.- 1.10 Sheaves.- 1.11 Structure of Sheaves on Spectra.- 1.12 Quasi-coherent Sheaves and Coherent Sheaves.- 1.13 Reduced Affine Schemes and Integral Affine Schemes.- 1.14 Morphism of Affine Schemes.- 1.15 Definition of Schemes and First Properties.- 1.16 Subschemes.- 1.17 Glueing Schemes.- 1.18 Projective Spaces.- 1.19 S-Schemes and Automorphism of Schemes.- 1.20 Product of S-Schemes.- 1.21 Base Extension.- 1.22 Graphs of Morphisms.- 1.23 Separated Schemes.- 1.24 Regular Functions and Rational Functions.- 1.25 Rational Maps.- 1.26 Morphisms of Finite Type.- 1.27 Affine Morphisms and Integral Morphisms.- 1.28 Proper Morphisms and Finite Morphisms.- 1.29 Algebraic Varieties.- 2 Normal Varieties.- 2.1 Normal Rings.- 2.2 Normal Points on Schemes.- 2.3 Unique Factorization Domains.- 2.4 Primary Decomposition of Ideals.- 2.5 Intersection Theorem and Complete Local Rings.- 2.6 Regular Local Rings.- 2.7 Normal Points on Algebraic Curves and Extension Theorems.- 2.8 Divisors on a Normal Variety.- 2.9 Linear Systems.- 2.10 Domain of a Rational Map.- 2.11 Pullback of a Divisor.- 2.12 Strictly Rational Maps.- 2.13 Connectedness Theorem.- 2.14 Normalization of Varieties.- 2.15 Degree of a Morphism and a Rational Map.- 2.16 Inverse Image Sheaves.- 2.17 The Pullback Theorem.- 2.18 Invertible Sheaves.- 2.19 Rational Sections of an Invertible Sheaf.- 2.20 Divisors and Invertible Sheaves.- 3 Projective Schemes.- 3.1 Graded Rings.- 3.2 Homogeneous Spectra.- 3.3 Finitely Generated Graded Rings.- 3.4 Construction of Projective Schemes.- 3.5 Some Properties of Projective Schemes.- 3.6 Chow’s Lemma.- 4 Cohomology of Sheaves.- 4.1 Injective Sheaves.- 4.2 Fundamental Theorems.- 4.3 Flabby Sheaves.- 4.4 Cohomology of Affine Schemes.- 4.5 Finiteness Theorem.- 4.6 Leray’s Spectral Sequence.- 4.7 Cohomology of Affme Morphisms.- 4.8 Riemann-Roch Theorem (in the Weak Form) on a Curve.- 5 Regular Forms and Rational Forms on a Variety.- 5.1 Modules of Regular Forms and Canonical Derivations.- 5.2 Lemmas.- 5.3 Sheaves of Regular Forms.- 5.4 Birational Invariance of Genera.- 5.5 Adjunction Formula.- 5.6 Ramification Formula.- 5.7 Generalized Adjunction Formula and Conductors.- 5.8 Serre Duality.- 6 Theory of Curves.- 6.1 Riemann-Roch Theorem.- 6.2 Fujita’s Invariant ? (C, D).- 6.3 Degree of a Curve.- 6.4 Hyperplane Section Theorem.- 6.5 Hyperelliptic Curves.- 6.6 ?-Gap Sequence and Weierstrass Points.- 6.7 Wronski Forms.- 6.8 Theorems of Hurwitz and Automorphism Groups of Curves.- 7 Cohomology of Projective Schemes.- 7.1 The Homomorphism ?M.- 7.2 The Homomorphism ? ?.- 7.3 Cohomology Groups of Coherent Sheaves on PnR.- 7.4 Ample Sheaves.- 7.5 Projective Morphisms.- 7.6 Unscrewing Lemma and Its Applications.- 7.7 Projective Normality.- 7.8 Etale Morphisms.- 7.9 Theorems of Bertini.- 7.10 Monoidal Transformations.- 8 Intersection Theory of Divisors.- 8.1 Intersection Number of Curves on a Surface.- 8.2 Riemann-Roch Theorem on an Algebraic Surface.- 8.3 Intersection Matrix of a Divisor.- 8.4 Intersection Numbers of Invertible Sheaves.- 8.5 Nakai’s Criterion on Ample Sheaves.- 9 Curves on a Nonsingular Surface.- 9.1 Quadric Transformations.- 9.2 Local Properties of Singular Points.- 9.3 Linear Pencil Theorem.- 9.4 Dual Curves and Plucker Relations.- 9.5 Decomposition of Birational Maps.- 10 D-Dimension and Kodaira Dimension of Varieties.- 10.1 D-Dimension.- 10.2 The Asymptotic Estimate for l(mD).- 10.3 Fundamental Theorems for D-Dimension.- 10.4 D-Dimensions of a K3 Surface and an Abelian Variety.- 10.5 Kodaira Dimension.- 10.6 Types of Varieties.- 10.7 Subvarieties of an Abelian Variety.- 11 Logarithmic Kodaira Dimension of Varieties.- 11.1 Logarithmic Forms.- 11.2 Logarithmic Genera.- 11.3 Reduced Divisor as a Boundary.- 11.4 Logarithmic Ramification Formula.- 11.5 Étale Endomorphisms.- 11.6 Logarithmic Canonical Fibered Varieties’.- 11.7 Finiteness of the Group SBir(V).- 11.8 Some Applications.- References.