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 ofBirational 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. Language: English