Algebraic Geometry

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.