Table of Contents

I. Riemann Surfaces and Algebraic Curves.- 1. Riemann Surfaces.- §1. Basic Notions.- 1.1. Complex Chart; Complex Coordinates.- 1.2. Complex Analytic Atlas.- 1.3. Complex Analytic Manifolds.- 1.4. Mappings of Complex Manifolds.- 1.5. Dimension of a Complex Manifold.- 1.6. Riemann Surfaces.- 1.7. Differentiable Manifolds.- § 2. Mappings of Riemann Surfaces.- 2.1. Nonconstant Mappings of Riemann Surfaces are Discrete.- 2.2. Meromorphic Functions on a Riemann Surface.- 2.3. Meromorphic Functions with Prescribed Behaviour at Poles.- 2.4. Multiplicity of a Mapping; Order of a Function.- 2.5. Topological Properties of Mappings of Riemann Surfaces . ..- 2.6. Divisors on Riemann Surfaces.- 2.7. Finite Mappings of Riemann Surfaces.- 2.8. Unramified Coverings of Riemann Surfaces.- 2.9. The Universal Covering.- 2.10. Continuation of Mappings.- 2.11. The Riemann Surface of an Algebraic Function.- § 3. Topology of Riemann Surfaces.- 3.1. Orientability.- 3.2. Triangulability.- 3.3. Development; Topological Genus.- 3.4. Structure of the Fundamental Group.- 3.5. The Euler Characteristic.- 3.6. The Hurwitz Formulae.- 3.7. Homology and Cohomology; Betti Numbers.- 3.8. 3.8. Intersection Product; Poincaré Duality.- § 4. Calculus on Riemann Surfaces.- 4.1. Tangent Vectors; Differentiations.- 4.2. Differential Forms.- 4.3. Exterior Differentiations; de Rham Cohomology.- 4.4. Kähler and Riemann Metrics.- 4.5. Integration of Exterior Differentials; Green’s Formula .....- 4.6. Periods; de Rham Isomorphism.- 4.7. Holomorphic Differentials; Geometric Genus; Riemann’s Bilinear Relations.- 4.8. Meromorphic Differentials; Canonical Divisors.- 4.9. Meromorphic Differentials with Prescribed Behaviour at Poles; Residues.- 4.10. Periods of Meromorphic Differentials.- 4.11. Harmonic Differentials.- 4.12. Hilbert Space of Differentials; Harmonic Projection.- 4.13. Hodge Decomposition.- 4.14. Existence of Meromorphic Differentials and Functions .....- 4.15. Dirichlet’s Principle.- § 5. Classification of Riemann Surfaces.- 5.1. Canonical Regions.- 5.2. Uniformization.- 5.3. Types of Riemann Surfaces.- 5.4. Automorphisms of Canonical Regions.- 5.5. Riemann Surfaces of Elliptic Type.- 5.6. Riemann Surfaces of Parabolic Type.- 5.7. Riemann Surfaces of Hyperbolic Type.- 5.8. Automorphic Forms; Poincar7#x00E9; Series.- 5.9. Quotient Riemann Surfaces; the Absolute Invariant.- 5.10. Moduli of Riemann Surfaces.- § 6. Algebraic Nature of Compact Riemann Surfaces.- 6.1. Function Spaces and Mappings Associated with Divisors . ..- 6.2. Riemann-Roch Formula; Reciprocity Law for Differentials of the First and Second Kind.- 6.3. Applications of the Riemann-Roch Formula to Problems of Existence of Meromorphic Functions and Differentials . ..- 6.4. Compact Riemann Surfaces are Projective.- 6.5. Algebraic Nature of Protective Models; Arithmetic Riemann Surfaces.- 6.6. Models of Riemann Surfaces of Genus 1.- 2. Algebraic Curves.- §1. Basic Notions.- 1.1. Algebraic Varieties; Zariski Topology.- 1.2. Regular Functions and Mappings.- 1.3. The Image of a Projective Variety is Closed.- 1.4. Irreducibility; Dimension.- 1.5. Algebraic Curves.- 1.6. Singular and Nonsingular Points on Varieties.- 1.7. Rational Functions, Mappings and Varieties.- 1.8. Differentials.- 1.9. Comparison Theorems.- 1.10. Lefschetz Principle.- § 2. Riemann-Roch Formula.- 2.1. Multiplicity of a Mapping; Ramification.- 2.2. Divisors.- 2.3. Intersection of Plane Curves.- 2.4. The Hurwitz Formulae.- 2.5. Function Spaces and Spaces of Differentials Associated with Divisors.- 2.6. Comparison Theorems (Continued).- 2.7. Riemann-Roch Formula.- 2.8. Approaches to the Proof.- 2.9. First Applications.- 2.10. Riemann Count.- 3. Geometry of Projective Curves.- 3.1. Linear Systems.- 3.2. Mappings of Curves into—n.- 3.3. Generic Hyperplane Sections.- 3.4. Geometrical Interpretation of the Riemann-Roch Formula ..- 3.5. Clifford’s Inequality.- 3.6. Castelnuovo’s Inequality.- 3.7. Space Curves.- 3.8. Projective Normality.- 3.9. The Ideal of a Curve; Intersections of Quadrics.- 3.10. Complete Intersections.- 3.11. The Simplest Singularities of Curves.- 3.12. The Clebsch Formula.- 3.13. Dual Curves.- 3.14. Plücker Formula for the Class.- 3.15. Correspondence of Branches; Dual Formulae.- 3. Jacobians and Abelian Varieties.- §1. Abelian Varieties.- 1.1. Algebraic Groups.- 1.2. Abelian Varieties.- 1.3. Algebraic Complex Tori; Polarized Tori.- 1.4. Theta Function and Riemann Theta Divisor.- 1.5. Principally Polarized Abelian Varieties.- 1.6. Points of Finite Order on Abelian Varieties.- 1.7. Elliptic Curves.- § 2. Jacobians of Curves and of Riemann Surfaces.- 2.1. Principal Divisors on Riemann Surfaces.- 2.2. Inversion Problem.- 2.3. Picard Group.- 2.4. Picard Varieties and their Universal Property.- 2.5. Polarization Divisor of the Jacobian of a Curve; Poincaré Formulae.- 2.6. Jacobian of a Curve of Genus 1.- II. Algebraic Varieties and Schemes.- 1. Algebraic Varieties: Basic Notions.- §1. Affine Space.- 1.1. Base Field.- 1.2. Affine Space.- 1.3. Algebraic Subsets.- 1.4. Systems of Algebraic Equations; Ideals.- 1.5. Hilbert’s Nullstellensatz.- §2. Affine Algebraic Varieties.- 2.1. Affine Varieties.- 2.2. Abstract Affine Varieties.- 2.3. Affine Schemes.- 2.4. Products of Affine Varieties.- 2.5. Intersection of Subvarieties.- 2.6. Fibres of a Morphism.- 2.7. The Zariski Topology.- 2.8. Localization.- 2.9. Quasi-affine Varieties.- 2.10. Affine Algebraic Geometry.- §3. Algebraic Varieties.- 3.1. Projective Space.- 3.2. Atlases and Varieties.- 3.3. Gluing.- 3.4. The Grassmann Variety.- 3.5. Projective Varieties.- §4. Morphisms of Algebraic Varieties.- 4.1. Definitions.- 4.2. Products of Varieties.- 4.3. Equivalence Relations.- 4.4. Projection.- 4.5. The Veronese Embedding.- 4.6. The Segre Embedding.- 4.7. The Plücker Embedding.- §5. Vector Bundles.- 5.1. Algebraic Groups.- 5.2. Vector Bundles.- 5.3. Tautological Bundles.- 5.4. Constructions with Bundles.- § 6. Coherent Sheaves.- 6.1. Presheaves.- 6.2. Sheaves.- 6.3. Sheaves of Modules.- 6.4. Coherent Sheaves of Modules.- 6.5. Ideal Sheaves.- 6.6. Constructions of Varieties.- § 7. Differential Calculus on Algebraic Varieties.- 7.1. Differential of a Regular Function.- 7.2. Tangent Space.- 7.3. Tangent Cone.- 7.4. Smooth Varieties and Morphisms.- 7.5. Normal Bundle.- 7.6. Tangent Bundle.- 7.7. Sheaves of Differentials.- 2. Algebraic Varieties: Fundamental Properties.- § 1. Rational Maps.- 1.1. Irreducible Varieties.- 1.2. Noetherian Spaces.- 1.3. Rational Functions.- 1.4. Rational Maps.- 1.5. Graph of a Rational Map.- 1.6. Blowing up a Point.- 1.7. Blowing up a Subscheme.- § 2. Finite Morphisms.- 2.1. Quasi-finite Morphisms.- 2.2. Finite Morphisms.- 2.3. Finite Morphisms Are Closed.- 2.4. Application to Linear Projections.- 2.5. Normalization Theorems.- 2.6. The Constructibility Theorem.- 2.7. Normal Varieties.- 2.8. Finite Morphisms Are Open.- § 3. Complete Varieties and Proper Morphisms.- 3.1. Definitions.- 3.2. Properties of Complete Varieties.- 3.3. Protective Varieties Are Complete.- 3.4. Example of a Complete Nonprojective Variety.- 3.5. The Finiteness Theorem.- 3.6. The Connectedness Theorem.- 3.7. The Stein Factorization.- § 4. Dimension Theory.- 4.1. Combinatorial Definition of Dimension.- 4.2. Dimension and Finite Morphisms.- 4.3. Dimension of a Hypersurface.- 4.4. Theorem on the Dimension of the Fibres.- 4.5. The Semi-continuity Theorem of Chevalley.- 4.6. Dimension of Intersections in Affine Space.- 4.7. The Generic Smoothness Theorem.- § 5. Unramified and Étale Morphisms.- 5.1. The Implicit Function Theorem.- 5.2. Unramified Morphisms.- 5.3. Embedding of Projective Varieties.- 5.4. Étale Morphisms.- 5.5. Étale Coverings.- 5.6. The Degree of a Finite Morphism.- 5.7. The Principle of Conservation of Number.- § 6. Local Properties of Smooth Varieties.- 6.1. Smooth Points.- 6.2. Local Irreducibility.- 6.3. Factorial Varieties.- 6.4. Subvarieties of Higher Codimension.- 6.5. Intersections on a Smooth Variety.- 6.6. The Cohen-Macaulay Property.- § 7. Application to Birational Geometry.- 7.1. Fundamental Points.- 7.2. Zariski’s Main Theorem.- 7.3. Behaviour of Differential Forms under Rational Maps .....- 7.4. The Exceptional Variety of a Birational Morphism.- 7.5. Resolution of Singularities.- 7.6. A Criterion for Normality.- 3. Geometry on an Algebraic Variety.- § 1. Linear Sections of a Projective Variety.- 1.1. External Geometry of a Variety.- 1.2. The Universal Linear Section.- 1.3. Hyperplane Sections.- 1.4. The Connectedness Theorem.- 1.5. The Ruled Join.- 1.6. Applications of the Connectedness Theorem.- § 2. The Degree of a Projective Variety.- 2.1. Definition of the Degree.- 2.2. Theorem of Bezout.- 2.3. Degree and Codimension.- 2.4. Degree of a Linear Projection.- 2.5. The Hubert Polynomial.- 2.6. The Arithmetic Genus.- §3. Divisors.- 3.1. Cartier Divisors.- 3.2. Weil Divisors.- 3.3. Divisors and Invertible Sheaves.- 3.4. Functoriality.- 3.5. Excision Theorem.- 3.6. Divisors on Curves.- § 4. Linear Systems of Divisors.- 4.1. Families of Divisors.- 4.2. Linear Systems of Divisors.- 4.3. Linear Systems without Base Points.- 4.4. Ample Systems.- 4.5. Linear Systems and Rational Maps.- 4.6. Pencils.- 4.7. Linear and Projective Normality.- § 5. Algebraic Cycles.- 5.1. Definitions.- 5.2. Direct Image of a Cycle.- 5.3. Rational Equivalence of Cycles.- 5.4. Excision Theorem.- 5.5. Intersecting Cycles with Divisors.- 5.6. Segre Classes of Vector Bundles.- 5.7. The Splitting Principle.- § 6. Intersection Theory.- 6.1. Intersection of Cycles.- 6.2. Deformation to the Normal Cone.- 6.3. Gysin Homomorphism.- 6.4. The Chow Ring.- 6.5. The Chow Ring of Projective Space.- 6.6. The Chow Ring of a Grassmannian.- 6.7. Intersections on Surfaces.- § 7. The Chow Variety.- 7.1. Cycles in—n.- 7.2. From Cycles to Divisors.- 7.3. From Divisors to Cycles.- 7.4. Cycles on Arbitrary Varieties.- 7.5. Enumerative Geometry.- 7.6. Lines on a Cubic.- 7.7. The Five Conies Problem.- 4. Schemes.- §1. Algebraic Equations.- 1.1. Real Equations.- 1.2. Equations over a Field.- 1.3. Equations over Rings.- 1.4. The Prime Spectrum.- 1.5. Comparison with Varieties.- § 2. Affine Schemes.- 2.1. Functions on the Spectrum.- 2.2. Topology on the Spectrum.- 2.3. Structure Sheaf.- 2.4. Functoriality.- 2.5. Example: the Affine Line.- 2.6. Example: the Abstract Vector.- § 3. Schemes.- 3.1. Definitions.- 3.2. Examples.- 3.3. Relative Schemes.- 3.4. Properties of Schemes.- 3.5. Properties of Morphisms.- 3.6. Regular Schemes.- 3.7. Flat Morphisms.- § 4. Algebraic Schemes and Families of Algebraic Schemes.- 4.1. Algebraic Schemes.- 4.2. Geometrization.- 4.3. Geometric Properties of Algebraic Schemes.- 4.4. Families of Algebraic Schemes.- 4.5. Smooth Families.- References.- References.