Basic Algebraic Geometry.

I. Algebraic Varieties in a Projective Space.- I. Fundamental Concepts.- § 1. Plane Algebraic Curves.- 1. Rational Curves.- 2. Connections with the Theory of Fields.- 3. Birational Isomorphism of Curves.- Exercises.- §2. Closed Subsets of Affine Spaces.- 1. Definition of Closed Subset.- 2. Regular Functions on a Closed Set.- 3. Regular Mappings.- Exercises.- § 3. Rational Functions.- 1. Irreducible Sets.- 2. Rational Functions.- 3. Rational Mappings.- Exercises.- § 4. Quasiprojective Varieties.- 1. Closed Subsets of a Projective Space.- 2. Regular Functions.- 3. Rational Functions.- 4. Examples of Regular Mappings.- Exercises.- § 5. Products and Mappings of Quasiprojective Varieties.- 1. Products.- 2. Closure of the Image of a Projective Variety.- 3. Finite Mappings.- 4. Normalization Theorem.- Exercises.- § 6. Dimension.- 1. Definition of Dimension.- 2. Dimension of an Intersection with a Hypersurface.- 3. A Theorem on the Dimension of Fibres.- 4. Lines on Surfaces.- 5. The Chow Coordinates of a Projective Variety.- Exercises.- II. Local Properties.- §1. Simple and Singular Points.- 1. The Local Ring of a Point.- 2. The Tangent Space.- 3. Invariance of the Tangent Space.- 4. Singular Points.- 5. The Tangent Cone.- Exercises.- §2. Expansion in Power Series.- 1. Local Parameters at a Point.- 2. Expansion in Power Series.- 3. Varieties over the Field of Real and the Field of Complex Numbers 88 Exercises.- § 3. Properties of Simple Points.- 1. Subvarieties of Codimension 1.- 2. Smooth Subvarieties.- 3. Factorization in the Local Ring of a Simple Point.- Exercises.- § 4. The Structure of Birational Isomorphisms.- 1. The ?-Process in a Projective Space.- 2. The Local ?-Process.- 3. Behaviour of Subvarieties under a ?-Process.- 4. Exceptional Subvarieties.- 5. Isomorphism and Birational Isomorphism.- Exercises.- §5. Normal Varieties.- 1. Normality.- 2. Normalization of Affine Varieties.- 3. Ramification.- 4. Normalization of Curves.- 5. Projective Embeddings of Smooth Varieties.- Exercises.- III. Divisors and Differential Forms.- § 1. Divisors.- 1. Divisor of a Function.- 2. Locally Principal Divisors.- 3. How to Shift the Support of a Divisor Away from Points.- 4. Divisors and Rational Mappings.- 5. The Space Associated with a Divisor.- Exercises.- § 2. Divisors on Curves.- 1. The Degree of a Divisor on a Curve.- 2. Bezout's Theorem on Curves.- 3. Cubic Curves.- 4. The Dimension of a Divisor.- Exercises.- §3. Algebraic Groups.- 1. Addition of Points on a Plane Cubic Curve.- 2. Algebraic Groups.- 3. Factor Groups. Chevalley's Theorem.- 4. Abelian Varieties.- 5. Picard Varieties.- Exercises.- §4. Differential Forms.- 1. One-Dimensional Regular Differential Forms.- 2. Algebraic Description of the Module of Differentials.- 3. Differential Forms of Higher Degrees.- 4. Rational Differential Forms.- Exercises.- § 5. Examples and Applications of Differential Forms.- 1. Behaviour under Mappings.- 2. Invariant Differential Forms on a Group.- 3. The Canonical Class.- 4. Hypersurfaces.- 5. Hyperelliptic Curves.- 6. The Riemann-Roch Theorem for Curves.- 7. Projective Immersions of Surfaces.- Exercises.- IV. Intersection Indices.- §1. Definition and Basic Properties.- 1. Definition of an Intersection Index.- 2. Additivity of the Intersection Index.- 3. Invariance under Equivalence.- 4. End of the Proof of Invariance.- 5. General Definition of the Intersection Index.- Exercises.- §2. Applications and Generalizations of Intersection Indices.- 1. Bezout's Theorem in a Projective Space and Products of Projective Spaces.- 2. Varieties over the Field of Real Numbers.- 3. The Genus of a Smooth Curve on a Surface.- 4. The Ring of Classes of Cycles.- Exercises.- § 3. Birational Isomorphisms of Surfaces.- 1. ?-Processes of Surfaces.- 2. Some Intersection Indices.- 3. Elimination of Points of Indeterminacy.- 4. Decomposition into ?-Processes.- 5. Notes and Examples.- Exercises.- II. Schemes and Varieties.- V. Schemes.- §1. Spectra of

I. Algebraic Varieties in a Projective Space.- I. Fundamental Concepts.- § 1. Plane Algebraic Curves.- 1. Rational Curves.- 2. Connections with the Theory of Fields.- 3. Birational Isomorphism of Curves.- Exercises.- §2. Closed Subsets of Affine Spaces.- 1. Definition of Closed Subset.- 2. Regular Functions on a Closed Set.- 3. Regular Mappings.- Exercises.- § 3. Rational Functions.- 1. Irreducible Sets.- 2. Rational Functions.- 3. Rational Mappings.- Exercises.- § 4. Quasiprojective Varieties.- 1. Closed Subsets of a Projective Space.- 2. Regular Functions.- 3. Rational Functions.- 4. Examples of Regular Mappings.- Exercises.- § 5. Products and Mappings of Quasiprojective Varieties.- 1. Products.- 2. Closure of the Image of a Projective Variety.- 3. Finite Mappings.- 4. Normalization Theorem.- Exercises.- § 6. Dimension.- 1. Definition of Dimension.- 2. Dimension of an Intersection with a Hypersurface.- 3. A Theorem on the Dimension of Fibres.- 4. Lines on Surfaces.- 5. The Chow Coordinates of a Projective Variety.- Exercises.- II. Local Properties.- §1. Simple and Singular Points.- 1. The Local Ring of a Point.- 2. The Tangent Space.- 3. Invariance of the Tangent Space.- 4. Singular Points.- 5. The Tangent Cone.- Exercises.- §2. Expansion in Power Series.- 1. Local Parameters at a Point.- 2. Expansion in Power Series.- 3. Varieties over the Field of Real and the Field of Complex Numbers 88 Exercises.- § 3. Properties of Simple Points.- 1. Subvarieties of Codimension 1.- 2. Smooth Subvarieties.- 3. Factorization in the Local Ring of a Simple Point.- Exercises.- § 4. The Structure of Birational Isomorphisms.- 1. The ?-Process in a Projective Space.- 2. The Local ?-Process.- 3. Behaviour of Subvarieties under a ?-Process.- 4. Exceptional Subvarieties.- 5. Isomorphism and Birational Isomorphism.- Exercises.- §5. Normal Varieties.- 1. Normality.- 2. Normalization of Affine Varieties.- 3. Ramification.- 4. Normalization of Curves.- 5. Projective Embeddings of Smooth Varieties.- Exercises.- III. Divisors and Differential Forms.- § 1. Divisors.- 1. Divisor of a Function.- 2. Locally Principal Divisors.- 3. How to Shift the Support of a Divisor Away from Points.- 4. Divisors and Rational Mappings.- 5. The Space Associated with a Divisor.- Exercises.- § 2. Divisors on Curves.- 1. The Degree of a Divisor on a Curve.- 2. Bezout’s Theorem on Curves.- 3. Cubic Curves.- 4. The Dimension of a Divisor.- Exercises.- §3. Algebraic Groups.- 1. Addition of Points on a Plane Cubic Curve.- 2. Algebraic Groups.- 3. Factor Groups. Chevalley’s Theorem.- 4. Abelian Varieties.- 5. Picard Varieties.- Exercises.- §4. Differential Forms.- 1. One-Dimensional Regular Differential Forms.- 2. Algebraic Description of the Module of Differentials.- 3. Differential Forms of Higher Degrees.- 4. Rational Differential Forms.- Exercises.- § 5. Examples and Applications of Differential Forms.- 1. Behaviour under Mappings.- 2. Invariant Differential Forms on a Group.- 3. The Canonical Class.- 4. Hypersurfaces.- 5. Hyperelliptic Curves.- 6. The Riemann-Roch Theorem for Curves.- 7. Projective Immersions of Surfaces.- Exercises.- IV. Intersection Indices.- §1. Definition and Basic Properties.- 1. Definition of an Intersection Index.- 2. Additivity of the Intersection Index.- 3. Invariance under Equivalence.- 4. End of the Proof of Invariance.- 5. General Definition of the Intersection Index.- Exercises.- §2. Applications and Generalizations of Intersection Indices.- 1. Bezout’s Theorem in a Projective Space and Products of Projective Spaces.- 2. Varieties over the Field of Real Numbers.- 3. The Genus of a Smooth Curve on a Surface.- 4. The Ring of Classes of Cycles.- Exercises.- § 3. Birational Isomorphisms of Surfaces.- 1. ?-Processes of Surfaces.- 2. Some Intersection Indices.- 3. Elimination of Points of Indeterminacy.- 4. Decomposition into ?-Processes.- 5. Notes and Examples.- Exercises.- II. Schemes and Varieties.- V. Schemes.- §1. Spectra of Rings.- 1. Definition of a Spectrum.- 2. Properties of the Points of a Spectrum.- 3. The Spectral Topology.- 4. Irreducibility, Dimension.- Exercises.- §2. Sheaves.- 1. Presheaves.- 2. The Structure Presheaf.- 3. Sheaves.- 4. The Stalks of a Sheaf.- Exercises.- §3. Schemes.- 1. Definition of a Scheme.- 2. Pasting of Schemes.- 3. Closed Subschemes.- 4. Reducibility and Nilpotents.- 5. Finiteness Conditions.- Exercises.- § 4. Products of Schemes.- 1. Definition of a Product.- 2. Group Schemes.- 3. Separation.- Exercises.- VI. Varieties.- §1. Definition and Examples.- 1. Definitions.- 2. Vector Bundles.- 3. Bundles and Sheaves.- 4. Divisors and Line Bundles.- Exercises.- § 2. Abstract and Quasiprojective Varieties.- 1. Chow’s Lemma.- 2. The ?-Process along a Subvariety.- 3. Example of a Non-Quasiprojective Variety.- 4. Criteria for Projectiveness.- Exercises.- §3. Coherent Sheaves.- 1. Sheaves of Modules.- 2. Coherent Sheaves.- 3. Dévissage of Coherent Sheaves.- 4. The Finiteness Theorem.- Exercises.- III. Algebraic Varieties over the Field of Complex Numbers and Complex Analytic Manifolds.- VII. Topology of Algebraic Varieties.- §1. The Complex Topology.- 1. Definitions.- 2. Algebraic Varieties as Differentiable Manifolds. Orientation.- 3. The Homology of Smooth Projective Varieties.- Exercises.- §2. Connectedness.- 1. Auxiliary Lemmas.- 2. The Main Theorem.- 3. Analytic Lemmas.- Exercises.- § 3. The Topology of Algebraic Curves.- 1. The Local Structure of Morphisms.- 2. Triangulation of Curves.- 3. Topological Classification of Curves.- 4. Combinatorial Classification of Surfaces.- § 4. Real Algebraic Curves.- 1. Involutions.- 2. Proof of Harnack’s Theorem.- 3. Ovals of Real Curves.- Exercises.- VIII. Complex Analytic Manifolds.- §1. Definitions and Examples.- 1. Definition.- 2. Factor Spaces.- 3. Commutative Algebraic Groups as Factor Spaces.- 4. Examples of Compact Analytic Manifolds that are not Isomorphic to Algebraic Varieties.- 5. Complex Spaces.- Exercises.- § 2. Divisors and Meromorphic Functions.- 1. Divisors.- 2. Meromorphic Functions.- 3. Siegel’s Theorem.- Exercises.- § 3. Algebraic Varieties and Analytic Manifolds.- 1. Comparison Theorem.- 2. An Example of Non-Isomorphic Algebraic Varieties that are Isomorphic as Analytic Manifolds.- 3. Example of a Non-Algebraic Compact Manifold with the Maximal Number of Independent Meromorphic Functions.- 4. Classification of Compact Analytic Surfaces.- Exercises.- IX. Uniformization.- § 1. The Universal Covering.- 1. The Universal Covering of a Complex Manifold.- 2. Universal Coverings of Algebraic Curves.- 3. Projective Embeddings of Factor Spaces.- Exercises.- §2. Curves of Parabolic Type.- 1. ?-Functions.- 2. Projective Embedding.- 3. Elliptic Functions, Elliptic Curves, and Elliptic Integrals 389 Exercises.- § 3. Curves of Hyperbolic Type.- 1. Poincaré Series.- 2. Projective Embedding.- 3. Algebraic Curves and Automorphic Functions.- Exercises.- § 4. On the Uniformization of Manifolds of Large Dimension.- 1. Simple Connectivity of Complete Intersections.- 2. Example of a Variety with a Preassigned Finite Fundamental Group.- 3. Notes.- Exercises.- Historical Sketch.- 1. Elliptic Integrals.- 2. Elliptic Functions.- 3. Abelian Integrals.- 4. Riemann Surfaces.- 5. The Inversion Problem.- 6. Geometry of Algebraic Curves.- 7. Many-Dimensional Geometry.- 8. The Analytic Theory of Manifolds.- 9. Algebraic Varieties over an Arbitrary Field. Schemes.- Bibliography for the Historical Sketch.- List of Notation.