Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes

ISBN-13: 9783540637059 / Angielski / Miękka / 1998 / 310 str.

V. I. Danilov; V. V. Shokurov; V. I. Danilov

Opis

From the reviews: "This volume... consists of two papers. The first, written by V.V. Shokurov, is devoted to the theory of Riemann surfaces and algebraic curves. It is an excellent overview of the theory of relations between Riemann surfaces and their models - complex algebraic curves in complex projective spaces. ... The second paper, written by V.I. Danilov, discusses algebraic varieties and schemes. ... I can recommend the book as a very good introduction to the basic algebraic geometry." European Mathematical Society Newsletter, 1996
..". To sum up, this book helps to learn algebraic geometry in a short time, its concrete style is enjoyable for students and reveals the beauty of mathematics." Acta Scientiarum Mathematicarum

Spis treści

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.

Blurb

From the reviews of the first printing, published as volume 23 of the Encyclopaedia of Mathematical Sciences:
"This volume... consists of two papers. The first, written by V.V.Shokurov, is devoted to the theory of Riemann surfaces and algebraic curves. It is an excellent overview of the theory of relations between Riemann surfaces and their models - complex algebraic curves in complex projective spaces. ... The second paper, written by V.I.Danilov, discusses algebraic varieties and schemes. ...
I can recommend the book as a very good introduction to the basic algebraic geometry."
European Mathematical Society Newsletter, 1996

"... To sum up, this book helps to learn algebraic geometry in a short time, its concrete style is enjoyable for students and reveals the beauty of mathematics."
Acta Scientiarum Mathematicarum, 1994