I. R. Shafarevich

Basic Algebraic Geometry

Übersetzung: Hirsch, K.A.

• Broschiertes Buch

Produktbeschreibung

Algebraic geometry occupied a central place in the mathematics of the last century. The deepest results of Abel, Riemann, Weierstrass, many of the most important papers of Klein and Poincare belong to this do mam. At the end of the last and the beginning of the present century the attitude towards algebraic geometry changed abruptly. Around 1910 Klein wrote: "When I was a student, Abelian functions -as an after-effect of Jacobi's tradition-were regarded as the undIsputed summit of mathe matics, and each of us, as a matter of course, had the ambition to forge ahead in this field. And now? The young generation hardly know what Abelian functions are." (Vorlesungen tiber die Entwicklung der Mathe matik im XIX. Jahrhundert, Springer-Verlag, Berlin 1926, Seite 312). The style of thinking that was fully developed in algebraic geometry at that time was too far removed from the set-theoretical and axio matic spirit, which then determined the development of mathematics. Several decades hadto lapse before the rise of the theory of topolo gical, differentiable and complex manifolds, the general theory of fields, the theory of ideals in sufficiently general rings, and only then it became possible to construct algebraic geometry on the basis of the principles of set-theoretical mathematics. Around the middle of the present century algebraic geometry had undergone to a large extent such a reshaping process. As a result, it can again lay claim to the position it once occupied in mathematics.

Produktdetails

• Springer Study Edition .
• Verlag: Springer, Berlin
• 1st ed. 1974. Rev. 3rd printing
• Erscheinungstermin: 4. Dezember 2012
• Englisch
• Abmessung: 18mm x 154mm x 232mm
• Gewicht: 675g
• ISBN-13: 9783540082644
• ISBN-10: 3540082646
• Artikelnr.: 25973145

Inhaltsangabe

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