Algebraic Geometry I - Danilov, V.I.;Shokurov, V.V.
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  • Broschiertes Buch

"... 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

Produktbeschreibung
"... 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
  • Produktdetails
  • Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
  • Softcover reprint of the original 1st ed. 1994
  • Seitenzahl: 324
  • Erscheinungstermin: 17. März 1998
  • Englisch
  • Abmessung: 235mm x 155mm x 17mm
  • Gewicht: 496g
  • ISBN-13: 9783540637059
  • ISBN-10: 3540637052
  • Artikelnr.: 24487680
Inhaltsangabe
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 Ide