From a review of the German edition: "The present book provides a completely self-contained introduction to complex plane curves from the traditional algebraic-analytic viewpoint. The arrangement of the material is of outstanding instructional skill, and the text is written in a very lucid, detailed and enlightening style ... Compared to the many other textbooks on (plane) algebraic curves, the present new one comes closest in spirit and content, to the work of E. Brieskorn and H. Knoerrer ... One could say that the book under review is a beautiful, creative and justifiable abridged version of this work, which also stresses the analytic-topological point of view ... the present book is a beautiful invitation to algebraic geometry, encouraging for beginners, and a welcome source for teachers of algebraic geometry, especially for those who want to give an introduction to the subject on the undergraduate-graduate level, to cover some not too difficult topics in substantial depth, but to do so in the shortest possible time." -- --Zentralblatt MATH
Author(s): Gerd Fischer
Series: Student Mathematical Library, V. 15 (Book 15)
Publisher: American Mathematical Society
Year: 2001
Language: English
Commentary: Front and back covers, OCR, 2 level bookmarks, paginated.
Pages: 248
Chapter 0. Introduction
0.1. Lines
0.2. Circles
0.3. The Cuspidal Cubic
0.4. The Nodal Cubic
0.5. The Folium of Descartes
0.6. Cycloids
0.7. Klein Quartics
0.8. Continuous Curves
Chapter 1. Affine Algebraic Curves and Their Equations
1.1. The Variety of an Equation
1.2. Affine Algebraic Curves
1.3. Study's Lemma
1.4. Decomposition into Components
1.5. Irreducibility and Connectedness
1.6. The Minimal Polynomial
1.7. The Degree
1.8. Points of Intersection with a Line
Chapter 2. The Projective Closure
2.1. Points at Infinity
2.2. The Projective Plane
2.3. The Projective Closure of a Curve
2.4. Decomposition into Components
2.5. Intersection Multiplicity of Curves and Lines
2.6. Intersection of Two Curves
2.7. Bezout's Theorem
Chapter 3. Tangents and Singularities
3.1. Smooth Points
3.2. The Singular Locus
3.3. Local Order
3.4. Tangents at Singular Points
3.5. Order and Intersection Multiplicity
3.6. Euler's Formula
3.7. Curves through Prescribed Points
3.8. Number of Singularities
3.9. Chebyshev Curves
Chapter 4. Polm·s and Hessian Curves
4.2. Properties of Polars
4.3. Intersection of a Curve with Its Polars
4.4. Hessian Curves
4.5. Intersection of the Curve with Its Hessian Curve
4.6. Examples
Chapter 5. The Dual Curve and the Pliicker Formulas
5.1. The Dual Curve
5.2. Algebraicity of the Dual Curve
5.3. Irreducibility of the Dual Curve
5.4. Local Numerical Invariants
5.5. The Bidual Curve
5.6. Simple Double Points and Cusps
5.7. The Plucker Formulas
5.8. Examples
5.9. Proof of the Plucker Formulas
Chapter 6. The Ring of Convergent Power Series
6.1. Global and Local Irreducibility
6.2. Formal Power Series
6.3. Convergent Power Series
6.4. Banach Algebras
6.5. Substitution of Power Series
6.6. Distinguished Variables
6.7. The Weierstrass Preparation Theorem
6.8. Proofs
6.9. The Implicit Function Theorem
6.10. Hensel's Lemma
6.11. Divisibility in the Ring of Power Series
6.12. Germs of Analytic Sets
6.13. Study's Lemma
6.14. Local Branches
Chapter 7. Parametrizing the Branches of a Curve by Puiseux Series
7.1. Formulating the Problem
7.2. Theorem on the Puiseux Series
7.3. The Carrier of a Power Series
7.4. The Quasihomogeneous Initial Polynomial
7.5. The Iteration Step
7.6. The Iteration
7.7. Formal Parametrizations
7.8. Puiseux's Theorem (Geometric Version)
7.9. Proof
7.10. Variation of Solutions
7.11. Convergence of the Puiseux Series
7.12. Linear Factorization of Weierstrass Polynomials
Chapter 8. Tangents and Intersection Multiplicities of Germs of Curves
8.1. Tangents to Germs of Curves
8.2. Tangents at Smooth and Singular Points
8.3. Local Intersection Multiplicity with a Line
8.4. Local Intersection Multiplicity with an Irreducible Germ
8.5. Local Intersection Multiplicity of Germs of Curves
8.6. Intersection Multiplicity and Order
8.7. Local and Global Intersection Multiplicity
Chapter 9. The Riemann Surface of an Algebraic Curve
9.1. Riemann Surfaces
9.2. Examples
9.3. Desingularization of an Algebraic Curve
9.4. Proof
9.5. Connectedness of a Curve
9.6. The Riemann-Hurwitz Formula
9.7. The Genus Formula for Smooth Curves
9.8. The Genus Formula for Plucker Curves
9.9. Max Noether's Genus Formula
Appendix 1. The Resultant
A.l.l. The Resultant and Common Zeros
A.1.2. The Discriminant
A.1.3. The Resultant of Homogeneous Polynomials
A.1.4. The Resultant and Linear Factors
Appendix 2. Covering Maps
A.2.1. Definitions
A.2.2. Proper Maps
A.2.3. Lifting Paths
Appendix 3. The Implicit Function Theorem
Appendix 4. The Newton Polygon
A.4.1. The Newton Polygon of a Power Series
A.4.2. The Newton Polygon of a Weierstrass Polynomial
Appendix 5. A Numerical Invariant of Singularities of Curves
A.5.1. Analytic Equivalence of Singularities
A.5.2. The Degree of a Singularity
A.5.3. The General Class Formula
A.5.4. The General Genus Formula
A.5.5. Degree and Order
A.5.6. Examples
Appendix 6. Harnack's Inequality
A.6.1. Real Algebraic Curves
A.6.2. Connected Components and Degree
A.6.3. Homology with Coefficients in Z/2Z
Bibliography
Subject Index
List of Symbols
