Plane Algebraic Curves

by Fischer, Gerd; Kay, Leslie

• ISBN13: 9780821821220
• ISBN10: 0821821229
• Format: Paperback
• Copyright: 2001-08-01
• Publisher: Amer Mathematical Society

Summary

The study of the zeroes of polynomials, which for one variable is essentially algebraic, becomes a geometric theory for several variables. In this book, Fischer looks at the classic entry point to the subject: plane algebraic curves. Here one quickly sees the mix of algebra and geometry, as well as analysis and topology, that is typical of complex algebraic geometry, but without the need for advanced techniques from commutative algebra or the abstract machinery of sheaves and schemes. In the first half of this book, Fischer introduces some elementary geometrical aspects, such as tangents, singularities, inflection points, and so on. The main technical tool is the concept of intersection multiplicity and Bézout's theorem. This part culminates in the beautiful Plücker formulas, which relate the various invariants introduced earlier. The second part of the book is essentially a detailed outline of modern methods of local analytic geometry in the context of complex curves. This provides the stronger tools needed for a good understanding of duality and an efficient means of computing intersection multiplicities introduced earlier. Thus, we meet rings of power series, germs of curves, and formal parametrizations. Finally, through the patching of the local information, a Riemann surface is associated to an algebraic curve, thus linking the algebra and the analysis. Concrete examples and figures are given throughout the text, and when possible, procedures are given for computing by using polynomials and power series. Several appendices gather supporting material from algebra and topology and expand on interesting geometric topics. This is an excellent introduction to algebraic geometry, which assumes only standard undergraduate mathematical topics: complex analysis, rings and fields, and topology. Reading this book will help the student establish the appropriate geometric intuition that lies behind the more advanced ideas and techniques used in the study of higher dimensional varieties. This is the English translation of a German work originally published by Vieweg Verlag (Wiesbaden, Germany

Table of Contents

Preface to the English Edition	xi
Preface to the German Edition	xiii
Introduction	1	(12)
Lines	1	(1)
Circles	2	(1)
The Cuspidal Cubic	3	(1)
The Nodal Cubic	4	(2)
The Folium of Descartes	6	(1)
Cycloids	7	(2)
Klein Quartics	9	(1)
Continuous Curves	10	(3)
Affine Algebraic Curves and Their Equations	13	(10)
The Variety of an Equation	13	(1)
Affine Algebraic Curves	14	(1)
Study's Lemma	15	(2)
Decomposition into components	17	(1)
Irreducibility and Connectedness	18	(1)
The Minimal Polynomial	18	(1)
The Degree	19	(1)
Points of Intersection with a Line	20	(3)
The Projective Closure	23	(12)
Points at Infinity	23	(1)
The Projective Plane	23	(2)
The Projective Closure of a Curve	25	(2)
Decomposition into Components	27	(1)
Intersection Multiplicity of Curves and Lines	28	(1)
Intersection of Two Curves	29	(2)
Bezout's Theorem	31	(4)
Tangents and Singularities	35	(24)
Smooth Points	35	(1)
The Singular Locus	36	(1)
Local Order	37	(3)
Tangents at Singular Points	40	(4)
Order and Intersection Multiplicity	44	(1)
Euler's Formula	45	(2)
Curves through Prescribed Points	47	(2)
Number of Singularities	49	(1)
Chebyshev Curves	50	(9)
Polars and Hessian Curves	59	(14)
Polars	59	(5)
Properties of Polars	64	(1)
Intersection of a Curve with Its Polars	64	(1)
Hessian Curves	65	(2)
Intersection of the Curve with Its Hessian Curve	67	(2)
Examples	69	(4)
The Dual Curve and the Plucker Formulas	73	(22)
The Dual Curve	73	(7)
Algebraicity of the Dual Curve	80	(1)
Irreducibility of the Dual Curve	81	(2)
Local Numerical Invariants	83	(2)
The Bidual Curve	85	(1)
Simple Double Points and Cusps	86	(2)
The Plucker Formulas	88	(2)
Examples	90	(1)
Proof of the Plucker Formulas	90	(5)
The Ring of Convergent Power Series	95	(30)
Global and Local Irreducibility	95	(1)
Formal Power Series	96	(3)
Convergent Power Series	99	(1)
Banach Algebras	100	(3)
Substitution of Power Series	103	(2)
Distinguished Variables	105	(2)
The Weierstrass Preparation Theorem	107	(2)
Proofs	109	(5)
The Implicit Function Theorem	114	(2)
Hensel's Lemma	116	(1)
Divisibility in the Ring of Power Series	117	(3)
Germs of Analytic Sets	120	(1)
Study's Lemma	121	(1)
Local Branches	122	(3)
Parametrizing the Branches of a Curve by Puiseux Series	125	(22)
Formulating the Problem	125	(1)
Theorem on the Puiseux Series	126	(1)
The Carrier of a Power Series	127	(2)
The Quasihomogeneous Initial Polynomial	129	(2)
The Iteration Step	131	(1)
The Iteration	132	(3)
Formal Parametrizations	135	(1)
Puiseux's Theorem (Geometric Version)	136	(2)
Proof	138	(3)
Variation of Solutions	141	(1)
Convergence of the Puiseux Series	142	(2)
Linear Factorization of Weierstrass Polynomials	144	(3)
Tangents and Intersection Multiplicities of Germs of Curves	147	(16)
Tangents to Germs of Curves	147	(2)
Tangents at Smooth and Singular Points	149	(1)
Local Intersection Multiplicity with a Line	150	(5)
Local Intersection Multiplicity with an Irreducible Germ	155	(2)
Local Intersection Multiplicity of Germs of Curves	157	(1)
Intersection Multiplicity and Order	158	(1)
Local and Global Intersection Multiplicity	159	(4)
The Riemann Surface of an Algebraic Curve	163	(18)
Riemann Surfaces	163	(2)
Examples	165	(3)
Desingularization of an Algebraic Curve	168	(2)
Proof	170	(5)
Connectedness of a Curve	175	(1)
The Riemann-Hurwitz Formula	175	(1)
The Genus Formula for Smooth Curves	176	(2)
The Genus Formula for Plucker Curves	178	(2)
Max Noether's Genus Formula	180	(1)
Appendix 1. The Resultant	181	(8)
A.1.1. The Resultant and Common Zeros	181	(2)
A.1.2. The Discriminant	183	(1)
A.1.3. The Resultant of Homogeneous Polynomials	184	(1)
A.1.4. The Resultant and Linear Factors	185	(4)
Appendix 2. Covering Maps	189	(4)
A.2.1. Definitions	189	(2)
A.2.2. Proper Maps	191	(1)
A.2.3. Lifting Paths	192	(1)
Appendix 3. The Implicit Function Theorem	193	(4)
Appendix 4. The Newton Polygon	197	(8)
A.4.1. The Newton Polygon of a Power Series	197	(2)
A.4.2. The Newton Polygon of a Weierstrass Polynomial	199	(6)
Appendix 5. A Numerical Invariant of Singularities of Curves	205	(12)
A.5.1. Analytic Equivalence of Singularities	205	(1)
A.5.2. The Degree of a Singularity	206	(4)
A.5.3. The General Class Formula	210	(1)
A.5.4. The General Genus Formula	211	(1)
A.5.5. Degree and Order	212	(2)
A.5.6. Examples	214	(3)
Appendix 6. Harnack's Inequality	217	(6)
A.6.1. Real Algebraic Curves	217	(1)
A.6.2. Connected Components and Degree	218	(3)
A.6.3. Homology with Coefficients in Z/2Z	221	(2)
Bibliography	223	(4)
Subject Index	227	(4)
List of Symbols	231