# Plane Algebraic Curves

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*Gerd Fischer*

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).

#### Readership

Advanced undergraduates, graduate students, and research mathematicians interested in algebraic geometry.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Plane Algebraic Curves

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface to the English Edition xi12 free
- Preface to the German Edition xiii14 free
- Chapter 0. Introduction 118 free
- Chapter 1. Affine Algebraic Curves and Their Equations 1330
- Chapter 2. The Projective Closure 2340
- Chapter 3. Tangents and Singularities 3552
- Chapter 4. Polars and Hessian Curves 5976
- Chapter 5. The Dual Curve and the Plücker Formulas 7390
- 5.1. The Dual Curve 7390
- 5.2. Algebraicity of the Dual Curve 8097
- 5.3. Irreducibility of the Dual Curve 8198
- 5.4. Local Numerical Invariants 83100
- 5.5. The Bidual Curve 85102
- 5.6. Simple Double Points and Cusps 86103
- 5.7. The Plücker Formulas 88105
- 5.8. Examples 90107
- 5.9. Proof of the Plucker Formulas 90107

- Chapter 6. The Ring of Convergent Power Series 95112
- 6.1. Global and Local Irreducibility 95112
- 6.2. Formal Power Series 96113
- 6.3. Convergent Power Series 99116
- 6.4. Banach Algebras 100117
- 6.5. Substitution of Power Series 103120
- 6.6. Distinguished Variables 105122
- 6.7. The Weierstrass Preparation Theorem 107124
- 6.8. Proofs 109126
- 6.9. The Implicit Function Theorem 114131
- 6.10. Hensel's Lemma 116133
- 6.11. Divisibility in the Ring of Power Series 117134
- 6.12. Germs of Analytic Sets 120137
- 6.13. Study's Lemma 121138
- 6.14. Local Branches 122139

- Chapter 7. Parametrizing the Branches of a Curve by Puiseux Series 125142
- 7.1. Formulating the Problem 125142
- 7.2. Theorem on the Puiseux Series 126143
- 7.3. The Carrier of a Power Series 127144
- 7.4. The Quasihomogeneous Initial Polynomial 129146
- 7.5. The Iteration Step 131148
- 7.6. The Iteration 132149
- 7.7. Formal Parametrizations 135152
- 7.8. Puiseux's Theorem (Geometric Version) 136153
- 7.9. Proof 138155
- 7.10. Variation of Solutions 141158
- 7.11. Convergence of the Puiseux Series 142159
- 7.12. Linear Factorization of Weierstrass Polynomials 144161

- Chapter 8. Tangents and Intersection Multiplicities of Germs of Curves 147164
- 8.1. Tangents to Germs of Curves 147164
- 8.2. Tangents at Smooth and Singular Points 149166
- 8.3. Local Intersection Multiplicity with a Line 150167
- 8.4. Local Intersection Multiplicity with an Irreducible Germ 155172
- 8.5. Local Intersection Multiplicity of Germs of Curves 157174
- 8.6. Intersection Multiplicity and Order 158175
- 8.7. Local and Global Intersection Multiplicity 159176

- Chapter 9. The Riemann Surface of an Algebraic Curve 163180
- 9.1. Riemann Surfaces 163180
- 9.2. Examples 165182
- 9.3. Desingularization of an Algebraic Curve 168185
- 9.4. Proof 170187
- 9.5. Connectedness of a Curve 175192
- 9.6. The Riemann-Hurwitz Formula 175192
- 9.7. The Genus Formula for Smooth Curves 176193
- 9.8. The Genus Formula for P Kicker Curves 178195
- 9.9. Max Noether's Genus Formula 180197

- Appendix 1. The Resultant 181198
- Appendix 2. Covering Maps 189206
- Appendix 3. The Implicit Function Theorem 193210
- Appendix 4. The Newton Polygon 197214
- Appendix 5. A Numerical Invariant of Singularities of Curves 205222
- Appendix 6. Harnack's Inequality 217234
- Bibliography 223240
- Subject Index 227244
- List of Symbols 231248
- Back Cover Back Cover1249