Plane Algebraic CurvesAmerican Mathematical Soc., 2001 - 231 Seiten 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 |
Inhalt
Chapter 0 Introduction | 1 |
Chapter 1 Affine Algebraic Curves and Their Equations | 13 |
Chapter 2 The Projective Closure | 23 |
Chapter 3 Tangents and Singularities | 35 |
Chapter 4 Polars and Hessian Curves | 59 |
Chapter 5 The Dual Curve and the Plücker Formulas | 73 |
Chapter 6 The Ring of Convergent Power Series | 95 |
Chapter 7 Parametrizing the Branches of a Curve by Puiseux Series | 125 |
Appendix 2 Covering Maps | 189 |
Appendix 3 The Implicit Function Theorem | 193 |
Appendix 4 The Newton Polygon | 197 |
Appendix 5 A Numerical Invariant of Singularities of Curves | 205 |
Appendix 6 Harnacks Inequality | 217 |
| 223 | |
| 227 | |
List of Symbols | 231 |
Chapter 8 Tangents and Intersection Multiplicities of Germs of Curves | 147 |
Chapter 9 The Riemann Surface of an Algebraic Curve | 163 |
Appendix 1 The Resultant | 181 |
Back Cover | 232 |
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Häufige Begriffe und Wortgruppen
algebraic curve C C arbitrary assume Bézout's theorem biholomorphic bitangents called carrier of f Chapter coefficients compute connected component construction convergent coordinates Corollary corresponding covering map curve C C P2(C curve of degree cuspidal cubic cuspidal tangent defined definition deg f dual curve equation equivalent examples exist factor of f Figure finite geometric given gives Hence Hessian curve homeomorphism homogeneous of degree hypocycloid implicit function theorem inflection points initial polynomial intentionally left blank intersection multiplicity invariant minimal polynomial mult(C neighborhood Newton polygon nodal cubic ord f ordinary double points plane Plücker curve Plücker formulas point at infinity points of intersection polar polynomial f polynomial of degree preparation theorem projective Proof prove Puiseux parametrization Puiseux series quadric rational parametrization real numbers resultant Riemann surface Section 8.1 simple cusp simple double point singular point smooth points topology unique factorization domain Weierstrass polynomial