Plane Algebraic Curves (Student Mathematical Library, V. 15)

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 PlÃÂ1⁄4cker 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. 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 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 Chapter 0. Introduction Chapter 1. Affine Algebraic Curves And Their Equations Chapter 2. The Projective Closure Chapter 3. Tangents And Singularities Chapter 4. Polars And Hessian Curves Chapter 5. The Dual Curve And The Plücker Formulas Chapter 6. The Ring Of Convergent Power Series Chapter 7. Parametrizing The Branches Of A Curve By Puiseux Series Chapter 8. Tangents And Intersection Multiplicities Of Germs Of Curves Chapter 9. The Riemann Surface Of An Algebraic Curve Appendix 1. The Resultant Appendix 2. Covering Maps Appendix 3. The Implicit Function Theorem Appendix 4. The Newton Polygon Appendix 5. A Numerical Invariant Of Singularities Of Curves Appendix 6. Harnack's Inequality Gerd Fischer ; Translated By Leslie Kay. Includes Bibliographical References (p. 223-225) And Index. ''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 establish the geometric intuition that lies behind the more advanced ideas and techniques used in the study of higher-dimensional varieties.''-- Site de l'éditeur 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 establish the geometric intuition that lies behind the more advanced ideas and techniques used in the study of higher-dimensional varieties.