Plane Algebraic Curves

Gerd Fischer

230 pages, parution le 22/01/2002

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Résumé

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

Contents

Introduction
Affine algebraic curves and their equations
The projective closure
Tangents and singularities
Polars and Hessian curves
The dual curve and the Plücker formulas
The ring of convergent power series
Parametrizing the branches of a curve by Puiseux series
Tangents and intersection multiplicities of germs of curves
The Riemann surface of an algebraic curve
The resultant
Covering maps
The implicit function theorem
The Newton polygon
A numerical invariant of singularities of curves
Harnack's inequality
Bibliography
Subject index
List of symbols

Caractéristiques techniques

	PAPIER
Éditeur(s)	American Mathematical Society (AMS)
Auteur(s)	Gerd Fischer
Parution	22/01/2002
Nb. de pages	230
Format	13,8 x 21,5
Couverture	Broché
Poids	250g
Intérieur	Noir et Blanc
EAN13	9780821821220
ISBN13	978-0-8218-2122-0