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Plane Algebraic Curves
Gerd Fischer, Heinrich-Heine-Universität, Düsseldorf, Germany
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Student Mathematical Library
2001; 231 pp; softcover
Volume: 15
ISBN-10: 0-8218-2122-9
ISBN-13: 978-0-8218-2122-0
List Price: US$44 Institutional Members: US$35.20
All Individuals: US\$35.20
Order Code: STML/15

Elementary Algebraic Geometry - Klaus Hulek

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.

Reviews

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

• 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