Plane Algebraic Curves (Student Mathematical Library, V. 15) by Gerd Fischer (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review 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

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I am so glad that I bought this book. It is full of great information to strengthen our understanding of Algebraic Geometry. Contains many examples that will nourish your mathematical experience. There may be no royal road to Algebraic Geometry but this book is a fine footpath for those intrepid sojourners in the math sciences along with Ueno’s excellent Introduction to Algebraic Geometry (which has exercises). Good luck to you all.

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