Overview
- Employs proven conception of teaching topics in commutative algebra through a focus on their applications to algebraic geometry, a significant departure from other works on plane algebraic curves in which the topological-analytic aspects are stressed
- Requires only a basic knowledge of algebra, with all necessary algebraic facts collected into several appendices
- Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook
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Table of contents (17 chapters)
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Front Matter
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Plane Algebraic Curves
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Front Matter
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Back Matter
Keywords
About this book
This work treats an introduction to commutative ring theory and algebraic plane curves, requiring of the student only a basic knowledge of algebra, with all of the algebraic facts collected into several appendices that can be easily referred to, as needed.
Kunz's proven conception of teaching topics in commutative algebra together with their applications to algebraic geometry makes this book significantly different from others on plane algebraic curves. The exposition focuses on the purely algebraic aspects of plane curve theory, leaving the topological and analytical viewpoints in the background, with only casual references to these subjects and suggestions for further reading.
Most important to this text:
* Emphasizes and utilizes the theory of filtered algebras, their graduated rings and Rees algebras, to deduce basic facts about the intersection theory of plane curves
* Presents residue theory in the affine plane and its applications to intersection theory
* Methods of proof for the Riemann–Roch theorem conform to the presentation of curve theory, formulated in the language of filtrations and associated graded rings
* Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook
From a review of the German edition:
"[T]he reader is invited to learn some topics from commutative ring theory by mainly studying their illustrations and applications in plane curve theory. This methodical approach is certainly very enlightening and efficient for both teachers and students… The whole text is a real masterpiece of clarity, rigor, comprehension, methodical skill, algebraic and geometric motivation…highly enlightening, motivating and entertaining at the same time… One simply cannot do better in writing such a textbook."
—ZentralblattMATH
Reviews
"This text stands out by the author's...writing style characterized by its systematic representations, didactical perfection, comprehensiveness, mathematical rigor, thematic determination, and striving for self-containedness. Like in most of his other textbooks on algebra and algebraic geometry [the author] focuses on the inseparable interplay between those two branches of mathematics, and again he presents and hits for further reading. There is no doubt that the international mathematical community, including students and teachers, will welcome the overdue English edition of this masterly textbook as a very special and useful addition to the great standard texts on plane curves." —Zentralblatt MATH
"The translation of the book is impeccable, one would never imagine that the book was written in another language. Moreover, the exposition is very clear and the reading flows nicely. The book is a very good choice for a first course in algebraic geometry. As a prerequisite the reader needs some basic notions of algebra; the rest of the needed algebraic requirements are developed in the appendices." —MAA Reviews
From a review of the German edition: "[T]he reader is invited to learn some topics from commutative ring theory by mainly studying their illustrations and applications in plane curve theory. This methodical approach is certainly very enlightening and efficient for both teachers and students…The whole text is a real masterpiece of clarity, rigor, comprehension, methodical skill, algebraic and geometric motivation…highly enlightening, motivating and entertaining at the same time…One simply cannot do better in writing such a textbook." —Zentralblatt MATH
Authors and Affiliations
Bibliographic Information
Book Title: Introduction to Plane Algebraic Curves
Authors: Ernst Kunz
DOI: https://doi.org/10.1007/0-8176-4443-1
Publisher: Birkhäuser Boston, MA
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Birkhäuser Boston 2005
Softcover ISBN: 978-0-8176-4381-2Published: 15 August 2005
eBook ISBN: 978-0-8176-4443-7Published: 10 June 2007
Edition Number: 1
Number of Pages: XIV, 294
Number of Illustrations: 52 b/w illustrations
Additional Information: Based on the original German edition, "Ebene algebraische Kurven", Der Regensburg Trichter, 23, Universität Regensburg, ISBN 3-88246-167-5, © 1991 Ernst Kunz; English translation by Richard G. Belshoff.
Topics: Algebraic Geometry, Algebraic Topology, Applications of Mathematics, Commutative Rings and Algebras, Associative Rings and Algebras, Field Theory and Polynomials