Introduction to Plane Algebraic Curves

by Ernst Kunz, Richard G. Belshoff (Translator)

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

* Studies algebraic curves over an algebraically closed field K and those of prime characteristic, which can be applied to coding theory and cryptography

* Covers filtered algebras, the associated graded rings and Rees rings to deduce basic facts about intersection theory of plane curves, applications of which are standard tools of computer algebra

* Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook

Product Details

ISBN-13:	9780817643812
Publisher:	Birkhäuser Boston
Publication date:	08/15/2005
Edition description:	2005
Pages:	294
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

Plane Algebraic Curves.- Ane Algebraic Curves.- Projective Algebraic Curves.- The Coordinate Ring of an Algebraic Curve and the Intersections of Two Curves.- Rational Functions on Algebraic Curves.- Intersection Multiplicity and Intersection Cycle of Two Curves.- Regular and Singular Points of Algebraic Curves. Tangents.- More on Intersection Theory. Applications.- Rational Maps. Parametric Representations of Curves.- Polars and Hessians of Algebraic Curves.- Elliptic Curves.- Residue Calculus.- Applications of Residue Theory to Curves.- The Riemann-Roch Theorem.- The Genus of an Algebraic Curve and of Its Function Field.- The Canonical Divisor Class.- The Branches of a Curve Singularity.- Conductor and Value Semigroup of a Curve Singularity.

Editorial 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

From the Publisher