Introduction to Plane Algebraic Curves

ISBN-13: 9780817643812 / Angielski / Miękka / 2005 / 294 str.

Ernst Kunz; Richard G. Belshoff

Opis

* 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

Recenzje

"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

Spis treści

* Preface * Conventions and Notation * Part I: Plane Algebraic Curves * Affine 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 * Part II: Algebraic Foundations * Algebraic Foundations * Graded Algebras and Modules * Filtered Algebras * Rings of Quotients. Localization * The Chinese Remainder Theorem * Noetherian Local Rings and Discrete Valuation Rings * Integral Ring Extensions * Tensor Products of Algebras * Traces * Ideal Quotients * Complete Rings. Completion * Tools for a Proof of the Riemann–Roch Theorem * References * Index * List of Symbols

Blurb

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."

—Zentralblatt MATH