Introduction to plane algebraic curves
著者
書誌事項
Introduction to plane algebraic curves
Birkhäuser, c2005
- タイトル別名
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Ebene algebraische Kurven
- 統一タイトル
-
Ebene algebraische Kurven
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注記
Includes bibliographical references and index
"Based on th original German edition, Ebene algebraische Kurven"--T.p. verso
内容説明・目次
内容説明
* 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
目次
* 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
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