Description:
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 employed herein 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. Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook. TOC:Preface * Some Conventions and Notations * Affine algebraic curves * Projective algebraic curves * The coordinate ring * Rational functions on algebraic curves * Intersection multiplicity * Regular and singular points * 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 intersection theory * The Riemann-Roch Theorem * The genus of an algebraic curve and its function field * The canonical divisor * The branches of a curve singularity * Conductor and value groups of a curve singularity * Algebraic Foundations * A: Graded algebras and modules * B: Filtered algebras * C: Rings of quotients. Localization * D: The Chinese remainder theorem * E: Noetherian local rings * F: Integral ring extensions * G: Tensor products of algebras * H: Traces * I: Ideal quotients * J: Complete rings. Completion * K: On the proof of the Riemann-Roch theorem * Bibliography