Introduction to Plane Algebraic Curves

* 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