* 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