* 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