The author of Introduction to Plane Algebraic Curves remarks in the preface that the best way to introduce commutative algebra is to simultaneously present applications in algebraic geometry. The book, however, is an introduction to algebraic geometry which simultaneously presents the theory of commutative algebra. The volume is divided in two parts. Part I is an introduction to the geometry of plane algebraic curves while Part II is a set of appendices which develops the commutative algebra background necessary for Part I.
The approach to plane curves is strictly algebraic (the author refers the more analytically inclined reader to other suitable texts, such as Plane Algebraic Curves, by Brieskorn and Knorrer) with great emphasis on intersection theory. The book is intended as a first course so the first topics cover the basics of affine and projective algebraic curves, coordinate rings and rational functions. Other topics covered by the book: regular and singular points; parametric representations; polar and hessians; intersection multiplicity, cycles and applications of intersection theory (like Pascal's theorem); introduction to elliptic curves; residue calculus and applications to intersection theory; the Riemann-Roch theorem, the genus and the function field; the canonical divisor class; the branches, conductor and value semigroup of curve singularities. The approach to some of the more advanced topics differs from other more traditional and classical texts since the proofs rely on the theory of filtered algebras, graded and Rees rings.
The second part (the appendices) is aimed to introduce the theory of graded, filtered and Rees algebras. Some other topics which are covered are rings of quotients and localizations; noetherian local rings, discrete valuation rings, complete rings and completions; integral ring extensions; tensor products of algebras; traces; ideal quotients, among others.
The translation of the book is impeccable, one would never imagine that the book was written in another language. Moreover, the exposition is very clear and the reading flows nicely. The book is a very good choice for a first course in algebraic geometry. As a prerequisite the reader needs some basic notions of algebra; the rest of the needed algebraic requirements are developed in the appendices. Sometimes, the fact that the text deals with curves as opposed to more general varieties seems unnecessarily restrictive. Finally, I did wish that the exposition of the commutative algebra and the algebraic geometry was truly simultaneous, instead of relegating both subjects to different parts of the book. However, a truly simultaneous treatment may be unrealistic in some ways and distracting to the student who wants to learn geometry.
Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.
* 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
* Ideal Quotients
* Complete Rings. Completion
* Tools for a Proof of the Riemann–Roch Theorem
* List of Symbols