Undergraduate Algebraic Geometry (London Mathematical Society Student Texts, Series Number 12)

This book is intended to provide us with a short (135 pages), down to earth and fluently motivated introduction to algebraic geometry. And it does a great job. While the author does not clearly state his intentions in advance, I think it would be safe to assume that this is meant to accompany a more standard text on the subject (Hartshorne, Harris, Shafarevich, etc), and that the author's main goal was to give the quickest possible route to the heart of the subject, making sure the reader stays interested throughout rather than that he is presented with the firmest logical structure. I would like to stress that despite of what I wrote so far, this book does present rigorous proofs and clear definitions. The style is friendly, straightforward and unpretentious. Everything is well motivated, and one occasionally gets to hear the author's personal perspective or view about a certain topic. I will quote two examples. When discussing the Zariski topology, the author writes "The Zariski topology may cause trouble to some students; since it is only being used as a language, and has almost no content, the difficulty is likely to be psychological rather than technical". This was very calming for me to read, as I have been previously struggling with the "deep meaning" of the Zariski topology, and no book has had the honesty to tell me that I shouldn't worry that much about it. As a second example of the author's style, after a Q.E.D. in page 53 the author explains that "The proof of (b) is a typical algebraist's proof: it's logically very neat, but almost completely hides the content: the real point is that ..." Chapter 1 begins with the concrete example of conics, intended to motivate the later definitions of the projective plane. Next elliptic curves and their group law is discussed. The chapter ends with a brief survey of the genus of curves. Chapter 2 is more technical; its purpose is to build the algebraic foundations needed for Hilbert's Nullstellensatz. Among topics covered are Noetherian rings, Hilbert's Basis Theorem, algebraic sets, the Zariski topology, prime ideals and a nice motivation for the Nullstellensatz. Next coordinate rings, morphisms, varieties and other standard topics are introduced. Chapter 3, titled "Applications", uses the previous material to discuss some nice geometric topics. I especially enjoyed the section on the 27 lines on a cubic surface. I would highly recommend this book to anyone not very familiar with algebraic geometry; for instance, it could be a good reading to decide if you want to take a more serious study (e.g. a university course) of the subject. If I were to suggest only one text for someone who just wants to know what algebraic geometry is all about, it would definitely be this one. Algebraic Geometry Is, Essentially, The Study Of The Solution Of Equations And Occupies A Central Position In Pure Mathematics. This Short And Readable Introduction To Algebraic Geometry Will Be Ideal For All Undergraduate Mathematicians Coming To The Subject For The First Time. With The Minimum Of Prerequisites, Dr Reid Introduces The Reader To The Basic Concepts Of Algebraic Geometry Including: Plane Conics, Cubics And The Group Law, Affine And Projective Varieties, And Non-singularity And Dimension. He Is At Pains To Stress The Connections The Subject Has With Commutative Algebra As Well As Its Relation To Topology, Differential Geometry, And Number Theory. The Book Arises From An Undergraduate Course Given At The University Of Warwick And Contains Numerous Examples And Exercises Illustrating The Theory. Miles Reid. Includes Index. Series: London Mathematical Society Student Texts......Page 2 Title: Undergraduate Algebraic Geometry......Page 4 Copyright......Page 5 Preface......Page 6 Contents......Page 8 §0. Woffle......Page 10 §1. Plane conics......Page 18 §2. Cubics and the group law......Page 36 Appendix to Chapter 1. Curves and their genus......Page 52 §3. Affine varieties and the Nullstellensatz......Page 57 §4. Functions on varieties......Page 75 §5. Projective and birational geometry......Page 88 §6. Tangent space and nonsingularity, dimension......Page 103 §7. The 27 lines on a cubic surface......Page 111 §8. Final comments......Page 123 Index......Page 138 Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. With the minimum of prerequisites, Dr. Reid introduces the reader to the basic concepts of algebraic geometry, including: plane conics, cubics and the group law, affine and projective varieties, and nonsingularity and dimension. He stresses the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book contains numerous examples and exercises illustrating the theory. An introduction to algebraic geometry which covers the basic concepts of plane conics, cubics and group law, affine and projective varieties, and non-singularity and dimension, the connections with commutative algebra as well as its relation to topology, differential geometry and number theory