Undergraduate Algebraic Geometry (London Mathematical Society Student Texts) [Paperback]

Miles Reid (Author)

Book Description

ISBN-10: 0521356628 | ISBN-13: 978-0521356626 | Publication Date: January 27, 1989

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.

Editorial Reviews

Review

"Before Reid's volume there was hardly anything to recommend at the undergraduate level...Reid's book is fun; it is filled with examples, applications, asides, gossip...What it does, it does well, and there is nothing comparable." Choice

"...at a level advanced undergraduates will understand and appreciate." Mathematics Magazine

"...the author leads the student on a lively, interesting, down-to-earth tour of the fundamental algebraic geometry...with some welcome, provocative comments..." American Mathematical Monthly

Product Details

• Paperback: 140 pages
• Publisher: Cambridge University Press (January 27, 1989)
• Language: English
• ISBN-10: 0521356628
• ISBN-13: 978-0521356626
• Product Dimensions: 8.8 x 6 x 0.5 inches
• Shipping Weight: 7.8 ounces (View shipping rates and policies)
• Average Customer Review: 3.3 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #670,673 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

42 of 48 people found the following review helpful:

4.0 out of 5 stars baked just right for the first timers !, July 12, 2000

By A Customer

This review is from: Undergraduate Algebraic Geometry (London Mathematical Society Student Texts) (Paperback)

There are many good books on the subject of algebraic geometry, so what was the use of one more - asks the author in the preface to this book. But there are none -at the UG level- which for the first time reveal to the younger mathematicians the secrets of this vast and growing subject. The book treats every new concept with the rigour that keeps in mind the level it is meant for, and yet maintains its mathematical "beauty" - setting firmly the basics for those who would want to take up this course at an advanced level as well as keeping the more casual mathematics reader interested.

8 of 10 people found the following review helpful:

5.0 out of 5 stars Exceptionally good book; but make sure this is what you need, March 20, 2010

By 

Dan - See all my reviews

This review is from: Undergraduate Algebraic Geometry (London Mathematical Society Student Texts) (Paperback)

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.

4 of 8 people found the following review helpful:

1.0 out of 5 stars Unreadable but well-organized, January 21, 2010

By 

Michael Abrahams - See all my reviews
(REAL NAME)   

Amazon Verified Purchase

This review is from: Undergraduate Algebraic Geometry (London Mathematical Society Student Texts) (Paperback)

It is difficult to see who this book is aimed at. Perhaps the extremely gifted undergraduate who can fill in sketchy, incomplete, difficult proofs, but has also taken courses? My professor (a topologist) even had a difficult time presenting the material as-is and solving the exercises, as very few examples were given, hence it was unclear exactly what was required for a satisfactory proof of the questions as stated. Reid, probably in an effort to save space, delegates difficult steps of proofs to the reader by declaring them "obvious," making the book practically unreadable to the average undergraduate student. The notation is used strangely and the typesetting is awkward.

The proof of the 27-lines theorem is interesting and a decent capstone for the introductory subject. However, I did not feel as though I had deepened my knowledge of algebraic geometry as a result, only having learned the bare minimum to approach one useless (albeit entertaining) theorem.

If you have to use this book I recommend buying another one to supplement the background knowledge and to figure out how to complete the proofs.