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Undergraduate Algebraic Geometry (London Mathematical Society Student Texts), by Miles Reid
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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.
- Sales Rank: #202753 in Books
- Brand: Brand: Cambridge University Press
- Published on: 1989-01-27
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .31" w x 5.98" l, .50 pounds
- Binding: Paperback
- 140 pages
- Used Book in Good Condition
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
Most helpful customer reviews
15 of 17 people found the following review helpful.
Exceptionally good book; but make sure this is what you need
By Dan
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.
15 of 19 people found the following review helpful.
Unreadable but well-organized
By Mike
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.
3 of 3 people found the following review helpful.
Very Ambitious
By Don Quixote
I picked this up as a self study entry point into the subject. Its a short read, but not terse at all, just a bit less formal than the more rigorous graduate level texts (I consider this book ambitious since this is generally considered a graduate level topic). Rather than throwing several complicated ideas at you and leaving it up to you to make sense of it all, it dives straight to the conclusions of what the author considers most important ideas. While some commutative algebra is an obvious prerequisite, I found myself having to backtrack a bit and take some detours into projective geometry. This text isn't intended to get you far, its just a starting point, and a great one at that considering its undergraduate level audience.
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