Undergraduate Algebraic Geometry

London Mathematical Society Student Texts

by Miles Reid

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, 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.

Genres: Mathematics, Reference

140 pages, Paperback

First published December 15, 1988

Reviews

[David]

The subject matter is quite difficult, but the author (Miles Reid) explains it so well that I was able to read all of this book in just two weeks. He even throws in a few jokes, just for fun!

Algebraic geometry is about sets of points (varieties) defined by one or more polynomial equation. It tends to use complex projective space as a setting, so that the varieties are as large and interesting as possible. Also, Bezout's Theorem holds in this setting. Miles Reid explains all of this with great lucidity, using conic sections (quadratic) and elliptic curves (cubic) as concrete examples.

Structures from abstract algebra (rings and ideals) are useful in analysing the structure of varieties. The author gives a concise summary of all the commutative algebra that the reader needs. This leads to Hilbert's Nullstellensatz (theorem about zeros of polynomials) which is a very important and powerful result.

Towards the end, he even mentions an application to designign the shape of a cooling tower at a power station! Algebraic geometry is of central importance in modern maths, and if often used in number theory.

I think that everyone with an interest in maths should know a little about this topic. Miles Reid's book is an essential and accessible introduction, and all undergraduate students should benefit from reading it. (There are several other books at a more advanced level.)

[JT]

I finally listened to what people kept telling me and read this. It really is as good as everyone says. If only he'd write another book just like it about schemes.

[Hollis Williams]

Recommended if you have an interest in algebraic geometry, but it's not an easy read and you have to put some work in to get something back out.

[Lucille Nguyen]

Readable little book for a complex subject. Author focuses on computation more than theory, which I think was not a good choice or should have been stated as such more early.