The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and their Jacobians (Lecture Notes in Mathematics) [Paperback]

David Mumford (Author), E. Arbarello (Contributor)

Book Description

ISBN-10: 354063293X | ISBN-13: 978-3540632931 | Publication Date: October 29, 1999 | Edition: 2nd exp. ed.

Mumford's famous "Red Book" gives a simple, readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra. It is aimed at graduates or mathematicians in other fields wishing to quickly learn aboutalgebraic geometry. This new edition includes an appendix that gives an overview of the theory of curves, their moduli spaces and their Jacobians -- one of the most exciting fields within algebraic geometry.

Editorial Reviews

Review

"This is the second edition of a famous and well-known introduction to algebraic geometry, written to show that the language of schemes is fundamentally geometrical and clearly expressing the intuition of algebraic geometry. ... This book can strongly be recommended to anybody interested in algebraic geometry and willing to learn about varieties and schemes and their main problems." EMS Newsletter, Vol. 37, Sept. 2000

Product Details

• Paperback: 304 pages
• Publisher: Springer; 2nd exp. ed. edition (October 29, 1999)
• Language: English
• ISBN-10: 354063293X
• ISBN-13: 978-3540632931
• Product Dimensions: 9.2 x 6.1 x 0.7 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,166,355 in Books (See Top 100 in Books)

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26 of 26 people found the following review helpful:

5.0 out of 5 stars The nearly Royal Road, December 10, 2001

By 

Colin McLarty (Chardon, OH USA) - See all my reviews

This review is from: The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and their Jacobians (Lecture Notes in Mathematics) (Paperback)

In a nutshell, reading this book is like reading the mind of a great mathematician as he thinks about a great new idea. Anyone interested in schemes should read it. But a review needs more detail:

The RED BOOK is a concise, brilliant survey of schemes, by one of the first mathematicians to learn of them from Grothendieck. He gives wonderfully intuitive pictures of schemes, especially of "arithmetic schemes" where number theory appears as geometry. The geometry shines through it all: as in differentials, and etale maps, and how unique factorization relates to non-singularity. There is a bravura discussion of Zariski's Main Theorem (the algebraic property of being "normal" implies that a variety has only one branch at each point) comparing forms of it from older algebraic geometry, topology, power series, and schemes. Mumford cites proofs of these but does not give them. In fact, this theorem was one of the first things Mumford could use, to get Zariski to respect schemes.

Many accomplished algebraic geometers say this book got them started. But you probably cannot learn to work in the subject from this book alone--you either have to work with people who work with it, or use some other books besides (maybe both). The other book would probably be Hartshorne ALGEBRAIC GEOMETRY, which is far more detailed, has far more examples, goes very much farther into cohomology--and is very much longer and denser (though also clearly written).

Eisenbud and Harris GEOMETRY OF SCHEMES covers a lot of the same ground as THE RED BOOK, with fewer advanced topics but many more details and examples, including classical geometric constructions like blow-ups and duals to projective plane curves. They use slightly more category theory than Mumford, more like Grothendieck.

Probably none of these books will work for you unless you already know some algebraic geometry: how polynomials define a variety, the Zariski topology, what regular and birational maps are. There is more than enough in Myles Reid's humorously titled UNDERGRADUATE ALGEBRAIC GEOMETRY and UNDERGRADUATE COMMUTATIVE ALGEBRA with vividly geometric ideas in slightly scheme-theoretic language.

The RED BOOK now includes the Michigan lectures, which are reputedly terrific, but I have not worked through them.

22 of 22 people found the following review helpful:

5.0 out of 5 stars Before Hartshorne, November 30, 1999

By 

Jeffrey Bolden (Princeton, NJ) - See all my reviews
(REAL NAME)   

This review is from: The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and their Jacobians (Lecture Notes in Mathematics) (Paperback)

There is a problem in getting going with alg. geo. To learn the geometry you need commutative algebra and to contextualize commutative algebra you need algebraic geometry. Mumford is an excellent book to get going without the need for the heavy prereqs of the more classic books like Hartshore or G&H. A really good read.

This is not however a terrific reference text, you'll need something else as a reference. Its much to expository and their is no index.