Principles of Algebraic Geometry (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts) [Hardcover]

Phillip Griffiths (Author), Joseph Harris (Author)

Book Description

ISBN-10: 0471327921 | ISBN-13: 978-0471327929 | Publication Date: October 1978 | Edition: 1

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.

Editorial Reviews

From the Inside Flap

Changes in algebraic geometry have made it a subject that, over the past few years, has become increasingly inaccessible to all but the specialist. This comprehensive, self-contained treatment presents some of the main, general results of the theory accompanied by (and with emphasis on) their applications to the study of interesting examples and to the development of computational tools. It establishes a geometric intuition and a working facility with specific geometric practices, providing mathematicians and physicists with a greater accessibility to the field. The effective utilization of the techniques of elementary complex analysis and topology synthesize the classical and the modern—the geometric and the abstract-into a cohesive presentation. Coverage ranges from analytic to geometric along classical lines. Basic techniques and results of complex manifold theory are treated, focusing on results applicable to projective varieties. Further discussions include the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex, and special topics in complex manifolds.

From the Back Cover

Other volumes in the Pure and Applied Mathematics series— The Algebraic Structure of Group Rings Donald S. Passman This book offers a comprehensive, self-contained treatment of group rings of infinite groups. It begins with basic definitions and contains background material on group theory and ring theory. Major topics considered include: the trace map, the augmentation ideal and dimension subgroups, linear and polynomial identities and their relationship to the center, semisimplicity and primitivity, polycyclic-by-finite groups and Philip Hall’s problem, zero divisors, and isomorphism questions. 1977 Topological Uniform Structures Warren Page Here is an overall unifying theme of topologies compatible with increasingly enriched algebraic structures, showing the rich interplay among mathematics’ diverse areas. It studies mathematics as a structured, coherent, and harmonious whole, giving a detailed examination of uniform spaces, topological groups, topological vector spaces, topological algebras, and abstract harmonic analysis. Also includes a section on topological vector-valued measure spaces and numerous problems and examples. The text is virtually self-contained, presenting detailed proofs, stressing readability and motivation, and covering much advanced material. 1978 Applied Abstract Analysis Jean-Pierre Aubin Discusses all the main theorems of topology by introducing and studying principal topics in the elementary framework of metric spaces. Considers various applications in differential equations, dynamic systems, game theory, and economics, illustrating the advantages of using an abstract approach to solve problems of a more concrete nature. Also includes a concise review of essential results, a set of exercises and problems, and a terminological index. 1977

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Product Details

• Hardcover: 832 pages
• Publisher: Wiley-Interscience; 1 edition (October 1978)
• Language: English
• ISBN-10: 0471327921
• ISBN-13: 978-0471327929
• Product Dimensions: 8.9 x 6.4 x 2 inches
• Shipping Weight: 2.4 pounds
• Average Customer Review: 4.9 out of 5 stars  See all reviews (8 customer reviews)
• Amazon Best Sellers Rank: #3,223,066 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

38 of 39 people found the following review helpful:

5.0 out of 5 stars The Book, February 15, 2001

By A Customer

This review is from: Principles of Algebraic Geometry (Paperback)

Just wanted to add the following:

1) The mathematics in this book is some of the most beautiful stuff I've ever seen. I don't in any way mean to deny the beauty of the Spec of a Ring, but - even if you have always planned on working in Grothendeick's world - I think this is worth reading for any algebraic geometer (regardless of what field you're living over).

With their bare hands, Griffiths and Harris prove some of the greatest results in maths. I learned more reading Chapter O than I did taking the entire collection of "first- year" grad courses (algebra & analysis). The material was more interesting, and it tied together in a way that had you remember all of it. From elliptic operator theory to the representation of sl(2), in the same chapter!

2) For string theorists trying to learn some of the math lingo, this is a necessary first step, though I would also highly recommend Candelas's notes, and Aspinwall's great paper, "K3 Surfaces and String Duality". Also, Brian Greene's notes are very nice. T. Hubsch's book is also great for the big picture, but I was disappointed by several non-trivial errors in his explanations of math concepts. I recommend all of the above to mathematicians as well - I am a mathematician, and I learned a lot of valuable side material from these physics sources. Especially in trying to understand mirror symmetry. Of course, Cox and Katz's newish book is also excellent for this.

3) My favorite parts: chap 1: divisors and line bundles, the exp sheaf sequence. read this, and then skip to the same picture for line bundles on a torus. the same type of bouncing back and forth works for getting the analogs between Reimann surfaces and complex surfaces...

actually, every page of this huge book has something valuable. I can't imagine what it was like to learn this field before this book came along. The price is exorbitant, but in the grand scheme of things, I've spent hundreds (thousands?) on math books that lie on my shelf, never to be explored. this one has given me years of enjoyment.

43 of 48 people found the following review helpful:

5.0 out of 5 stars More geometry, less algebra., August 13, 1997

By A Customer

This review is from: Principles of Algebraic Geometry (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts) (Hardcover)

This book is a throwback to the time when algebraic geometry was a branch of geometry rather than category theory. As wonderful as the books by Mumford and Hartshorne are, they are rather long on abstract nonsense and short on geometry. This book is a refreshing exception to the 'modern' trend. Actually, there is a renaissance in applications of algebraic geometry to surprizing fields such as encryption and string field theory, and these are more in the spirit of this book than those of the Grothendieck school. Except for the obscenely high price and occasional typos, I highly recommend this book, especially to geometrically inclined mathematicians who don't really care about the category of schemes over an arbitrary field

27 of 29 people found the following review helpful:

4.0 out of 5 stars A review from a graduate student, March 15, 2004

By 

Abed Elhashash "[e-mail]" (Philadelphia, PA USA) - See all my reviews
(REAL NAME)   

This review is from: Principles of Algebraic Geometry (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts) (Hardcover)

If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following:

1. Complex Analysis
2. Differential Geometry and calculus on manifolds
3. Homology-Cohomology Theory
4. Undergraduate Algebraic Geometry

Do not expect chapter 0, "Foundational Material", to be the place where you are supposed to build your "foundation". You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.

However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.

So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.