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Principles of Algebraic Geometry Paperback – August 16, 1994

ISBN-13: 978-0471050599 ISBN-10: 0471050598 Edition: 1st

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Principles of Algebraic Geometry + Algebraic Geometry (Graduate Texts in Mathematics) + Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics)
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Product Details

  • Paperback: 832 pages
  • Publisher: Wiley-Interscience; 1 edition (August 16, 1994)
  • Language: English
  • ISBN-10: 0471050598
  • ISBN-13: 978-0471050599
  • Product Dimensions: 6 x 1.8 x 9 inches
  • Shipping Weight: 3.1 pounds (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (9 customer reviews)
  • Amazon Best Sellers Rank: #169,435 in Books (See Top 100 in Books)

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. --This text refers to an out of print or unavailable edition of this title.

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 --This text refers to an out of print or unavailable edition of this title.

Customer Reviews

4.6 out of 5 stars
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This is an amazing book with an amazing subject (complex algebraic geometry).
The presence of meromorphic functions will alert the astute reader as to the role of Riemann surfaces in the study of complex algebraic varieties.
Dr. Lee D. Carlson
It is very clearly written and ideas are always motivated by examples and problems.

Most Helpful Customer Reviews

43 of 44 people found the following review helpful By A Customer on February 15, 2001
Format: 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.
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37 of 39 people found the following review helpful By Abed Elhashash on March 15, 2004
Format: 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.
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47 of 52 people found the following review helpful By A Customer on August 13, 1997
Format: 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
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32 of 35 people found the following review helpful By Dr. Lee D. Carlson HALL OF FAMEVINE VOICE on July 9, 2002
Format: Paperback
Once thought to be highly esoteric and useless by those interested in applications, algebraic geometry has literally taken the world by storm. Indeed, coding theory, cryptography, steganography, computer graphics, control theory, and artificial intelligence are just a few of the areas that are now making heavy use of algebraic geometry.
This book would probably be the most useful one for those interested in applications, for it is an overview of algebraic geometry from the complex analytic point of view, and complex analysis is a subject that most engineers and scientists have had to learn at some point in their careers. But one must not think that this book is entirely concrete in its content. There are many places where the authors discuss concepts that are very abstract, particularly the discussion of sheaf theory, and this might make its reading difficult. The complex analytic point of view however is the best way of learning the material from a practical point of view, and mastery of this book will pave the way for indulging oneself in its many applications.
Algebraic geometry is an exciting subject, but one must master some background material before beginning a study of it. This is done in the initial part of the book (Part 0), wherein the reader will find an overview of harmonic analysis (potential theory) and Kahler geometry in the context of compact complex manifolds. Readers first encountering Kahler geometry should just view it as a generalization of Euclidean geometry in a complex setting. Indeed, the so-called Kahler condition is nothing other than an approximation of the Euclidean metric to order 2 at each point.
The authors choose to introduce algebraic varieties in a projective space setting in chapter 1, i.e.
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