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Basic Algebraic Geometry I (Springer Study Edition)

3.6 out of 5 stars 5 customer reviews
ISBN-13: 978-0387548128
ISBN-10: 0387548122
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Editorial Reviews


From the reviews: "To my best knowledge, only this book manages to describe so many advanced constructions while still being accessible for researchers outside the field of algebraic geometry. This book is indeed a tremendous achievement." (Newsletter on Computational and Applied Mathematics) --This text refers to an out of print or unavailable edition of this title.

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Product Details

  • Series: Springer Study Edition
  • Paperback: 303 pages
  • Publisher: Springer-Verlag (December 1994)
  • Language: English
  • ISBN-10: 0387548122
  • ISBN-13: 978-0387548128
  • Product Dimensions: 0.8 x 6.2 x 9.2 inches
  • Shipping Weight: 15.2 ounces
  • Average Customer Review: 3.6 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Best Sellers Rank: #3,555,030 in Books (See Top 100 in Books)

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Top Customer Reviews

Format: Paperback
In the 1950's algebraic geometry was tedious and hard to grasp because it was mostly commutative algebra, developed by Zariski and Weil and their schools to fill logical gaps in the Italian arguments of the previous half century. The rich geometric texture of the italian school was lost. In the 1960's Serre and Grothendieck introduced homological algebra to the subject and greatly expanded and enhanced it to embrace also arithmetic, but the abstraction level went WAY up, so again it was hard to grasp and relate to geometry. Hartshorne is a member of both Zariski and Grothendieck's schools and appreciates down to earth objects like space curves, but his book has a long beginning section on schemes and cohomology that can definitely throw a beginner off the horse.

Pardon the delay in getting here, but the point is that Shafarevich's book has none of the tediousness of the previous generation, yet benefits from the rigorous foundations via commutative algebra of Zariski's works. I would say Shafarevich's book, is a geometrically oriented explanation of the material that can be explained using Zariski's methods. I.e., it has a rich geometric feel, is very well explained, includes many easy examples, and is rigorous in its use of commutative algebra. This book allowed many of us who were stymied by the huge amount of algebra needed for 1960's Grothendieck style AG, to finally gain admission to the subject. I love this book.

There are three themes one can mention in algebraic geometry, 1) projective varieties, 2) schemes 3) cohomology. The first topic concerns the objects most geometers are interested in. The second one is of more interest to number theorists, but also has value for geometers in understanding limits of varieties.
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Format: Paperback
The central issue with Shafarevich is that it repeatedly blends the algebraic and geometric aspects of the subject together in such a way that you can no longer be sure what is actually a trivial geometric corollary of a deep algebraic result and vice versa. An example occurs in the first few pages, where Shafarevich presents the most confusing treatment of local parameters I've seen. After reading this, I was at a complete loss; only by consulting Fulton's excellent but short Algebraic Curves was I able to cut the issue down to its fundamental algebraic and geometric components.

Shafarevich aims to develop algebraic geometry by starting from the ultra-concrete and slowly moving, over the duration of two volumes, to the ultra-abstract, and his strategy, which is utilized most explicitly in the first part of this book, is to occasionally revise all preceding definitions in light of some new observation, increasing their generality and occasionally abstractness a little bit each time. This is a reasonable idea which I think could have worked marvelously. But as it stands Shafarevich was far too sloppy to have succeeded in this plan. It's never clear when Shafarevich has decided to make a jump to the next level of generality, and when he does it's as if he tries his hardest to make it as difficult as it could be for you to see exactly how the definitions are revised. Most definitions are just dumped into the middles of paragraphs with nothing to set them off in the text. Concepts which are presented in one order early on are revised in a contradictory order later on. The first part will leave you wondering how the word "variety" has any meaning left in Shafarevich's world after having been revised countless times, each revision only applying in certain contexts.
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Format: Paperback Verified Purchase
DISCLAIMER: I am a lowly first year graduate student trying to learn algebraic geometry for the first time from this book. What follows below is just my opinion. If you love this book you are free to blame my own shortcomings for the difficulty I am having learning from this book.

People say this is the best book to use as an introduction to algebraic geometry. If that is true then algebraic geometry pedagogy is in a sorry state of affairs.

Here is a list of reasons I hate this book:

It doesn't assume any point set topology. Often proofs of basic point set topological facts are done only in the Zariski topology, which is silly and obscures how trivial and general they are. If you don't know point set topology you really shouldn't be learning algebraic geometry yet anyway.

It is weak on algebra and category theory. The correspondences between algebra and geometry are not made clear enough and not exploited enough. I wish the book would be more clear on the sheaf structure of varieties, which would help prepare the reader better for the modern approach to algebraic geometry using schemes.

The book is organized terribly. Important definitions are placed randomly in the middle of paragraphs and are difficult to find if you forget what they are. Also, each definition is made over and over again at varying levels of generality and the explanations as to why they are consistent are either poor or omitted. For a beginner this is horribly confusing. In my opinion it is one of the biggest mistakes you can make writing a book for beginners.

Many of the exercises are completely unclear. Often the notation is not specified and the terms are not defined.
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