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Algebraic Geometry: A First Course (Graduate Texts in Mathematics) (v. 133) Hardcover


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Algebraic Geometry: A First Course (Graduate Texts in Mathematics) (v. 133) + Algebraic Geometry: Part I: Schemes. With Examples and Exercises (Advanced Lectures in Mathematics)
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Product Details

  • Series: Graduate Texts in Mathematics (Book 133)
  • Hardcover: 330 pages
  • Publisher: Springer; Corrected edition (December 21, 1995)
  • Language: English
  • ISBN-10: 0387977163
  • ISBN-13: 978-0387977164
  • Product Dimensions: 9.6 x 6.3 x 0.9 inches
  • Shipping Weight: 1.5 pounds (View shipping rates and policies)
  • Average Customer Review: 3.0 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Best Sellers Rank: #305,818 in Books (See Top 100 in Books)

Editorial Reviews

Review

J. Harris

Algebraic Geometry

A First Course

"This book succeeds brilliantly by concentrating on a number of core topics (the rational normal curve, Veronese and Segre maps, quadrics, projections, Grassmannians, scrolls, Fano varieties, etc.) and by treating them in a hugely rich and varied way. The author ensures that the reader will learn a large amount of classical material and perhaps more importantly, will also learn that there is no one approach to the subject. The essence lies in the range and interplay of possible approaches. The author is to be congratulated on a work of deep and enthusiastic scholarship."—MATHEMATICAL REVIEWS


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33 of 37 people found the following review helpful By Dr. Lee D. Carlson HALL OF FAMEVINE VOICE on August 19, 2001
Format: Hardcover
If one is planning to do work in coding theory, cryptography, computer graphics, digitial watermarking, or are hoping to become a mathematician specializing in algebraic geometry, this book will be of an enormous help. The author does a first class job in introducing the reader to the field of algebraic geometry, using a wealth of examples and with the goal of building intuition and understanding. It is great that a mathematician of the author's caliber would take the time to write these lectures here put into book form. It is rare to find a book on algebraic geometry that attempts to make the subject concrete and understandable, and yet points the way to more modern "scheme-theoretic" formulations.
In lecture 1, the author introduces affine and projective varieties over algebraically closed fields. Linear subspaces of n-dimensional projective space P(n) are shown to be varieties, along with any finite subset of P(n). He delays giving rigorous definitions of degree and dimension, emphasizing instead concrete examples of varieties. The twisted cubic is given as the first example of a concrete variety that is not a hypersurface, along with their generalizations, the rational normal curves.
The Zariski topology, considered by the newcomer to the subject as being a rather "strange" topology, is introduced in lecture 2. The author does a great job though explaining its properties, and introduces the regular functions on affine and projective varieties. The Nullstellensatz theorem, needed to prove that the ring of regular functions is the coordinate ring, is deferred to a later lecture. Rational normal curves are further generalized to Veronese maps in this lecture, and the properties of the corresponding Veronese varieties discussed in some detail.
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17 of 18 people found the following review helpful By Mark Arjomandi on December 3, 2004
Format: Hardcover
Last year we used this title as a main reference for the first two quarters of a year-long introductory sequence on algebraic geometry, at the beginning graduate student level (2nd year). Our professor --who was himself a former student of Harris, and a specialist in the Mori program-- backed up the presentation with personal lecture notes, moving mostly in parallel to the book's topics. In the third quarter of the sequence, we moved up to cover the theory of sheaves and schemes from Robin Hartshorne's advanced treatise. Prior to this point, my only exposure to the subject was from the corresponding chapter in Dummit and Foote's algebra, and also from a recent introductory text, "An Invitation to Algebraic Geometry" by Karen Smith et al. At this stage of my studies I was mainly testing the waters: My ineterest on one hand was driven by a curiosity for the subject itself, which has a reputation for being difficult and hard to grasp, and on the other hand from its pivotal role in the formulation of new physical theories of mirror symmetry and string theory (for more in this direction, see the 1999 AMS title by A. Cox and S. Katz).

Back to the present text, as the editorial notes correctly point out, this Harris book emphasizes the classical algebraic geometry from the 19th & early 20th centuries prior to the introduction of highly abstract machinery, due to the work of A. Groethendieck in the 50's and 60's. Therefore it's quite natural to base the treatment mostly on the examples and concrete constructions, which were the guiding principles of the abstract development in the first place.
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Format: Hardcover
If you were hoping, for some reason, that this book is any more appropriate for a first course as is his more well-known "first course" in representation theory, you're going to be disappointed. Would be very difficult, as a quick perusal of the table of contents - and how many pages each topic is allotted - would indicate to anybody at least vaguely familiar with varieties. Other books 'recommended' in comments are random. Hartshorne is standard for good reasons; your second-best option is to look for some lecture notes on affine and projective varieties, the basic study of which would comprise a big chunk of a first course in algebraic geometry, but which gets about 20 pages in this tome. My recommendation would be Hartshorne, maybe augmented with some other text just to get a second perspective, followed by Eisenbud's 'Geometry of Schemes'; this is a fairly doable path.
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21 of 42 people found the following review helpful By Michael Graber on December 19, 2004
Format: Hardcover
This book is not a good first text for a person trying to learn algebraic geometry. Prof. Harris gives very limited explanations and few clear examples. Try Shafarevich, Basic Algebraic Geometry vols I & II or Miles Reid, Undergraduate Algebraic Geometry. For both books, you would need commutative algebra, at least at the level of Miles Reid, Undergraduate Commutative Algebra.
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6 of 49 people found the following review helpful By "boian" on January 16, 2002
Format: Hardcover
I was confused by the many examples. Most of the times I did not see how they fit together in the theory. Perhaps I would have appreciate it more, had I known some algebraic geometry first.
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