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Elementary Algebraic Geometry (Student Mathematical Library, Vol. 20) (Student Mathematical Library, V. 20)

4.5 4.5 out of 5 stars 15 ratings

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This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra.

The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory.

The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry.


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Editorial Reviews

Review

"The book balances theory and examples well and the exercises are well-chosen to further illustrate the basic concepts. All in all, the book does an excellent job of explaining what algebraic geometry is about, what are the basic results, and it invites the reader to continue exploring the subject ... I would definitely recommend it as reading material to a bright undergraduate who has taken a basic course on rings and fields and has read about Noetherian rings. It is certainly suitable for a one-semester graduate course ... Mathematicians from other areas will also enjoy the book ... [It] reminds me of more old-fashioned books on algebraic geometry ... but updated to our modern standards of rigor and shorter attention span." ---- MAA Online

From a review for the German Edition: "The introduction contains numerous examples which illustrate and motivate the discussed theory and which reappear, as the course develops, handled in a precise and clear manner ... Each section ends with interesting and doable (!) exercises ... the author makes a great effort to prove most of the theorems in the rigorous way ... Precision and clarity are distinguished features of the reviewed test." ---- MathSciNet, Mathematical Reviews on the Web

"The book remains one of the very best introductory texts on algebraic geometry. The last chapter is a masterpiece of didactic art ... absolutely unique for such an elementary textbook." -- Zentralblatt MATH ---- Zentralblatt MATH

Product details

  • Publisher ‏ : ‎ American Mathematical Society (February 13, 2003)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 213 pages
  • ISBN-10 ‏ : ‎ 0821829521
  • ISBN-13 ‏ : ‎ 978-0821829523
  • Item Weight ‏ : ‎ 8.1 ounces
  • Dimensions ‏ : ‎ 5.75 x 0.5 x 8.5 inches
  • Customer Reviews:
    4.5 4.5 out of 5 stars 15 ratings

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Klaus Hulek
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Customer reviews

4.5 out of 5 stars
4.5 out of 5
15 global ratings

Top reviews from the United States

Reviewed in the United States on January 25, 2023
Linda's evaluation below reflects the difficulty of this entire subject matter rather than the quality of this text.
Linda should look at Plane Algebraic Curves by Gerd Fischer for more of an introduction to the basis of the subject. Also, Algebraic Geometry has invented its own special language that takes time to acquire and become comfortable with and is dependent on much advanced Commutative Algebra. Best wishes to you, don't give up. Another great truly introductory text is by Prof. Keith Kendig A Guide To Plane Algebraic Curves which can give the reader a firm basis for discovery.

This text is strongly recommended in the book 3264 and All That: Second Course of Algebraic Geometry by Eisenbud and Harris as a first course in Algebraic Geometry which impacts and is enriched by many other fields of Mathematics.
Reviewed in the United States on March 29, 2018
Nicely written book; just a few typos so be sure to get the errata notes. Otherwise the exposition flows very nicely, good for self study.
2 people found this helpful
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Reviewed in the United States on August 5, 2017
A very good introduction to AG.
One person found this helpful
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Reviewed in the United States on October 24, 2015
I thought this book was great. Elementary, yet conceptually deep and terrific background for Hartshorne's Algebraic Geometry. I used it as a graduate student in a course in algebraic geometry.
5 people found this helpful
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Linda Sutherland
1.0 out of 5 stars Absolutely dreadful pedagogy
Reviewed in the United Kingdom on February 9, 2022
I can't fault the mathematics in this book. Everything is there, though you will have to make some effort to fill in the details of the proofs. However, as pedagogy it is absolutely dreadful.

In my opinion, any book with the word "elementary" or "introduction" in the title ought to proceed at a gentle pace, at least in the first few sections. This book does not. It is exceedingly concise right from the start.

Moreover, the material is poorly organised: the author may state a theorem and then go off on a long meander through other topics before returning to it. The other topics eventually turn out to be related to the theorem, but that only becomes apparent after the event. That is no way to write an introduction to a subject. In a supposedly elementary text, if I have to struggle through a long digression, I would like the author to do me the courtesy of explaining, in advance, why it is relevant.

The author plays this same trick even in Chapter 0, entitled Introduction, where there is a digression involving Weirstrass's rho-function. It's nice mathematics, but fairly indigestible if you haven't encountered that function before, and, whether you have or not, Chapter 0 is absolutely the wrong place to put it. (Also, its relevance to the subject will only be apparent to those readers who have studied elliptic curves, and many readers will not have done. The author does not bother to point out the connection.)

The author also has a bad habit of introducing terms long before they are defined, and pointing the reader to some section of the book chapters ahead. This might be acceptable if there was a good informal discussion when the undefined term is introduced - but there rarely is. This is very bad practice.

Most readers tackling a field of mathematics that is new to them require a little time to familiarise themselves with the basic ideas of that field. That is true even if they already have some mathematical maturity. This book is not written for them. It seems to have been written for readers who already know the subject, but want a concise text as a refresher. For those readers it may be a very good book, but for other readers it is very poor.

Algebraic geometry is not an easy topic to enter, but the author has compounded the unavoidable mathematical difficulties by his unwillingness to present material in a clear and logical way. (Perhaps the word "unwillingness" does him too much of a compliment. "Inability", which suggests a degree of incompetence, may be more accurate.)

I have worked my way through large parts of this book several times, and every time I get extremely annoyed with the author.

The main mathematical prequisite for this book is a good background in abstract algebra. It is absolutely essential.
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