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(undergraduate) Algebraic Geometry Textbook Recomendations

I am interested in algebraic number theory and I am recently acquainted with the theory of valuations, which further leads to Riemann-Roch theory, and which is closely related to algebraic geometry, and the algebraic-K-theory.
Therefore, my problem is:

Are there excellent introductory books of the theory of Algebraic Geometry to recommend?

Since I know in general nothing about this theory, I may want a book which explains the ideas as clear as possible and which at the mean time contains as much material as possible.
If I am asking too much, then any good book in your view suffices.
Thanks very much.

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marked as duplicate by t.b., Qiaochu Yuan Mar 1 '11 at 13:35

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Possible duplicate: math.stackexchange.com/questions/1748/… –  Jesse Madnick Mar 1 '11 at 12:18
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It seems better to collect these recommendations at one place, therefore I'm voting to close. –  t.b. Mar 1 '11 at 12:39
    
Perhaps we shall make an area for this kind of questions... –  awllower Mar 1 '11 at 13:00
    
And I cannot find my community button, hence I will appreciate everyone who makes this community wiki, thanks. –  awllower Mar 1 '11 at 13:09

1 Answer 1

up vote 3 down vote accepted

As I mentioned at another post Teaching myself differential topology and differential geometry.

If you are interested in learning Algebraic Geometry I recommend the books of my Amazon list. They are in recommended order to learn from the beginning by yourself:

http://www.amazon.com/lm/RHQS8Y3V7LJRQ/ref=cm_pdp_lm_title_1

In particular, from that list, a quick path to understand basic Algebraic Geometry would be to read Bertrametti et al. "Lectures on Curves, Surfaces and Projective Varieties", Shafarevich's "Basic Algebraic Geometry" vol. 1, 2 and Perrin's "Algebraic Geometry an Introduction". But then you are entering the world of abstract algebra.

There is no a single complete book and much less explaining the ideas as clearly as possible. If you are starting from the very beginnig, I recommend these in this order: Karen Smith's, Beltrametti, Hulek, Safarechiv vol. 1, Perrin, Shafarevich vol. 2 and then scheme theory with Ueno's three volumes.... then you can jump with enough background to the bible by Hartshorne, or Griffiths/Harris for the more complex geometric side.

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Alvarez: As I said, I have some foundation to read the algebraic number theory while lack that of the algebraic geometry. And it seems that the books recommended here are all pretty suitable, and I cannot understand the Bible of Hartshorn indeed, anyway, thanks very much. –  awllower Mar 1 '11 at 12:53
    
And I will wait for other answers; nevertheless, if it turns out to be the best answer, I will accept it, I just want to wait, sorry. –  awllower Mar 1 '11 at 13:07
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would you mind moving this answer to the other question? –  Qiaochu Yuan Mar 1 '11 at 15:31
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I will elaborate more and move it, no problem. –  Javier Álvarez Mar 1 '11 at 16:32

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