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Possible Duplicate:
(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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awllower
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    Possible duplicate: http://math.stackexchange.com/questions/1748/undergraduate-algebraic-geometry-textbook-recomendations – Jesse Madnick Mar 01 '11 at 12:18
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    Also: http://math.stackexchange.com/questions/998/best-algebraic-geometry-text-book-other-than-hartshorne – Jesse Madnick Mar 01 '11 at 12:18
  • It seems better to collect these recommendations at one place, therefore I'm voting to close. – t.b. Mar 01 '11 at 12:39
  • Perhaps we shall make an area for this kind of questions... – awllower Mar 01 '11 at 13:00
  • And I cannot find my community button, hence I will appreciate everyone who makes this community wiki, thanks. – awllower Mar 01 '11 at 13:09
  • I uesd to be able to do this, there must be some errors..... – awllower Mar 01 '11 at 13:09
  • @awllower: You can't do it yourself for questions, you can only do it yourself for answers. Next time, you should flag your question for moderator attention click on "other" and ask them to turn the question into Community Wiki. I've done it for you this time. – t.b. Mar 01 '11 at 13:17
  • @Theo Burhler: I see, and thank very much. – awllower Mar 02 '11 at 12:33

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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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Javier Álvarez
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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 01 '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 01 '11 at 13:07
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    would you mind moving this answer to the other question? – Qiaochu Yuan Mar 01 '11 at 15:31
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    I will elaborate more and move it, no problem. – Javier Álvarez Mar 01 '11 at 16:32