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I need to learn about Algebraic Geometry (perhaps from in the context of finite fields) and am looking for a recommendation for a text. Now, I've already done a search and checked out what was suggested. However, they weren't what I was looking for. To be honest, I felt that the suggested textbooks were terrible. (This even included some famous ones like Hartshorne.) They all followed the format of Theorem - Proof - Theorem - Proof - etc... Although this format makes for a good reference, it has no educational value for me; in the end, I will know how to prove some specialised set of theorems but not understand what any of it really means or why anybody cares in the first place. So, I would like to focus more about the motivation and history of the field and its concepts in order to get a big-picture understanding. (I will worry about the fine technical details on my own.)

Does anyone know of such a book? I really appreciate your help!

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Have you looked at Shafarevich? He provides a lot more motivation than Hartshorne. Also, after you have read som AG, Eisenbud and Harris' "The Geometry of Schemes" is wonderful. –  Fredrik Meyer Jul 23 '12 at 22:59
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Did you look at Ravi Vakil's notes? More or less every algebraic geometry textbook that I know about has been listed in one of the questions you probably found; you're going to have to find one that works for you, and unfortunately it's going to be hard for someone to say which one that will be. –  Dylan Moreland Jul 23 '12 at 23:01
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And I second the recommendation of Shafarevich. Miles Reid, who translated the book, says in the preface “My experience is that some graduate students can work hard for a year or two on Chapters II–III of Hartshorne and still know more or less nothing at the end of it.” So certainly their hearts are in the right place. It would also help to know what it is you want to learn and for what purpose. Algebraic geometry is gigantic. –  Dylan Moreland Jul 23 '12 at 23:06
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I think it's definitely worth taking a look at Gathmann's notes (mentioned by Joachim below). While I have not studied them, I do have them saved to use in the future. There is a fair amount of discussion, the first chapter is solely motivational, and overall it seems quite pleasant. Here's the link: mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf –  Alex Petzke Jul 24 '12 at 0:53
    
@Dylan I'm also a big fan of Vakil's notes and are monitoring thier development closely-but I think these notes are way too difficult and abstract for the questioner's purposes. –  Mathemagician1234 Jul 24 '12 at 8:26
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6 Answers 6

I'd like to leave a comment instead of an answer, since my "answer" wont be definite. However, i'm quite new so i don't have the reputation to leave a comment. But i think i have something useful to say, so i'll take this opportunity.

To start, could you be more specific on what you already know about? What do you know about differential and complex geometry? How is your knowledge of commutative algebra? Are you an undergraduate or grad student? And are you a mathematician or do you study something else and need to learn algebraic geometry? If i'm honest, i must admit i am a little confused by your dislike of the theorem-proof structure, since it is quite standard.

Now the books.

A professor at my uni recommended Miles Reid - Undergraduate Algebraic Geometry. As the title gives away, its written for undergraduates. As i was already a grad student when i got it, i didn't read much of it. However, i can say that there is an excellent introduction focusing both on the history and motivation that you mentioned (it helped me even when i already learned a big part of Hartshorne). Also the down-to-earth approach helped me in some cases where books like Hartshorne were too abstract to grasp. The treatment of the tangent space of a variety is excellent, for example.

When i was learning about schemes, a classmate of mine was enthousiastic about the notes of Andreas Gathmann. They are supposed to contain motivation and examples, but i have rarely used them myself so this is not my personal experience. As i mentioned, i have no idea about your level, but i just took a quick look and Gathmann starts at the basis i believe (varieties, Zariski topology...)

I could also recommend Ravi Vakil's notes. I haven't used them a lot but they tend to show up everywhere and i guess that's for a reason. I found the style very accesible.

Lastly, please reconsider Hartshorne, but you should use it in the right way. The style of writing is really compact, so if you read it, get used to using a lot of other resources. For example, when i learned about "injective objects in a category", it took my professor a substantial part of the lecture, however Hartshorne spends a few sentences about the definition. So when reading Hartshorne, don't: expect a lot of intuition and background. Do: consider it your atlas, your guidance. Use google a lot (loads of teachers put their lectures online), be prepared to look for background and spend a lot of time on just a few sentences. But everything's there, it's incredibly complete. Spend a LOT of time on the exercises, the book is written for the active learner and a substantial part of the theory is contained in the exercises. Once you use it in the right way, you'll fall in love. The book is beautiful. By the way, this guy put a very complete collection of solutions to the exercises online, check it out at http://www.math.northwestern.edu/~jcutrone/Work/Hartshorne%20Algebraic%20Geometry%20Solutions.pdf

Lastly, if you're interested in surfaces, Beauville: algebraic surfaces is an excellent treatment that contains a lot of historical notes. However, it is a little specialized so i doubt that it is what you're looking for.

Whenever i did not state a reference, just google.

I hope this helps. Joachim

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At any rate, this would not fit in a comment :) –  Dylan Moreland Jul 24 '12 at 0:09
    
Yeah when i wrote the first sentence, i wasn't expecting it to become this big.. =) –  Joachim Jul 24 '12 at 0:13
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Dear Joachim, This is a nice answer. I recommend all my beginning graduate students in AG to start with Miles Reid's book if they haven't already read it. The word "undergraduate" in the title is somewhat misleading; it is great for anyone who doesn't yet have a strong feeling of what algebraic geometry is about. Regards, –  Matt E Jul 24 '12 at 3:17
    
Thanks for your reply. To provide the background you requested: I am a mathematics graduate student who is studying algebraic coding theory. I am currently working on Reed-Solomon codes, but I will be moving toward AG codes shortly, so I require a background in algebraic geometry. I immensely dislike the theorem-proof format because it tells me nothing. All I can do is verify that a particular statement is correct, but I gain no insight or understanding about the ideas of the field. I'll definitely consider your suggestions and try to find a good fit for me. –  roninpro Jul 25 '12 at 3:35
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You may want to look at Dieudonne's book History of algebraic geometry, for historical motivation. There is also his shorter note, The historical development of algebraic geometry . Note that this latter article discusses finite fields and the Weil conjectures towards the end. Note also that it was written 40 or so years ago, and so its concluding remars no longer reflect the state of the art. Nevertheless, especially for a historical survey, it is hard to beat, I think.

You can read this note of Dieudonne, together with Miles Reid's Undergraduate algebraic geometry, in a fairly short amount of time, and if you do both, you will have some sense of what algebraic geometry is about both at a beginning technical level and a broader conceptual level. You could then move on to more substantial texts, with better preparation than you would have approaching them cold. (Once you get to this stage, Mumford's Red book has always been regarded as something to read alongside Hartshorne as being at a similar technical level, but providing more movitation and intuition.)

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Here are three books (in increasing order of technicity) that haven't yet been mentioned in the great comments and answers above.

1) Jenner's Rudiments of Algebraic Geometry is a charming almost fifty year old booklet (99 pages of actual text!) packed with large, simple, edifying pictures, just like any book with "geometry" in the title should (Yes, I'm talking to you, dear Grothendieck!).
The level is that of undergraduate linear algebra but Jenner manages to make some non-trivial calculations on Grassmann varieties and Plücker embeddings.

2) Algebraic Geometry by Smith et al. is an excellent survey of classical algebraic geometry at the intermediate level.
It contains serious material like Hilbert functions or blowing-up of ideals and calculations of just the right kind: neither trivial nor too technical.
Its strong points are the numerous, illuminating pictures.

3) Since you are interested in finite fields, you could have a look at Niederreiter-Xing's Algebraic Geometry in Coding Theory and Cryptography
It is an excellent example of a recent kind of books that handle algebraic geometry as a tool for applications, as the very title says.
This means that the perspective is narrow, that there are no pictures, but that the text only uses elementary concepts and that specialization allows the authors to quickly prove very advanced results which most algebraic geometers do not know (I certainly don't).
For example already on page 122 they prove an improvement due to Serre of the Hasse-Weil bound for global function fields , using only elementary tools: no schemes, no sheaves hence no cohomology, no Hilbert functions,...
The attraction of such a book is a very subjective matter: I'll leave it to you to decide whether this is the sort of book that you are looking for.

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Thanks for replying. I actually did pull up a copy of Niederreiter-Xing since, as I recently commented above, I am looking to work on AG-codes in the near future. The algebraic geometry part of the book was still very mysterious to me, since there is very little exposition. However, I will hang on to it for the coding part. –  roninpro Jul 25 '12 at 3:47
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Funny you should ask that. Audun Holme of the University of Bergen in Norway has just published A Royal Road To Algebraic Geometry, which appears to be exactly what you're looking for. It's a historically grounded text on algebraic geometry split into 2 halves: the first is on classical algebraic geometry via curves and varieties; the second is on the modern theory via schemes. I've only read some of it, but it looks like an incredible book that will clarify the entire field and is exactly what most nonexperts need.

The subject of algebraic geometry is both an extraordinarily dense and difficult subject and one of growing importance for all mathematicians due to its' applications in computer graphics and coding. Part of the problem is that beyond some simple problems about polynomial plane curves, it's very difficult to create any real motivation for modern AG-it just looks like a very alien and strange subject with no real purpose.This is a problem due to the field's frontier status. A historical development will assist enormously in clarifying the concepts and overall structure of algebraic geometry and that's why I was very excited to see this book. I think it's a book any non-expert should check out.

By the way-I agree wholeheartedly with the recommendation of Reid's classic Undergraduate Algebraic Geometry as well. It tries very hard to motivate the subject through classical problems in the plane with minimal prerequisites and is beautifully written. However, it's entirely classical and once one finishes with it,it's very hard to go to a modern treatment. This is the advantage of Holmes' book-it seeks to actively bridge classical to modern aspects of the subject. You should definitely check out both.

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There is no "University of Norway". Audun Holme is a professor at the University of Bergen (which is the second largest city in Norway). –  Fredrik Meyer Jul 24 '12 at 6:57
    
@Frederik Ok,I feel like an idiot now. I corrected the error,thanks for pointing it out. –  Mathemagician1234 Jul 24 '12 at 8:23
    
Thanks for the suggestion! Although the title seems kind of high-brow for me, I will definitely check it out! –  roninpro Jul 25 '12 at 3:43
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The book Introduction To Commutative Algebra by Atiyah and MacDonald, is as the title says an introduction to Commutative Algebra. However, if one goes through the exercises it is also a nice introduction to Algebraic Geometry.

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Dear A.D., Given the OP's complaint about lacking the motivation behind the technical development of the theory, the exercises in AM don't seem to be the ideal answer to his request for a more elightening treatment. Regards, –  Matt E Jul 25 '12 at 2:21
    
I see your point, but on the other hand since this is exercises the proving parts are left to the reader - so it is not the Bourbaki style thm-proof.. thm-proof. –  AD. Jul 25 '12 at 21:21
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You need:

I need to learn about Algebraic Geometry (perhaps from in the context of finite fields)

and:

So, I would like to focus more about the motivation and history of the field and its concepts in order to get a big-picture understanding.

It is hard to beat Mumford's Red Book for these requirements. It is a tailor-made introduction to schemes with arithmetical applications too in mind. Just replace all instances of "prescheme" with "scheme" and you are good.

If you want to go further, you can check out Mumford-Oda.

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