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I have just finished my undergraduate studies and i want to study some algebraic geometry. i have taken 1 year analysis, topology, algebra courses and finished 1 semester complex analysis, homological algebra, fourier analysis, differential geometry.

with this background i'm planning to study atiyah's commutative algebra. then next, where should be the starting point for studying the A.G? hartshorne is surely too tough for me. i want to read some A.G books which are not too hard to follow, and kind of self contained.

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    $\begingroup$ Javier Ávarez has posted nice answers with lots of recommendations; you can find them among this list (that you can narrow down by also including the tag reference-request and similar) $\endgroup$ – pjs36 Jan 12 '17 at 1:52
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    $\begingroup$ I recommend you "Undergraduate Algebraic Geometry" by Miles Reid. $\endgroup$ – Xam Jan 12 '17 at 1:54
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    $\begingroup$ Here are two of the answers mentioned by pjs36: 1, 2. I personally recommend Fulton's Algebraic Curves. It is classical, concise, and best of all, freely available. $\endgroup$ – André 3000 Jan 12 '17 at 2:05
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(My answer ended up being too long for a comment.)

Hi there. Just finished up my first semester as a graduate student focusing heavily on geometry and topology at Colorado State University. Didn't know any algebraic geometry until this past spring and have...grown since then haha.

Given your background, you would probably appreciate Rick Miranda's Algebraic Curves and Riemann Surfaces. It is almost entirely done in the context of complex analysis, but all the things you want to be true in the algebraic geometry context are true if you just replace "holomorphic" and "meromorphic" with "regular" and "rational." (In fact, Dr. Miranda notes this in his "Algebraic Sheaves" section toward the end.) You get a great introduction to curves, divisors, sheaves, line bundles, etc., all with examples using the Riemann Sphere (Complex projective line), tori, and a general complex projective (or affine) curve. You also might be able to find this book as a PDF if you know how to Google well. And you don't need all the deep commutative algebra to understand this book, because it leaves everything mostly complex analytic (in a fun way) and really just relies on a little differential topology in some spots that you might not have. If you do choose that book, feel free to get ahold of me if you want some discussion on the concepts or a bit of help with some of the exercises. I'm just finishing up my first readthrough of that one over the winter break and have been doing the problem in that book for a couple years now. (I think my contact info should be on my Google Drive. If not, you can ask me for it on here or probably find it under the CSU graduate students list.)

After a final review of the rest of my recommendations, I really would put Dr. Miranda's book at the top for you. That book makes you feel like you're doing complex analysis with some differential topology but sneaks you into algebraic geometry until you realize that's what you were doing the whole time. And, in fact, there is a paper called Algebraic Geometry and Analytic Geometry--GAGA because French--which states that basically these two are one in the same; complex analytic geometry and complex algebraic geometry really aren't different. (Cf. Chow's Theorem and Dr. Miranda mentions this paper as well.) I think this book would be ideal for your background.

It's a good idea to get some of the intuition in Algebraic Geometry before trying to get all the pure commutative algebra down. Otherwise, you're just doing pure algebra...which, like...why bother :P And for that purpose I would also recommend reading some of Dieudonne's The Historical Development of Algebraic Geometry. Get a feel for what problems people were trying to solve when developing all this theory and an (even better) appreciation for why Riemann was such a badass. Diuedonne, if you don't know, was part of the Bourbaki group and was (if I remember right) the scribe of the group for some time. He was around when all the exciting developments in alg. geo. were happening and knew enough of history to chronicle those developments in context.

If, however, you are wanting to do some algebra before anything else, check out Dummit and Foote's Algebraic Geometry part/chapter/section (whatever they call it) which goes through the Hilbert Basis Theorem and other very basic results which we use(d) in a classical algebraic geometry intro. class. You also learn enough about localization (especially the correspondence between prime ideals in a localization and prime ideals in your original ring).

Other popular algebraic geometry books include Milne's Notes (just google the words "Milne Algebraic Geometry"--freely available online by the author), obviously Hartshorne (though I get that that's a bit tough to read), and maybe some Harris and/or Eisenbud stuff (e.g. Geometry of Schemes which you might be able to find online). You might be able to find Shafarevich's Basic Algebraic Geometry online, too, and that one has plenty of examples. You really want a lot of examples when studying algebraic geometry.

A book with some heavy category theory and basically all the algebraic geometry in the world, you could look at Vakil's The Rising Sea. He publishes these (I hesitate to call them) notes online every year or so after edits, and this book is just a massive outpouring of alg. geo. with a 100-page grounding in category theory (and some commutative algebra in category theoretic terms) as the foundation for sheaves and schemes. Along the way, Ravi Vakil gives some great pictures of what we are trying to describe with Spec(-) under the Zariski topology and the affine scheme. I honestly have only gotten through maybe 150 pages of that book after about 4 or 5 months of time with it, because of its density and thoroughness. There are examples throughout, and you should really do all the examples you can. The only way I've found to get through alg. geo. is by examples--otherwise you're lost in a sea of abstract meaninglessness.

One note that cannot be stressed enough: DO PROBLEMS Do problems while you're awake. Do problems while you're half-awake. Do problems on the train. Do problems on the bus. Do problems in your sleep. Do problems at the gym. Do problems literally all of the time every time all day every day you possibly can. And if you're having trouble with some problems, ask the community! They've been great while I learn algebraic geometry, and I'll be around to post when I can as well! :)

Another book that some of my peers found useful while trudging through our first alg. geo. class this fall is Karen Smith's An Invitation to Algebraic Geometry.

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    $\begingroup$ Side note, Javier Alvarez is one of my peers at CSU and is in the same office with me. He's basically the person at whom we can throw any math question and get a helpful answer. $\endgroup$ – Tanner Strunk Jan 12 '17 at 11:34
  • $\begingroup$ i appreciate your detailed, kind answer so much. how sweet! from my some experience of googling and your answer, i'm thinking of starting miranda and next doing milne or gathmann's note, or fulton's algebraic curves. which book would you recommend among those 3 books? ( milne, gathmann, fulton) thank you. $\endgroup$ – Mathcho Jan 13 '17 at 5:14
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    $\begingroup$ Never heard of Gathmann's notes or Fulton's book, so that would be up to you. Miranda's book took me a while to grasp well, and I think some of the ideas are still sinking in with time. (That's for me, though. You might be different, of course.) I am going to go ahead and take down my Google drive link now, because I thought about it and don't really want that out in public like this. Glad to have been some help. $\endgroup$ – Tanner Strunk Jan 15 '17 at 20:35
  • $\begingroup$ I think Fulton's is a nice book, we used it for our course. $\endgroup$ – Rick May 21 at 9:37
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I recommend the books "Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Authors: Cox, David A, Little, John, Oshea, Donal" and

"Computational Commutative Algebra and Algebraic Geometry Course and exercises with solutions. Author: I. Yengui".

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  • $\begingroup$ I wish this answer would get more attention. The Cox, Little, O'Shea book does a great job of presenting the actual utility of algebraic geometry, and I didn't get to it until nearly a year after the post I made on this thread. It was pretty easy to read after all the abstract junk, but I think their book makes an even better introduction and motivation for the subject. $\endgroup$ – Tanner Strunk May 29 at 3:42

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