Undergraduate roadmap to algebraic geometry? Hello,
I'm sorry if this question isn't posted correctly. I hope that it is (since other questions regarding roadmaps have been allowed). Now to my question:
From what I've heard from professors and such, algebraic geometry seems like an interesting branch of mathematics. I'd like to learn some basic results and maybe do some kind of thesis in a few years on the subject. So, what I'm curious about is you have any tips on what books to read? Say that one has read Artin's Algebra and Herstein's Topics in Algebra, and also has the basic courses in real analysis and topology, complex variables etc. down, where should one go to learn? What books? I'm also curious if algebraic geometry (at an "easy level") requires deep knowledge about other fields of mathematics too, so that one might have to read books that at first seems to have no relevance to algebraic geometry?
Best regards.
 A: After having picked up some basic ideas - from sources mentioned above or maybe even from Kenji Ueno's Introduction to Algebraic Geometry, which according to the introduction is aimed at "scientists", not just mathematicians, and thus has almost zero prerequisites - you should not be afraid to take the next step and study schemes.
For this I want to recommend warmly the 3 little, undergraduate readable, books by Kenji Ueno, Algebraic Geometry 1-3, which appeared in the AMS series "Translations of Mathematical Monographs" as nr. 185, 197 and 218. Each of these tomes has less than 200 pages - and the pages are small. Yet the author manages to cover all the basic topics of scheme theory, painlessly and in a very friendly style. To read the book you do not need to have studied Commutative Algebra before - instead you can have a copy of Matsumura ready while reading Ueno; for all the results he uses he gives precise references in H. Matsumura, Commutative Algebra, 2nd edition, Benjamin 1980.
It is no accident that the AMS decided to have this translated this from Japanese, I find the text an extraordinary combination of good content and friendliness
A: I think the best way is to read a book on commutative algebra (Atiyah & MacDonald) and then, you can start reading Hartshorne's. Chapter 1 will give you a fairly concrete idea of what classical alg. geo. is about. You should also try to do all the exercises even though that will be time consuming. I think this is the best way one can do it.
A: I believe there have been similar questions, but not one exactly of this flavor.
To answer your last question, it is true that you need to know many different areas of mathematics in order to delve deeply into algebraic geometry. On the other hand, to get a basic grounding in the field, one need only have a basic understanding of abstract algebra.
That being said, I will give my recommendations.
If you have already done complex variables, and I'm not sure that every student in your position will have completed this, I recommend Algebraic Curves and Riemann Surfaces by Rick Miranda. Although this book also develops a complex analytic point of view, it also develops the basics of the theory of algebraic curves, as well as eventually reaching the theory of sheaf cohomology. Multiple graduate students have informed me that this book helped them greatly when reading Hartshorne later on.
If you want a very elementary book, you should go with Miles Reid's Undergraduate Algebraic Geometry. This book, as its title indicates, has very few prerequisites and develops the necessary commutative algebra as it goes along. More advanced students may complain that this book does not get very far, but I think it may very well satisfy what you are looking for.
Another book you might want to check out is the book Algebraic Curves by William Fulton, which you can thankfully find online for free.
If you would not mind a computational approach, and furthermore a book which requires even fewer algebraic prerequisites than you seem to have, you might want to check out Ideals, Varieties, and Algorithms by Cox and O'Shea.
Thierry Zell's suggestion is also supposed to be good.
That being said, if you decide that you like algebraic geometry and decide to go more deeply into the subject, I highly recommend that you learn some commutative algebra (such as through Commutative Algebra by Atiyah and Macdonald). But for the moment, I think the above recommendations will suit you well.
A: I had the opportunity to teach an undergraduate course in algebraic geometry, and let me tell you that finding a book at that level was not easy. First, be aware that most books will not be self-contained and will farm out the necessary commutative algebra to outside references.
For a first taste, I would go with An Invitation to Algebraic Geometry by Karen Smith et al. (Springer) It's not really designed for undergrads, and I'm not sure how it would work as a textbook for a course; rather, it was written for people with a certain mathematical maturity who don't know anything about the subject and want to learn the basics. I recommend it because it's short, recent, good for self-study, contains a lot of the classical stuff, and it can give you an idea of the flavor of the subject.
There are tons of algebraic geometry books out there, so I'm sure other good recommendations are forthcoming.
A: There is an excellent book on algebraic geometry entitled Algebraic Geometry: A First Course by Joe Harris. This book, however, emphasizes the classical roots of the subject but if you have not yet seen too much of algebraic geometry, it is worthwhile getting this book and reading a few lectures. (The book is split into "lectures" rather than "chapters".) There are many beautiful constructions in classical algebraic geometry that can be understood without too much background (and which lay the foundations for some aspects of modern algebraic geometry) and this can perhaps give you a rough indication of the geometric intuitions in algebraic geometry. And in my opinion, the book does an excellent job of conveying the beauty and elegance of algebraic geometry. 
The prerequisites for reading this book (according to Harris) are: linear algebra, multilinear algebra and modern algebra. However, since this is a "Graduate Texts in Mathematics" book, there are some places where it is very helpful (but not essential to the point that you cannot read the book otherwise) to have a basic knowledge of commutative algebra, complex analysis and point-set topology. (E.g., basic facts about topological spaces, local rings, basic constructions in commutative algebra, holomorphic functions etc.) Atiyah and Macdonald's an An Introduction to Commutative Algebra should furnish more than enough preparation. (You can also concurrently read commutative algebra if that is your preference.)
Since you are an undergraduate student, you should not worry too much about learning "background material" just yet before at least seeing what classical algebraic geometry is about. If at some point you decide to specialize in the subject, you will need to learn the "modern tools" such as, for example, schemes, sheaves and sheaf cohomology. The "classic book" for this is Robin Hartshorne's Algebraic Geometry but since that does require a solid background in commutative algebra (or at least the mathematical maturity to accept facts without proofs), you might want to try other books. (But this is, I hasten to add, an excellent book if you do have the background to understand it.) 
As Bcnrd (on MathOverflow) recommended to me, Qing Liu's Algebraic Geometry and Arithmetic Curves seems to be an excellent book on the subject. Most of the background material in commutative algebra is developed from scratch, and the first six chapters furnish a good introduction to the "modern tools". The last three chapters focus more on the arithmetic side of algebraic geometry, but you can always omit that if you so desire. (But if you are interested in number theory, definitely take a look at that!)
Succinctly, I recommend: Take a look at Atiyah and Macdonald and at least read the first few chapters. (The book is roughly 120 pages so covering the first few chapters is not too hard. Though be warned: Some people say that Atiyah and Macdonald is "dense", but I personally found it a very readable book and I think the majority find that so as well.) Then you should have the right background to read Harris and I hope that that will show you how fascinating the subject of algebraic geometry is. Good luck!
A: Try Shafarevich's Basic Algebraic Geometry I. It includes a good amount of examples from geometry in the exposition, as well as the necessary commutative algebra. I also like that differentials, divisors, and other gadgets are introduced without schemes (of course, you should eventually learn about schemes). The book moves pretty quickly, but if you have gone through Artin and Herstein then you should find it approachable.
A: Broadly speaking, algebraic geometry is the geometric study of solutions to polynomial equations. To begin with, you would start by working with solutions in affine space $\mathbb{A}_k^n= k^n$, where $k$ is an algebraically closed field (e.g. $\mathbb{C})$. Eventually, it becomes advantageous to add points at infinity by working in projective space $\mathbb{P}^n_k=\mathbb{A}^n_k\cup (\text{hyperplane at }\infty)$. After a while, 
you may move beyond even this.
All of this is spelled out in the basic books  on algebraic geometry.
Although there are quite a number of these, very few 
are explicitly at the undergraduate level. If I had to come up with a list, it would probably be the similar to Davidac897's. Out of the lot, I'd recommend Reid's "Undergraduate
Algebraic Geometry" as a good starting point: it is short, to the point, with minimal prerequisites. Once you've gotten through it, you'll have enough of a foundation to tackle
something more ambitious if you're still so inclined. 
One other thought. I have a sense that you are planning to read this on your own. Things are easier if you find a group of fellow students to "share the pain". Have fun.
