What are the best algebraic geometry textbooks for undergraduate students?
(answer moved from closed duplicated question)
If you are interested in learning Algebraic Geometry I recommend the books of my Amazon lists. Most of them, at the beginning of the lists, are in increasingly difficulty and recommended order to learn from the beginning by yourself.
In particular, from those lists, 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" and the beautiful new Holme's "A Royal Road to Algebraic Geometry" . But then you are entering the world of abstract algebra. For that it is advisable to master the book of Miles Reid "Undergraduate Commutative Algebra" accompanied by the new Singh's "Basic Commutative Algebra". Both are very readable and thorough at their level, the former a geometry-oriented introduction and the latter a purely formal reference.
My personal opinion is to avoid as main text, books like Harris - "Algebraic Geometry: A First Course" or Cox et al., above all if you have limited time to learn. In my experience these kind of books will not get you very far within algebraic geometry by themselves, although they are good companions as sources for examples and computations. In particular Harris' is a very nice companion to the others as a more "literary" supplement.
Focusing only at the undergraduate level the best books in order of sophistication are: Smith et al. "An invitation to Algebraic Geometry", Reid "Undergraduate Algebraic Geometry", Hulek "Elementary Algebraic Geometry", Beltrametti et al. "Lectures on Curves, Surfaces and Projective Varieties".
There are several free online pdf courses. Concretely, the introductory notes by Dolgachev "Introduction to Algebraic Geometry" (along with his "Topics on Classical Algebraic Geometry" to be published soon) are very algebraic and may be read without problems after or along with a book like Hulek's "Elementary Algebraic Geometry". For introductions which cover very nicely just the basic up to even schemes and cohomology the very best are by far the notes by Gathmann's "Algebraic Geometry" and Holme's "A Royal Road to Algebraic Geometry". A nice introductory course focusing on algebraic curves is Fulton's "Algebraic Curves - An introduction to Algebraic Geometry".
When one dives into graduate level, it is very important to remember that Algebraic Geometry is a huge subject and there are different approaches to start. In my opinion, the word "geometry" is fundamental and foundational, so people should not forget that in the end there must be some geometric content or analogue. Therefore the bible book by Hartshorne "Algebraic Geometry" MUST be studied only after some mastering of the basics like Beltrametti et al., Shafarevich and Perrin and above all Mumford's "Algebraic Geometry: Complex Projective Varieties" which is a masterpiece (you can continue to schemes by Mumford with the unpublished notes here). In order to supplement Hartshorne's with another schematic point of view, the best books are Mumford's "The Red Book of Varieties and Schemes" and the three volumes by Ueno "Algebraic Geometry I. From Algebraic Varieties to Schemes", "Algebraic Geometry II. Sheaves and Cohomology", "Algebraic Geometry III. Further theory of Schemes". ONLY after all this materials one can understand the geometry behind the extremely algebraic but also good and interesting books like Liu Qing - "Algebraic Geometry and Arithmetic Curves" (another approaches into Arithmetic Geometry might be Lorenzini - "An Invitation to Arithmetic Geometry", and Hindry/Silverman "Diophantine Geometry" which starts with a review on algebraic geometry and proves Mordell and Faltings theorems among others without the use of schemes).
My personal learning path is this: Beltrametti et al.'s for classic and basic geometric foundations, Perrin's for a more algebraic introduction with a nice treatment of Riemann-Roch, Mumford's "projective varieties" for foundations along with a flavor on complex algebraic geometry, then study notes by Gathmann. After this, I would start approaching Mumford's red book and Ueno to supplement Hartshorne on schemes. With all this background mastering Hartshorne should not be a problem, BUT you must DO all the exercises you can since they are the most important stuff in Hartshorne's. Other sources on this regard is the new book by Görtz/Wedhorn "Algebraic Geometry I, Schemes wiith Examples and Exercises" and its future volume two. If one needs a different approach including some of the sheaf theory and homology needed along with Riemann surfaces, there is Harder's "Lectures on Algebraic Geometry vol. 1 & 2".
Further references about algebraic curves, surfaces, higher-dimensional varieties and other subjects can be found in my Amazon lists.
For those interested in the Complex Geometric side (Kähler, Hodge...) I recommend Moroianu's "Lectures on Kähler Geometry", Ballmann's "Lectures on Kähler Manifolds" and Huybrechts' "Complex Geometry" above all. To connect this with Analysis of Several Complex Variables I recommend trying Fritzsche/Grauert "From Holomorphic Functions to Complex Manifolds" and also Wells' "Differential Analysis on Complex Manifolds". Or, to connect this with algebraic geometry, try, in this order, Miranda's "Algebraic Curves and Riemann Surfaces", or the new excellent introduction by Arapura - "Algebraic Geometry over the Complex Numbers", Voisin's "Hodge Theory and Complex Algebraic Geometry" vol. 1 and Griffiths/Harris "Principles of Algebraic Geometry".
See the MathOverflow question about exactly this: Best Algebraic Geometry text book?
Personally, from what little I've seen, Ideals, Varieties and Algorithms by Cox, Litle and O'Shea is a very nice book for undergraduates. It requires little background, and has a computational perspective that's especially good for those with concrete or computational interests. I don't know how good it is as a preparation for further "serious" study in algebraic geometry, but it's certainly engaging.
Since today (when I received it) I think that the new book Introduction to Algebraic Geometry is the best undergraduate textbook on the market.
The book has an extremely motivated presentation of concepts, many excellent examples and a lot of exercises, half of them solved at the back (a unique feature at this level).
This thick volume, more than 600 pages, starts from elementary algebraic subsets of $\mathbb A^n$ and goes on to schemes and projective varieties.
Another unique feature is that the book contains in its four appendices all the necessary prerequisites, including 149 pages of commutative algebra.
As an icing on the cake, the volume is dirt-cheap at $25 in the US (and 21.10 euros in France, where I bought it).
Moreover our friend @Justin Smith lets us download it for free from his site, even after the publication of his book (see link above): thank you for your generosity, Justin, and for having written such a great book .
Cox, Little, and O'Shea is an excellent book. I'd also like to recommend Miles Reid's Undergraduate Algebraic Geometry, which unlike Hartshorne or many other standard references was written specifically with undergraduates in mind. You should also check out his book on commutative algebra; I found both very enlightening.
Everyone is linking to that MO question, so allow me to link to another MO question which is more relevant to this specific request.
Also, I should mention that it partially depends on what you are looking for. Should this be for self study with little help from professors? Then your best bet would be to follow Ravi Vakil's course next semester or use his old notes here.
If you intend on having a professor over see the self study then I think that Ueno's books: Alg Geom 1 and 2, which I think are great for self study.(I should mention that his lower level book is not nearly as good and I would NOT recommend it.) Another strong candidate for led self study would be Harris' nice intro.
I think that covers the basics. :D
EDIT: Two more comments, this book is frustrating and I do not think it is appropriate to look at for undergrads trying to learn algebraic geometry, especially if they are moving to modern scheme theory. And another book with a very different feeling, but a really fun way to start thinking about algebraic geometry is Katz' Enumerative Geometry book. It was written especially for undergrads, and covers some really neat material that can give some good motivation to think about algebraic geometry and topology.
Hulek's «Elementary algebraic geometry» is a really nice text for undergraduates. (It's part of the AMS's student mathematical library.)
Bix - Conics and Cubics is probably the most accessible. I think a high school student who did the BC calculus syllabus could read this and get a lot out of it. But I'd agree with the others about Cox, Little, O'Shea - Ideals Varieties and Algorithms being the 'best' undergraduate text.
karen smith wrote a little book an invitation to algebraic geometry that has lots of pictures and is readable by an undergraduate.
There also is a quite readble old introductory book by Jenner Rudiments of algebraic geometry.
If you are also interested in the computational aspects, the book
"Rational Algebraic Curves, A Computer Algebra Approach" by Sendra, J. Rafael, Winkler, Franz, Pérez-Diaz, Sonia
and the new book "Computational Commutative Algebra and Algebraic Geometry Course and exercises with solutions" by I. Yengui
might interest you.