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”.