I think Algebraic Geometry is too broad a subject to choose only one book. Maybe if one is a beginner then a clear introductory book is enough or if algebraic geometry is not ones major field of study then a self-contained reference dealing with the important topics thoroughly is enough. But Algebraic Geometry nowadays has grown into such a deep and ample field of study that a graduate student has to focus heavily on one or two topics whereas at the same time must be able to use the fundamental results of other close subfields. Therefore I find the attempt to reduce his/her study to just one book (besides Hartshorne's) too hard and impractical. That is why I have collected what I consider the best books for each stage and topic of study, my personal choices for the BEST BOOKS are then:Beltrametti et al. "Lectures on Curves, Surfaces and Projective Varieties"
which starts from the very beginning with a classical geometric style. Very complete (proves Riemann-Roch for curves in an easy language) and concrete in classic constructions needed to understand the reasons about why things are done the way they are in advanced purely algebraic books. There are very few books like this and they should be a must to start learning the subject. (Check out Dolgachev's review
- "Basic Algebraic Geometry"
vol. 1 and 2. They are the most complete on foundations and introductory into Schemes so they are very useful before more abstract studies. But the problems are almost impossiblevery hard for many beginners
ONLINE NOTES:Gathmann - "Algebraic Geometry" which can be found here. Just amazing notes; short but very complete, dealing even with schemes and cohomology and proving Riemann-Roch and even hinting Hirzebruch-R-R. It is the best free course in my opinion, to get enough algebraic geometry background to understand the other more advanced and abstract titles.Liu Qing
- "Algebraic Geometry and Arithmetic Curves"
. It is a very complete book even introducing some needed commutative algebra and preparing the reader to learn arithmetic geometry like Mordell's conjecture, Faltings' or even Fermat-Wiles Theorem (If it were not out of stock, I would have said Iitaka
's "Algebraic Geometry. Birational Geometry of Algebraic Varieties"
because it is a book very similar to Hartshorne but it proves results that the latter leaves as problems). Nevertheless it lacks geometric motivations
- "Principles of Algebraic Geometry"
. By far the best for a complex-geometry-oriented mind. Also useful coming from studies on several complex variables .
ONLINE NOTES:Gathmann - "Algebraic Geometry" which can be found here. Just amazing notes; short but very complete, dealing even with schemes and cohomology and proving Riemann-Rochor differential geometry. It is the best free book you need to get enough develops a lot of algebraic geometry to understand without so much advanced commutative and homological algebra as the other titlesmodern books tend to emphasize.Görtz; Wedhorn
- Algebraic Geometry I, Schemes with Examples and Exercises
. Tons of stuff on schemes; more complete than Mumford's Red Book (For an online free alternative check Mumfords' Algebraic Geometry II unpublished notes on schemes.)
. It does a great job complementing Hartshorne's treatment of schemes, above all because of the more solvable exercises.Fulton
- "Algebraic Curves, an Introduction to Algebraic Geometry"
which can be found here
. It is a classic and although the flavor is clearly of typed concise
notes, it is by far the shortest and manageable but thorough
book on curves, which serves as a very nice introduction to the whole subject. It does everything that is needed to prove Riemann-Roch for curves and introduces many concepts useful to motivate more advanced courses
- "Algebraic Surfaces"
. Excellent complete and advanced reference for surfaces. Very well done and indispensable for those needing a companion, but above all an expansion, to Hartshorne's chapter.
ON HODGE THEORY AND TOPOLOGY:Voisin - Hodge Theory and Complex Algebraic Geometry vols. I and II. The first volume can serve almost as an introduction to complex geometry and the second to its topology. They are becoming more and more the standard reference on these topics, fitting nicely between abstract algebraic geometry and complex differential geometry.
INTRODUCTORY ON MODULI AND INVARIANTS: Mukai - An Introduction to Invariants and Moduli. Excellent but extremely expensive hardcover book. When a cheaper paperback edition is released by Cambridge Press any serious student of algebraic geometry should own a copy since, again, it is one of those titles that help motivate and give conceptual insights needed to make any sense of abstract monographs like the next ones.
ON MODULI SPACES AND DEFORMATIONS:Fulton - "Intersection Theory". It is the standard reference and is also cheap compared to others. It deals with almost all the material needed on intersections for a serious student going beyond HartshorneHartshorne's appendix; it is a good reference for the use of the language of characteristic classes in algebraic geometry, proving Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch among many interesting results.
ON SINGULARITIES:Kollár - Lectures on Resolution of Singularities. Great exposition, useful contents and examples on topics one has to deal with sooner or later. As a fundamental complement check Hauser's wonderful paper on the Hironaka theorem.
ON POSITIVITY:Lazarsfeld - Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series and Positivity in Algebraic Geometry II: Positivity for Vector Bundles and Multiplier Ideals. Amazingly well written and unique on the topic, summarizing and bringing together lots of information, results, and many many examples.
INTRODUCTORY ON HIGHER-DIMENSIONAL VARIETIES:Debarre - "Higher Dimensional Algebraic Geometry". The main alternative to this title is Kollar/Mori the new book by Hacon/Kovács' "Birational Geometry Classifiaction of Higher-dimensional Algebraic Varieties" but which includes recent results on the classification problem and is regarded intended as much a graduate topics course.
ADVANCED ON HIGHER-DIMENSIONAL VARIETIES:Kollár; Mori - Birational Geometry of Algebraic Varieties. Considered as harder to understand learn from by many some students(and even professors), it has become the standard reference on birational geometry.