Lifted from Mathoverflow:
I think (almost) everyone agrees that Hartshorne’s Algebraic Geometry is still the best.
Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc.
One suggestion per answer please. Also, please include an explanation of why you like the book, or what makes it unique or useful.
I think Algebraic Geometry is too broad a subject to choose only one book. But my personal choices for the BEST BOOKS are
Beltrametti et al. “Lectures on Curves, Surfaces and Projective Varieties” (errata) 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.
Shafarevich – “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 impossible.
GRADUATE FOR ALGEBRISTS AND NUMBER THEORISTS:
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. Filled with exercises.
GRADUATE FOR GEOMETERS:
Griffiths; Harris – “Principles of Algebraic Geometry”. By far the best for a complex-geometry-oriented mind. Also useful coming from studies on several complex variables.
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. It is the best free book you need to get enough algebraic geometry to understand the other titles.
BEST ON SCHEMES:
Görtz; Wedhorn – Algebraic Geometry I, Schemes with Examples and Exercises. Tons of stuff on schemes; more complete than Mumford’s Red Book. It does a great job complementing Hartshorne’s treatment of schemes, above all because of the more solvable exercises. A second volume is on its way on cohomology.
UNDERGRADUATE ON ALGEBRAIC CURVES:
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 notes, it is by far the shortest and manageable 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.
GRADUATE ON ALGEBRAIC CURVES:
Arbarello; Cornalba; Griffiths; Harris – “Geometry of Algebraic Curves” vol 1 and vol 2. This one is focused on the reader, therefore many results are stated to be worked out. So some people find it the best way to really master the subject. Besides, the vol. 2 has finally appeared making the two huge volumes a complete reference on the subject.
INTRODUCTORY ON ALGEBRAIC SURFACES:
Beauville – Complex Algebraic Surfaces. I have not found a quicker and simpler way to learn and clasify algebraic surfaces. The background already needed is minimum compared to other titles.
ADVANCED ON ALGEBRAIC SURFACES:
Badescu – “Algebraic Surfaces”. For those needing a companion and expansion to Hartshorne’s chapter. Done with more advanced tools than Beauville.
ON INTERSECTION THEORY:
Fulton – Intersection Theory. By far the best and most complete book on the subject, from general Bézout’s theorem to Grothendieck-Riemann-Roch theorem. Lots of examples.
ON RESOLUTION OF SINGULARITIES:
Kollár – Lectures on Resolution of Singularities. Small but fundamental book on singularities, methods of resolution for curves and surfaces and proof of the cornerstone Hironaka’s theorem. The only main alternative is Cutkosky’s book.
ON MODULI SPACES AND DEFORMATIONS:
Hartshorne – “Deformation Theory”. Just the perfect complement to Hartshorne’s main book, since it did not deal with these matters, and other books approach the subject from a different point of view (e.g. oriented for complex geometry or for physicists) than what a student of AG from Hartshorne’s book may like to learn the subject. The alternative and easier title is Sernesi’s book on deformation of algebraic schemes
ON GEOMETRIC INVARIANT THEORY:
Mumford; Fogarty; Kirwan – “Geometric Invariant Theory”. Simply put, it is the original reference. Besides, Mumford himself developed the subject. The best alternative to this and the previous title, but more on the introductory side, is Mukai’s book on moduli and invariants.
ON HIGHER-DIMENSIONAL VARIETIES:
Debarre – “Higher Dimensional Algebraic Geometry”. The main alternative to this title is Kollar/Mori “Birational Geometry of Algebraic Varieties” but is regarded as much harder to understand by many students. This is a very active frontier of research, with new fundamental results proved in Hacon/Kovács’ “Classification of Higher-Dimensional Varieties”.