Question

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.

Answer 1

I think the best "textbook" is Ravi Vakil's notes:

http://math.stanford.edu/~vakil/0708-216/

http://math.stanford.edu/~vakil/0910-216/

Answer 2

I'm a fan of The Geometry of Schemes by Eisenbud and Harris. Its great for a conceptual introduction that won't turn people off as fast as Hartshorne. However, it barely even mentions the concept of a module of a scheme, and I believe it ignores sheaf cohomology entirely.

Answer 3

Liu wrote a nice book, which is a bit more oriented to arithmetic geometry. (The last few chapters contain some material which is very pretty but unusual for a basic text, such as reduction of algebraic curves.)

Answer 4

Perhaps this is cliché, but I recommend EGA (links to full texts: I, II, III(1), III(2), IV(1), IV(2), IV(3), IV(4)).

I know it's a scary 1800 pages of French, but

1. It's really easy French. I would describe myself as not knowing any French, but I can read EGA without too much trouble.
2. It's extremely clear. The proofs are usually very short because the results are very well organized.
3. It's the canonical reference for algebraic geometry. I assure you it is not 1800 pages of fluff.

I've found it quite rewarding to to familiarize myself with the contents of EGA. Many algebraic geometry students are able to say with confidence "that's one of the exercises in Hartshorne, chapter II, section 4." It's even more empowering to have that kind of command over a text like EGA, which covers much more material with fewer unnecessary hypotheses and with greater clarity. I've found this combined table of contents to be useful in this quest.

Answer 5

At a lower level then Hartshorne is the fantastic "Algebraic Curves" by Fulton. It's available on his website.

Answer 6

Shafarevich wrote a very basic introduction, it's used in undergraduate classes in algebraic geometry sometimes

Basic Algebraic Geometry 1: Varieties in Projective Space

also, for a more computational point of view

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra

And the followup by the same authors

Using Algebraic Geometry

Answer 7

I've been teaching an introductory course in algebraic geometry this semester and I've been looking at many sources. I've found that Milne's online book (jmilne.org) is excellent. He gives quite a thorough treatment of the theory of varieties over an algebraic closed field. The book is very complete and everything seems to be done "in the nicest way".

Answer 8

I enjoyed Griffiths-Harris a lot.

Answer 9

Kenji Ueno's three-volume "Algebraic Geometry" is well-written, clear, and has the perfect mix of text and diagrams. It's undoubtedly a real masterpiece- very user-friendly.

Answer 10

The book An Invitation to Algebraic Geometry by Karen Smith et al. is excellent "for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites," to quote from the product description at amazon.com.

Answer 11

Joe Harris's book Algebraic Geometry might be a good warm-up to Hartshorne.

Answer 12

Miles Reid's Undergraduate Algebraic Geometry is an excellent topical (meaning it does not intend to cover any substantial part of the whole subject) introduction. In particular, it's the only undergraduate textbook that isn't commutative algebra with a few pictures thrown in.

Answer 13

Computer Scientists, me included, seem to prefer Ideals, Varieties, and Algorithms by David A. Cox, John B. Little, Don O'Shea (http://www.cs.amherst.edu/~dac/iva.html)

Answer 14

I believe the issue of "which book is best" is extremely sensitive to the path along which one is moving into the subject. If your background is in differential geometry, complex analysis, etc, then Huybrechts' Complex Geometry is a good bridge between those vantage points and a more algebraic geometric landscape. Obviously I'm taking liberties with the question, as I wouldn't advertise Huybrechts' book as an algebraic geometry text in the strict sense. However, I think it can, for certain people, help to ease the transition into one. It's also very well written, in my opinion. (I should also emphasize that I'm not saying this is the only purpose of the book: its content is extremely valuable for other reasons, with material on vector bundles, SUSY, deformations of complex structures, etc.)

As for dedicated algebraic geometry texts other than Hartshorne, I also vote for Ravi Vakil's notes. They're excellent.

Answer 15

Also Eisenbud.

Every algebraic geometer needs to know at least some commutative algebra. And this is a very good introductory textbook, which teaches commutative algebra rigorously but at the same time provides a good geometric explanation.

Answer 16

I second Shafarevitch's two volumes on Basic Algebraic Geometry: the best overview of the subject I have ever read.

Another very nice book is Miranda's Algebraic Curves which manages to get a long way (Riemann-Roch etc) without doing sheaves and line bundles until the end. Of course, by then, you are really wanting sheaves and line bundles!

Answer 17

I've tried learning algebraic geometry several times. I asked around and was told to read Hartshorne. I started reading it several times and each time put it away. I realized that I could work through the sections and solve some of the problems, but I gained absolutely no intuition for reading Hartshorne. Discussing this with other people, I found that it was a common occurrence for students to read Hartshorne and afterwards have no idea how to do algebraic geometry. (I imagine this was the motivation for asking this question.)

After more poking around, I discovered Mumford's "Red book of Varieties and Schemes". While Mumford doesn't do cohomology, he motivates the definitions of schemes and and many of there basic properties while providing the reader with geometric intuition. This book isn't easy to read and you have to work out a lot, but the rewards are great. Another great feature of this book is that Mumford bought the rights to the book back from Springer and the book is available for free online.

Another book was supposed to be written that built on the "Red book" including cohomology. After many years, I think this is near completion; see Algebraic Geometry 2. Whlile many of the above books are excellent, it's a surprise that these books aren't the standard.

Answer 18

I liked Mumford's "Algebraic geometry I: Complex projective varieties" a lot, and also Griffiths' "Introduction to algebraic curves". Now I think I am falling in love with "Griffiths & Harris". For the record, I hate Hartshorne's.

Answer 19

Also lots of things on jmilne.org

Answer 20

If you know french, you might enjoy David Harari's course notes. These are the notes for a basic course in schemes and cohomology of sheaves. He combines the best parts of Hartshorne with the best parts of Liu's book. Hartshorne doesn't always do things in the nicest possible way, and the same is of course true for Liu.

I agree that Vakil's notes are great, since they also contain a lot of motivation, ideas and examples. But does anyone know where to get the files with this year's notes? I only found the notes of previous years on the web.

Answer 21

Macdonald "Algebraic geometry: Introduction to schemes" (not only about noetherian schemes), Dieudonné's two booklets with focus on the motivation and history, the first chapter in Demazure, Gabriel "Groupes algebraique I" (recommended here to undergraduates as intro text), Mumford's "red book".

Answer 22

Mukai's Introduction to Invariants and Moduli surely deserves to be on this list.

Answer 23

About HArtshorne and Griffith, I think a comparison between the two texts is misleading. The first is a introduction to the "Grothendieck ioga" where geometrical classical idea are "immersed" in the more large but abstract mathematical world of schemas. But also if the Complex differential manifold style of Griffith is "more concrete" is very different from the "Algebraic Geometry" idea, also if it is a deep study of it.

As categorist I love Grothendiek, but I find Grothendieck work fantastic in itself (like a music), also if I never understand nothing about Geometry Geometry while studing or readind EGA or some SGA.

FOr understant what is "ALgebraic Geometry" I had to read this

Beltrametti-Carletti-Gallarati-Monti Bragadin, Letture su curve, superficie e varietà proiettive speciali, Boringhieri

Beltrametti-Carletti-Gallarati-Monti Bragadin, Lezioni di geometria analitica e proiettiva, Boringhieri

(sorry, these are in Italian)

Answer 24

I've found something extraordinary and of equally extrordinary pedigree online recently. I mentioned it briefly in response to R. Vakil's question about the best way to introduce schemes to students. But this question is really where it belongs and I hope word of it spreads far and wide from here.

Last fall at MIT, Micheal Artin taught an introductory course in algebraic geometry that required only a year of basic algebra at the level of his textbook. The official text was William Fulton's Algebraic Curves, but Artin also wrote an extensive set of lecture notes and exercise sets. I found them quite wonderful and very much in the spirit of his classic textbook(By the way, simply can't wait for the second edition.).

Not only has he posted these notes for download, he's asked anyone working through them to email him any errors found and suggestions for improvements.All the course materials can be found at the MIT webpage. I've also posted the link at MathOnline, of course.

I don't know if most of the hardcore algebraic geometers here would recommend these materials for a beginning course. But for any student not looking to specialize in AG, I can't think of a better source to begin with. That's just my opinion. But it certainly belongs as a possible response to this question. Then again, it may be too softball for the experts,particularly those of the Grothendieck school.

Here's keeping our fingers crossed that this is the beginning of the gestation of a full blown text on the subject by Artin.