Question

Path for algebraic geometry?

Answer 1

Take a look at the book "Algebraic Geometry 1: Schemes" By Görtz and  Wedhorn. The presentation of this book is somewhat like an abridged  English version of EGA with exercises, I'm quite fond of it. Note that this book  does not cover cohomology so you will need to read about that somewhere else. It does have a chapter on descent theory though which is wonderful. The functorial point of view is also developed throughout, which really is essential to actually understanding schemes (something Hartshorne's book completely misses) so it's worth looking at if only for that IMO.

Answer 2

If you're comfortable with commutative algebra, I recommend just jumping in with an algebra reference or two by your side: Eisenbud's "Commutative Algebra" is great for that.

Algebraic geometry is hard. I think the best way to proceed is to use multiple sources and bounce back and forth as needed! I learned most of the technical stuff from Hartshorne, Vakil, and Liu (good if you have an interest in arithmetic geometry) but I used a ton of other books and notes as references for intuition and more. Some of my favorites are Mumford/Oda and Goertz/Wedhorn (they discuss the functorial point of view more than the other sources above, which in my opinion is really important for understanding what's really going on e.g. with projective space), and Mumford's red book and Eisenbud/Harris for intuition.

Above all, don't be afraid to shop around and test a lot of things out until you find some sources that work for you. And ask plenty of questions!

Answer 3

Take a look at Harris Algebraic Geometry and Mumford Complex Projective Varieties. See how you fare with the material on the first pages. If it's alien to you, go learn the required stuff - complex analysis, differential forms, some manifolds and for Harris refresh a lot on that Linear Algebra and Projective Geometry. Hell, I would even start by reading some old classical projective geometry something like the book Plane Algebraic Curves from the Birkhauser editorial.

The thing is, you can go right away into Vakil or Hartshorne and learn schemes, but you will inevitably need to be know a lot of differential geometry, topology and complex analysis if you want to do serious Algebraic Geometry. So I would start by learning some more general math before diving into AG, and when the time comes for you to really go into AG, things will make MUCH MORE SENSE.

Oh, and don't forget, computations, lots of computations, don't let the overwhelming theoretical aspect of Algebraic Geometry deviate you from knowing how to explicitly calculate things like a blow up of a point

Answer 4

*Basic Algebraic Geometry* by Shafarevich (it's two volumes). The first volume sticks to varieties, the second deals with schemes and complex algebraic geometry. Much more geometric than Hartshorne, although Shafarevich is less comprehensive (in particular, he doesn't cover a lot about cohomology).

Answer 5

I started with Dummit and Foote, then Atiyah MacDonald, then Hartshorne

Answer 6

If you like the more "scheme theoretic" Algebraic Geometry style, the canonical recommendation is Hartshorne, but as a first read it's a little hard to work through. Vakil's "The Rising Sea" is a great book, but it's over 900 pages and you need to get to the second half of the book to get to "the good stuff" so it'll take quite a bit of time. That being said, the exposition is extremely helpful; algebraic geometry is very technical and it's easy to get lost in the machinery. Vakil's book is the only source (to my knowledge) that actually breaks down why the machinery is useful and how it works.

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If you prefer to work in the language of varieties (more "classical" and less technical) Joe Harris's book(s) in Algebraic Geometry (a first course in algebraic geometry), and his second course on intersection theory (3264 & all that) are good recommendations.

Answer 7

David Cox - ideals, varieties and algorithms!

Answer 8

I highly recommend at least becoming familiar with Grothendieck's EGA.

It is enormous, but despite the reputation for difficulty, line-by-line it is easier to read than any other text, with the possible exception of Vakil's book (which other people have recommended).

Also, if you are interested in the number-theory and representation-theory applications of algebraic geometry, Deligne's SGA 4.5 is extremely interesting and a (perhaps surprisingly) pleasant read.

This is not to say that you should spend years of your life focused *solely* on EGA/SGA (it *will* take years for you to understand everything in them). But IMO many people ignore those sources because of ill-founded fears.

Answer 9

In addition to my other comment, there's one more thing which applies to any recommendations that have been given here: you MUST do the exercises from the sources you use! Hartshorne is a tough place to learn from, but the exercises are phenomenal for the most part -- even if you don't use it as your main reference, it would benefit you to work out problems from the book alongside whatever you use. Vakil's exercises are also good for getting a thorough treatment, the difference being that they're much friendlier and spaced throughout the text to do as you're going along, not at the end of each chapter.

Answer 10

Perrins book on algebraic geometry is a good start in my opinion