Algebraic Geometry Textbook Recommendations

jgens

So I have been meaning to learn a little algebraic geometry for some time now, but have never gotten around to it. Since classes are just now winding down for the year, I figured that it was an ideal time to self-study a bit.

Now for a bit about my background: I know that commutative algebra is provides the local tools in some sense for algebraic geometry, and in previous classes I have worked through Introduction to Commutative Algebra by Atiyah and MacDonald as well as Undergraduate Commutative Algebra by Miles Reid. I am hoping that these books provide enough background to begin learning a little algebraic geometry, but if not, then I am not opposed to reading a more voluminous work like Eisenbud's text on commutative algebra.

Looking around, most of the common introductions to algebraic geometry seem to be either the book by Hartshorne or the (red) book by Mumford. Are these good places to start or are there other texts that I should look into? I would like to add that the cost of the book is not an issue since I plan on checking them out from my university library.

mathwonk

try miles reid's undergrad alg geom, and rick miranda's algebraic curves and riemann surfaces, and shafarevich's basic algebraic geometry, also walker's plane algebraic curves, and fulton's algebraic curves (free), griffiths' algebraic curves, and joe harris's algebraic geometry. mumford is excellent but has too few examples, and hartshorne is a very advanced book for a third or fourth course.

remember when you took calculus? there were three or four elementary computational courses, and then an abstract analysis course, and an advanced calculus course, and then an abstract advanced course. it takes that long to learn alg geom also. hartshorne is a book for more or less the last general course, not the first course.

jgens

I will definitely look into the Miles Reid book! His Undergraduate Commutative Algebra is excellent. You and micromass have also convinced me to stay away from Hartshorne for now.

kloptok

I can't really say that I know any algebraic geometry so I don't know about the quality, but here are some lecture notes by David A. Cox of Amherst College:
http://www.cs.amherst.edu/~dac/lectures/intro.pdf

From Cox's website (you have some more stuff there also): http://www3.amherst.edu/~dacox/

mathwonk

If you read the introductory comments in the first notes there from Cox, he says they are intended for someone who already knows elementary algebraic geometry. David writes carefully and well and there are many good notes on his page, but none as elementary as Reid's book for beginners.

Chapter one of Hartshorne does have a lot of nice examples. mostly in the exercises. He also tends to take for granted a lot of commutative algebra already in section one, but you may know this algebra from Miles Reid. I prefer books that prove the algebra along with the geometry, like the first edition of Shafarevich, Fulton, and Mumford's red book. Joe Harris' book is a rich source of examples.

rudolfstr

tits