Selecting a book for a reading course in Algebraic Geometry

Question

I'm not sure if this is an appropriate question in this forum, but here is the situation.

I must begin by saying that I know basically nothing about Algebraic Geometry, but this semester I will be doing a reading course in Algebraic Geometry.

Of course, I will be discussing the choice of a book with my instructor, however, I wanted to have a better idea when it comes to selecting a book.

So far, I have thought of the following books:

1. Basic Algebraic Geometry, volume 1, by Shafarevich;
2. Algebraic Geometry, by Hartshorne;
3. Algebraic Geometry, a first course, by Joe Harris;
4. A Royal Road to Algebraic Geometry, Holme etc.

So, I guess my first question is, since there are a few branches of Algebraic Geometry, why would a person select one of these books over the others? Second, which of these books(or others that I don't know about),in your opinion, is the best book for someone who is just beginning to learn about Algebraic Geometry.

My background goes up to a course in Commutative Algebra (Atiyah MacDonald).

My goal is eventually to do a PhD thesis in Algebraic Geometry (I am a first year graduate student). Thanks!

Answer 1

Although it is not exactly Algebraic Geometry, I would really recommend that you read "Algebraic Curves and Riemann Surfaces" by Rick Miranda. In my opinion, you should really not start learning algebraic geometry until you have geometric intuition from Riemann surfaces. This book also does things at first in a more analytic way, but gradually shifts the style of proof to a more "algebraic" perspective. You will learn some of the fundamental theorems in the subject, like Abel's theorem and Riemann-Roch. You will also get acquainted with sheaves in a very down to earth way.

I took a course from Hartshorne before reading this book, and I have to say I gained very little. Returning to Hartshorne after this book, and I could appreciate a lot. It always helps to know what you are abstracting. In fact, I would recommend reading Joseph Taylor's book on several complex variables and algebraic geometry after this one. I am quite biased towards complex analysis however.

Answer 2

I had once done a reading course in Algebraic Geometry from the book Algebraic Curves by William Fulton.

I also had similar background (I had read parts of Atiyah Macdonald). I really enjoyed the book mainly because of the following reasons:

1) The book treats plane curves, and hence most of the techniques can be "visualized", while the major techniques are introduced to understand the basic problems underlying classical Algebraic Geometry.

2) The book's presentation is great. If this is your first reading course (as it was in my case) you would definitely want to select a book that you would enjoy reading.

3) The book has quite a few exercises, ranging from trivial to not so obvious (some exercises required a bit of thought). For a reading course, I feel it's important to solve a few exercises to know that you are going on the right track.

Of course, I must mention that I am not an Algebraic Geometrist. My research area is quite different- and I haven't used Algebraic Geometry in a while. I also haven't read any of the books that you mentioned; so you should take my advice with a pound of salt. My main motivations for doing the reading course was to gain familiarity with the commutative algebraic techniques that one uses, and also to get some geometric and algebraic intuition into the "calculus of curves". Fulton's book was ideal for me, and I greatly enjoyed the reading course. However depending on your aims and taste, you might want to look at different texts.