A guide to Algebraic Geometry

Question

I have completed one semester course on Commutative Algebra and Riemann Surfaces, and currently I am trying to read Algebraic Geometry. While reading from different books I feel that I must need a direction such that I can cover much topics in a semester. That is to say, what are the fundamental results or topics in Algebraic Geometry that I should read in a one semester course. Besides this I feel that I should follow some text books or some well written lecture notes as a prepratory metarial. It will be helpful to me if someone guide me by saying that which topics I have to read , from which books or notes. Thanks in advance.

Answer 1

In my sense, the first key topic should be to see examples, and a lot of examples. You've seen some with your class, but I would advice to go further in this direction and to read a lot of chapters in "A first course in algebraic geometry" by J.Harris. This is a self-contained account of basic algebraic geometry, wonderfully written and each chapter contains examples and exercises. If you spent a semester reading it, you will learn a lot (moreover you won't forget it, which might happens if you try directly a more abstract text) and also this can give you a good direction of what you want. More specifically chapters $1,2,4,5,7,10, 11$ and $14$ contains essential ideas and lot of beautiful examples so you can get your hands dirty.

---

After that, you'll have lot of examples in mind and will be ready to read more specific topics. Here are my personal suggestions :

1) M. Reid, Chapters on Algebraic surfaces : This is the natural step after curves, and a fascinating subject. This is the most comprehensive reference on surfaces, written by a master.

2) K.Lamotke, The topology of complex projective varieties after S.Lefschetz : This wonderful article explains in a modern langage the proof of Lefschetz hyperplane theorem, and give you all the tools to understand the topology of complex algebraic varieties. This a perfect motivation before studying Hodge theory or singular algebraic varieties.

3) S.Mukai, Introduction to Invariant and Moduli : A gentle introduction about ideas behind geometric invariant theory (i.e what should be a "geometric" quotient ?) and moduli spaces, with a special emphasis on classical groups, and moduli of vector bundles on curves.

4) W.Fulton, Toric varieties : Toric varieties are a special kind of algebraic varieties with a torus action. The action can be encoded combinatorially by a polytope or a fan. The study of these combinatorial objects contains rich informations about the variety itself, and toric geometry is a fertile ground to try new ideas in algebraic geometry.

5) D.Eisenbud and J.Harris, 3264 and all that, a second course in geometry : A really rich book explaining the basic of intersection theory and enumerative geometry, with as usual lot of examples, pictures and computations.