A guide to Algebraic Geometry

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.

• Well you could go the classical route and learn about varieties over an algebraically closed field, e.g. Milne's notes or first chapter of Hartshorne. That's doable in one semester if you know some commutative algebra. Or you could go straight to schemes, e.g. second chapter of Hartshorne. It's independent of the classical stuff, but maybe not well motivated in its absence. – D_S Nov 25 '18 at 6:20
• You didn't mention Galois theory and arithmetic of elliptic curves – reuns Nov 25 '18 at 6:54

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.