Reading group on Algebraic Geomtry
We provide some materials for a reading group for junior graduate students and some undergraduate students who are interested in algebraic geometry and related fields.
The aim of this one-semester reading group is to understand some basic knowledge for the foundation of algebraic geometry (part of Hartshorne's book Algebraic Geometry II and III).
The book the geometry of schemes by Einsenbud and Harris.
See also Ravi Vakil's notes on algebraic geometry, which is a very good reference.
Rougly speaking, we hope to cover (at least the first 20 pages) of this survey article.
Our aim is to develop languages and techniques about schemes, sheaves (quasi-coherent and coherent), derived functors, sheaf cohomology, Cech cohomology, and arrive at Serre's duality theorem.
Tasks.
Task 1: Sheaves (warm up).
A crush course = [Survey 1.1.1].
Read: [Hartshorne Chap II. 1];
See also [Vakil Chap 2].
Task 2: Schemes I: affine schemes.
A crush course = the first paragraph of [Survey 1.1.2]
Read: [EH] Affine schemes I.1. ([EH] = the geometry of schemes by Eisenbud-Harris.
See also [Hartshorne Chap II. 2 ] and [Vakil Chap 3-5].
Task 3: Scheme II: General schemes.
Read: [EH] Schemes in general I.2.
If you have more energy, go to [EH] III.1, III.2,
to learn a special class of schemes that we will be focusing on: projective schemes.
Task 4: Sheaves of O_X-modules.
Read: [Hartshorne Chap II 5] till the end of Cor 5.10. (Pay attention to Prop 5.6; compare it with Ex 1.8 (Chap II))
Task 5: Sheaves of O_X module II.
Read: the rest of [Hartshorne Chap II 5]. Pay attention to projective schemes, and the sheaves O_X(n). Important theorems: Thm 5.17 and Thm 5.19.
Fall Break.
Task 6: Derived functor and cohomology of sheaves.
Read: [Hartshorne Chap III 1, and 2 (before Thm 2.7)].
7 Task 7: Cohomology of sheaves.