MATH 7670: Algebraic Geometry: Sheaves and Schemes (Fall 2009)

Instructor: Michael Stillman

The language of modern algebraic geometry: Sheaves and schemes

Textbook: Eisenbud and Harris, Geometry of Schemes

In this course we will introduce sheaves and schemes, which form the language used for almost all modern algebraic geometry research. We will motivate the use of these concepts by giving many examples. We will also introduce many of the basic constructions in algebraic geometry via schemes.

Tentative outline: We will mostly follow Eisenbud and Harris, supplementing in a few places where I think they do not provide enough detail.

1. Affine schemes
2. Sheaves
3. General schemes (and morphisms between them)
4. Fiber products: families of varieties and schemes
5. Examples: non-reduced schemes, flat families of schemes, arithmetic schemes
6. Projective schemes (including tangent sheaves, Grassmannians, Cartier divisors and line bundles)
7. Geometric constructions: flexes of plane curves, blowups, Fano schemes, ...

Algebraic geometry is a subject where you must "learn by doing". I will assign regular homework, and we will discuss the solutions during class time.