Cornell Math - MATH 767, Fall 2004

MATH 767: Algebraic Geometry (Fall 2004)

Instructor: Michael Stillman

Meeting Time & Room

Prerequisites: A course in commutative algebra at the level of MATH 634 (Atiyah-Macdonald), and the first chapter of Hartshorne (especially the first 4 sections).

Textbook(s): Algebraic Geometry, by Robin Hartshorne; The Geometry of Schemes, by Eisenbud and Harris

Our goal is to understand the material on sheaves, schemes, and cohomology (chapters 2,3) from Hartshorne's book. (The course could be called: "Hartshorne: the terrifying chapters").

Each week, one class period will be used for students to present solutions to exercises (mostly taken from Hartshorne and Eisenbud/Harris).

One aspect of Hartshorne's book that makes it seem more difficult for students is a lack of motivation for the main ideas. We will present motivation and examples of all of the main concepts. On the other hand, students will be expected to read the sections being covered before class, so that we will not need to cover in class all of the proofs. However, one goal of the course is to insure that everyone understands the important techniques of proof in this area.

Tentative list of topics:

  1. Sheaves
  2. Affine schemes

Motivation, definition and basic properties
We will introduce many concepts first for affine schemes, then come
back to them for general schemes

  1. General schemes
  2. Maps between schemes
  3. Projective schemes
  4. Coherent sheaves, some of Serre's theorems
  5. Divisors and line bundles
  6. Linear systems and maps to projective space
  7. Differentials, blowups
  8. Cohomology of sheaves
  9. Cech cohomology
  10. global Ext and Serre duality
  11. Flat families of varieties and schemes
  12. The upper semicontinuity of cohomology
  13. Formal schemes
  14. Zariski's main theorem and the theorem on formal functions

We will also introduce many examples, as algebraic geometry is a subject with a wealth of beautiful examples.

Missing from this list is the discussion on separated and proper morphisms. We will cover this material if time permits, but later in the course than it appears in Hartshorne's book.