Professor A.J. de Jong, Columbia university, Department of Mathematics.

This is the webpage of the graduate course "Spring 2018 Mathematics GR6262 ARITH & ALGEBRAIC GEOMETRY".

Tuesday and Thursday, 10:10 AM - 11:25 PM in Room 407 Math.

Grading will be based on homework and a final exam. It is very important to attend the lectures. You will not be able to pick up the material by just reading this webpage.

The TA will most likely be Raymond Cheng. He will be in the help room ??.

We will use the Stacks project as a reference, but of course feel free to read elsewhere. If you see a four character alphanumeric code, like 0000, then this is a link to a chapter, section, exercise, or a result in the Stacks project.

Reading. Please keep up with the course by studying the following material as we go through it. We will go through material at a rapid pace and the assumption is that you'll spend a lot of time with the theory by yourself in order to understand and keep up with the course.

Topics. We will start discussing algebraic curves and their linear systems. Hopefully we will be able to discuss a bit about the theory of algebraic surfaces later in the course. For both topics the corresponding chapters of Hartshorne are an excellent source of material, but we will often go beyond what is said in there.

Exercises. Please do the exercises to keep up with the course:

  1. Due 1-25 in class: do 2 of the exercises from 0DJ0 and do one of 0EG6 and 0EG7 and do 0DAM
  2. Due 2-1 in class: do as much of the exercises from 0EG8 as you can; if you prefer to read about these questions elsewhere, then write a one page report about what you read.
  3. Due 2-8 in class: as much as you can from 0EGN
  4. Due 2-15 in class: as much as you can from 0EGR
  5. Due 2-22 in class: give a precise statement and proof of the base point free pencil trick.
  6. Due 3-1 in class: Let X be the incidence correspondence between lines and points in the projective plane. This means that X is given by the equation T_0S_0 + T_1S_1 + T_2S_2 = 0 in P^2 x P^2 with homogeneous coordinates T_0, T_1, T_2 and S_0, S_1, S_2. Find the Picard group of X and the intersection form on the Picard group (this is a cubic form as X has dimension 3).
  7. Due 3-8 in class: as much as you can from 02EC 02ED 02EF
  8. Due 3-22 in class: as much as you can from 02AS 0AAR 0AAS