Graduate Student Seminar, Spring 2015

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


  1. Please email me if you want to be on the associated mailing list.
  2. The talks will be roughly 45 + 15 + 30 minutes where 15 = break.
  3. Time and place: Fridays 10:30 -- 12:00 AM in Room 528.
  4. First organizational meeting: Friday, Jan 23.
  5. No lecture on: TBA.
  6. Last meeting: TBA.

Intersection theory

The idea is to carefully work through classical intersection theory using moving lemmas and using Serre's Tor formula to define local multiplicities. The idea is to follow the outline sketched in the discussion on intersection theory (see below) and LaTeX up the result as a chapter of the Stacks project (I will do most of the typing myself).

We will follow very closely the exposition given in the corresponding chapter of the Stacks project. Here are two links:

  1. pdf: intersection.pdf
  2. html: Chapter Tag 0AZ6


Here the idea is to take an adequate equivalence relation on algebraic cycles and carefully discuss how one constructs a category of motives from such a gadget. It is always fun to explain the content of Jannsen's paper on numerical equivalence and semi-simplicity as an application so if we have time we will do so.

List of talks

Tentative list of talks on Intersection Theory:

  1. January 23: Organizational
  2. January 30, Qixiao Ma: Sections 1 -- 9 of the Chapter above. This will cover: Cycles, cycle associate to closed subscheme, cycle associated to a coherent sheaf, proper pushforward of cycles, flat pullback of cycles, rational equivalence in 2 ways
  3. February 6: I am away, no talk
  4. February 13, Remy van Dobben de Bruyn: Sections 10 and 11 of the Chapter above. This will cover: proper pushforward and flat pullback factor through rational equivalence.
  5. Febuary 20, Joshua Seaton: Sections 12 -- 15 of the Chapter above. This will cover: proper intersections, intersection multiplicities using the Tor formula, algebraic multiplicities, computing intersection multiplicities.
  6. February 27, Dmitry Korb: Sections 16 -- 21 of the Chapter above. This will cover: exterior product, reduction to the diagonal, flat pullback and intersection products, projection formula for flat maps.
  7. March 6, No lecture.
  8. March 13, No lecture.
  9. March 20: Spring break, no lecture
  10. March 27, Ashwin Deopurkar: Sections 22 -- 27 of the Chapter above. This will cover: moving lemma, intersection products and rational equivalence, Chow rings.
  11. April 3, Johan de Jong: Some remarks on intersection theory

List of possible talks on Motives:

  1. April 10, Feiqi Jiang: Adequate equivalence relations and correspondences
  2. April 17, Johan de Jong: The category of motives and Uwe Jannsen's paper
  3. April 24, Tian An: Tannakian formalism and why motives are not neutral
  4. May 1, Michel van den Bergh: Non-commutative resolutions of quotient singularities

Old references

Here are some related documents: