Topics in Algebraic Geometry (Math 245): Intersection Theory

 Fall 2018

Monday Wednesday Friday 9-10:20 in Herrin T195

The goal of this class is to gain familiarity and comfort with intersection theory.

Prerequisites. Comfort with algebraic geometry (the language of schemes), combined with a willingness to work with things you haven't fully mastered.

The main reference I will follow is Fulton's Intersection theory. Another great reference that I will appeal to is Eisenbud and Harris' 3264 and all that.

Course email list. Please be sure you are on the course email list (by asking me).

Instructor: Ravi Vakil (office 383-Q, office hours TBA, but possibly just by appointment if I know everyone in the class well enough).

  • Week 1 (September 24-28): Welcome. Conventions, and desiderata. Group k-cycles, defined as formal sums of varieties, or else with enhancements to subschemes or coherent sheaves. Fundamental class. Proper pushforward of cycles. Rational equivalence, defined in terms of divisors of rational functions on varieties, or else with enhancements to subschemes. Chow groups as cycles modulo rational equivalence (where rational equivalence is defined in various ways). Proper pushforward of Chow.
  • Week 2 (October 1-5): Flat pullback of cycles, and of Chow classes. Excision exact sequence.
  • Week 3 (October 8-12): Intersecting pseudodivisors with cycles, and consequences. Chern and Segre classes.
  • Week 4 (October 15-19). More on Chern and Segre classes. Deformation to the normal cone.
  • Week 5 (October 22-26). End of deformation to the normal cone. Gysin pullback. Intersection theory on the Grassmannian, and the 27 lines on the cubic surface.
  • Week 6 (October 29 - November 2).
  • Week 7 (November 5-9).
  • Week 8 (November 12-16). (No class Friday November 16.)
  • Week 9 (November 26-30). (No class Monday November 26.)
  • Week 10 (December 3-7).

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