Topics in Algebraic Geometry (Math 245): Intersection Theory
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
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
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
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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