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Intersection theory

Course Type: 
PhD Course
Anno (LM): 
First Year
Second Year
Academic Year: 
60 h

The course is 60 hours, but it has a natural division into 40+20 so it makes sense to attend only the first part.It will take place starting in the second half of November and will finish at the end of March.

The course is based on parts of the book "Intersection Theory" by William Fulton, but is different enough that informal notes will be provided. The focus in the course and notes will be to include examples, motivations and highlight relationship with complex geometry; some technical proofs will be skipped.

The first 40 hours present the core of modern intersection theory for quasiprojective schemes over a fixed algebraically closed field, roughly corresponding to chapters 1 to 6 of the book: Chow cycles, proper pushforward and flat pullback, rational equivalences and Chow classes, first Chern class of a line bundle as operator on Chow classes, characteristic classes of vector bundles and the splitting principle, Chow groups for vector and projective bundles, MacPherson degeneration to the normal cone, Gysin pullback and its properties, lci pullback.

The last 20 hours will be devoted to more advanced material. A natural choice would be Grothendieck Riemann Roch theorem for projective morphism of smooth varieties; alternatively, we can cover perfect obstruction theories and virtual fundamental classes. The precise topic will be decided by me after a discussion with interested people.


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