Algebraic Geometry IV: Intersection theory and enumerative geometry
This is a continuation of algebraic geometry III. In that course, we finished up with the intersection theory for Cartier divisors and pseudo-divisors, which gave a well-defined pull-back map on the Chow groups for the inclusion of an effective Cartier divisor. In this continuation, we complete this to a good theory of pull-back maps on the Chow groups for lci morphisms. This leads to a functorial graded ring structure on the Chow groups of a smooth variety.
We will then look at refinements, extensions and applications. This includes the theory of Chern classes of vector bundles and their properties including the Whitney sum formula, excess intersection formulas, formulas for the class of degeneracy loci for maps of vector bundles, and applications to concrete problems in enumerative geometry. We will also prove the Grothendieck-Riemann-Roch theorem and give applications. For a more detailed program see the program.