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 Whit- ney sum formula, excess intersection formulas, formulas for the class of degeneracy loci for maps of vector bundles, and applications to concrete problems in enumer- ative geometry. We will also prove the Grothendieck-Riemann-Roch theorem and give applications. Here is a more detailed program. I. Chow groups and the Chow ring. We have developed the method of deformation to the normal bundle to extend the pull-back maps for the inclusion of a Cartier divisor to the case of regular embeddings, and from there the case of a general lci morphism. Applying this to the diagonal morphism for a smooth variety gives the Chow groups of a smooth variety the structure of a graded ring for which the pullback maps are graded ring homomorphisms. [3, Chap. 6, 7]. We will develop this further, defining and using the “refined” Gysin morphisms. We will introduce Chern classes and Segre classes of vector bundles and use these to give the excess intersection formula [3, Chap. 6, 9]. II. Grothendieck-Riemann-Roch. Using the method of Grothendieck [1], the pro- jective bundle formula gives rise to a theory of Chern classes of vector bundles and ∗ the Chern character homomorphism ch : K0(−) ! CH (−)Q. The GRR theorem ∗ compares the pushforward in K0 with that in CH via the Chern character. We use the method of the deformation to the normal bundle for the case of closed immer- sion and Panin's method of the diagonal for the case of a projection X × Pn ! Pn, both approaches relying on the relation of formal group laws with Todd classes [2]. We will introduce the Grassmann-graph construction and use this in the proof of the Riemann-Roch theorem for singular varieties. III. Degeneracy loci and intersection theory on Grassmannians. Many enumera- tive problems can be expressed in terms of computing the class in the Chow ring of the locus of points for which a map of vector bundles has rank ≤ m for some number m. The problem of computing the class of such degeneracy loci relies on understanding the Chow ring of Grassmann varieties via Chern classes and the Schubert calculus. [3, Chap. 14]. References [1] A. Grothendieck, La th´eoriedes classes de Chern. Bull. SMF tome 86 (1958) 137{154 [2] I. Panin, Riemann-Roch theorems for oriented cohomology [3] W. Fulton, Intersection Theory 1.