Intersection Theory - Chow groups and Their Applications

Question

I'm a beginner in the field of Algebraic Geometry, especially the Intersection Theory. I came to know that the Chow groups of schemes are analogs to the homology groups of manifolds. I know at least one motivation behind the study of homology groups; they provide invariants of the manifold that can be used for classification purposes. I believe there must be many such motivations/applications of chow groups but right now I am failed to find. Can any of you be kind enough to enlist few of these?

Answer 1

The book "3264 and all that, a second course in algebraic geometry" by Eisenbud and Harris has multiple applications and motivations, mainly to enumerative geometry. Here what is really interesting is how to intersect cycles, i.e what is the product on $A^*(X)$ so we can get concrete numbers for various enumerative problems.

To my knowledge we don't really use $A^*(X)$ as an invariant because it's really hard to compute ( I believe we don't know how to compute it even for quadrics !). Rather than that, we start with say an enumerative problem, translate into a Chow ring computation, and hope we'll be able to compute everything.

In some case we know how to compute $A^*(X)$, for example, if your space is covered by affine spaces of various dimension (e.g $\Bbb P^n$) then $A^*(X) \cong H^*(X)$ canonically, this is Totaro's theorem.

A famous concrete example : on a generic cubic surface in $\Bbb P^3$ there is $27$ lines. You can prove it pretty easily using Chern classes and the Chow ring of grassmannians.

If you want to have more fun you can even check the following quote of Dijkgraaf : "One number which every algebraic geometer knows is the number 2,875 because obviously, that is the number of lines on a quintic" :-)