Gysin morphism

When we have a vector bundle ( of finite rank ) over a scheme, that induces a flat pull back map of cycle classes, which can be shown to be isomorphism.

The surjectivity holds for affine bundle as well, this can be reduced to the case where the bundle is trivial using Noetherian induction : the cycle classes of a scheme can be “roughly” divided into the classes in an open subset and those in the complement ( this can be described by an exact sequence, which I would name the structural exact sequence ).

The injectivity is much harder to show, we need extra structures. First we need to complete the bundle projectively, with its projectivization added as the hyperplane at infinity. Then using Noetherian induction, we can show that there is an “O(1) decomposition” of each cycle class in the projective bundle as a ordered sum of classes : each of which is represented by the iterated action of the canonical line bundle on a pull back class. This fits nicely with the description of a projective space recursively as piecing an affine space with a projective space of 1 dimension less at infinity. Then the injectivity comes as a corollary of these added structure, using the “structural exact sequence” on the bundles, which are all compatible with the projection onto the base scheme.

There is a more descriptive description of the Gysin homomorphism using the the universal quotient bundle on the projective completion . To get the class on the base scheme via the Gysin morphism, first we take the closure of the class inside the projective completion of the bundle, intersect it with the highest Chern class of the universal quotient bundle, and the project it onto the base scheme.

Construction of Hilbert Schemes

We want to parametrize subschemes of projective space . Naturally, those with same Hilbert polynomial will cluster together, by the result which states that any continuous (flat ) family of closed projective subschemes have constant Hilbert polynomial. Now we will construct the moduli space of all subschemes with a fixed Hilbert polynomial . Here is the idea :
The value of the Hilbert polynomial at will coincide with the dimension of the graded piece of the coordinate ring of when is large. Another description of the Hilbert polynomial is the Euler characteristic of the structure sheaf of the scheme twisted by , by Serre vanishing theorem ( why?).
The idea is, if we choose large enough, then we hope to identify with the graded piece of its defining ideal . So each scheme can therefore be identified with a subspace of fixed dimension of the space of all monomials of degree . Which means that we have a map from our “moduli space” to the Grassmanian , and then hope that the image is actually a closed subscheme of the Grassmanian. Fortunately, we can do that, but need a lot of results.
First in order to cut out precisely those with Hilbert polynomial , we require that the image of the map in has dimension for all ( the dimension of the piece of the defining ideal ). That could be consider a linear map of two vector bundles over the Grassmanian. Unfortunately , this is the same as to require the map of vector bundles has constant rank, which is not in general a closed condition. The best that we can do is to require that the rank is alway not mroe than . There are three problems we need to solve :
1) As mentioned, this is not a closed condition.
2) We have to do this for infinitely many .
3) It is not clear how we can pick uniformly, so that the dimension of the graded pieces after behave as we expect.
There are two important results that have us resolve this :
Thm 1: We can in fact choose such a uniformly.
Thm 2 : (i) We can choose such a that for all , the multiplication by map increase the dimension at least 1 ( first test )
(ii) Once any has passed the first test, it will pass all the required tests:
has image of dimension at least .
Now, with all of the above, to cut out the moduli space, we can just put on the closed condition: the rank of the multiplication maps must not be more than “expected”, i.e. and that would cut out precisely the subschemes having Hilbert polynomial .