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 .