We start by defining the blow up of an affine Noetherian scheme along a closed subscheme . Since is given as the vanishing set of some ideal we have .
For a Noetherian affine scheme and a closed subscheme we define the blow up of along to be
where we view
as a graded
-algebra whose degree
Note first that this strange-looking algebra is in fact generated in degree by construction and also, by the Noetherian hypothesis, it is generated by finitely many elements, say . using this, we can give a more explicit model for it. The key point is to view not as a subset of but rather as the quotient module
are just formal variables and
is the surjective
In fact we can extend
to a surjective graded algebra homomorphism
where on the right hand side the non-zero term lies in position
and the map is
In many concrete cases we have a much more concise way of writing down the kernel of this map and thus get a handle on the algebra
. Let us see how we can compute our favourite example
with this new definition.
We consider and with and . So from the discussion above we have that
is the quotient of the graded algebra
by its ideal .
So in fact we have:
whose closed points are precisely
We now move on to give the general definition of a blow up.
Let be any Noetherian scheme and -- a closed subscheme. Then on every affine open in , is given as the (scheme-theoretic) vanishing set of some ideal which defines an ideal sheaf on . These local ideal sheaves glue together to give a sheaf . We can now consider the sheaf of graded algebras
. On each affine open subset of we can then define a scheme .
In fact these schemes glue together to define a new scheme which we denote . Moreover there is a natural morphism
Finally we state the most general definition of a blow up:
Let be a Noetherian scheme and a closed subscheme. Then we define the blow up of along to be
The exceptional divisor
is the natural morphism.