The general definition

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 .

Definition 2.6

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 piece is (with ).

 


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

where are just formal variables and is the surjective module homomorphism
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 linear. 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.

Example 2.2

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:

Definition 2.7

Let be a Noetherian scheme and a closed subscheme. Then we define the blow up of along to be

The exceptional divisor is , where is the natural morphism.

 
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