If is a closed subscheme of , with the defining sheaf of ideals, then the normal cone to in is defined by the spectrum of the sheaf of algebra defined by : .
The question is why it is defined this way? First of all, note that this is infact a cone over , as the zero-th grade of is just . Now let’s look a particularly nice case, when is a regular imbedding in of codim , i.e., locally is cut out by a regular sequences with elements. Then in this case, we have a normal bundle , the DUAL of the sheaf of sections of which is the locally free sheaf of dim . Thus is the spectrum of the algebra which is the symmetric algebra of . A well known result in algebra says that this algebra is just .
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