Deformation to the Normal Cone
If X is a closed subscheme of Y, there is a family of imbeddings X ↪ Y t , parametrized by t ∈ ℙ1, such that for t = 0 (in fact for t ≠ ∞) the imbedding is the given imbedding of X in Y, and for t = ∞ one has the zero section imbedding of X in the normal cone C X Y. The existence of such a deformation, together with the “principle of continuity” that intersection products should vary nicely in families, explains the prominent role to be played by the normal cone in constructing intersection products.
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