Suppose i : X \to Y is an embedding. This can be deformed to the embedding of X inside the normal cone C_{X/Y} (as the zero section) in the following sense: there is a flat family \pi : M^o_{X/Y} \to \mathbb{P}^1 with generic fiber Y and special fiber C_{X/Y} such that there exists a family of closed embeddings X \times \mathbb{P}^1 \hookrightarrow M^o_{X/Y} over \mathbb{P}^1 such that • Over any point t \in \mathbb{P}^1-\{0\} the associated embeddings are an embedding X\times\{t\} \hookrightarrow Y • The fiber over 0 \in \mathbb{P}^1 is the embedding of X \hookrightarrow C_{X/Y} given by the zero section. This construction defines a tool analogous to
differential topology where non-transverse intersections are performed in a tubular neighborhood of the intersection. Now, the intersection of X with a cycle Z in Y can be given as the pushforward of an intersection of X with the pullback of Z in C_{X/Y}.
Construction One application of this is to define intersection products in the
Chow ring. Suppose that
X and
V are closed subschemes of
Y with intersection
W, and we wish to define the intersection product of
X and
V in the Chow ring of
Y. Deformation to the normal cone in this case means that we replace the embeddings of
X and
W in
Y and
V by their normal cones
CY(
X) and
CW(
V), so that we want to find the product of
X and ''C'
W'V
in C'
X'Y''. This can be much easier: for example, if
X is
regularly embedded in
Y then its normal cone is a vector bundle, so we are reduced to the problem of finding the intersection product of a subscheme ''C'
W'V
of a vector bundle C'
X'Y
with the zero section X
. However this intersection product is just given by applying the Gysin isomorphism to C'
W'V''. Concretely, the deformation to the normal cone can be constructed by means of blowup. Precisely, let \pi: M \to Y \times \mathbb{P}^1 be the blow-up of Y \times \mathbb{P}^1 along X \times 0. The exceptional divisor is \overline{C_X Y} = \mathbb{P}(C_X Y \oplus 1), the projective completion of the normal cone; for the notation used here see . The normal cone C_X Y is an open subscheme of \overline{C_X Y} and X is embedded as a zero-section into C_X Y. Now, we note: • The map \rho: M \to \mathbb{P}^1, the \pi followed by projection, is flat. • There is an induced closed embedding \widetilde{i}: X \times \mathbb{P}^1 \hookrightarrow M that is a morphism over \mathbb{P}^1. •
M is trivial away from zero; i.e., \rho^{-1}(\mathbb{P}^1 - 0)= Y \times (\mathbb{P}^1 - 0) and \widetilde{i} restricts to the trivial embedding X \times (\mathbb{P}^1 - 0) \hookrightarrow Y \times (\mathbb{P}^1 - 0). • \rho^{-1}(0) as the divisor is the sum \overline{C_X Y} + \widetilde{Y} where \widetilde{Y} is the blow-up of
Y along
X and is viewed as an effective Cartier divisor. • As divisors \overline{C_X Y} and \widetilde{Y} intersect at \mathbb{P}(C), where \mathbb{P}(C) sits at infinity in \overline{C_X Y}. Item 1 is clear (check torsion-free-ness). In general, given X \subset Y, we have \operatorname{Bl}_V X \subset \operatorname{Bl}_V Y. Since X \times 0 is already an effective Cartier divisor on X \times \mathbb{P}^1, we get X \times \mathbb{P}^1 = \operatorname{Bl}_{X \times 0} X \times \mathbb{P}^1 \hookrightarrow M, yielding \widetilde{i}. Item 3 follows from the fact the blowdown map π is an isomorphism away from the center X \times 0. The last two items are seen from explicit local computation.
Q.E.D. Now, the last item in the previous paragraph implies that the image of X \times 0 in
M does not intersect \widetilde{Y}. Thus, one gets the deformation of
i to the zero-section embedding of
X into the normal cone. == Intrinsic normal cone ==