The normal cone (or rather its projective cousin) appears as a result of blow-up. Precisely, let
be the blow-up of Y along X. Then, by definition, the exceptional divisor is the pre-image ; which is the projective cone of . Thus,
The global sections of the normal bundle classify embedded infinitesimal deformations of Y in X; there is a natural bijection between the set of closed subschemes of Y ×kD, flat over the ring D of dual numbers and having X as the special fiber, and H0(X, NXY).
Let be a scheme of finite type over a field and a closed subscheme. If is of pure dimensionr; i.e., every irreducible component has dimension r, then is also of pure dimension r. (This can be seen as a consequence of #Deformation to the normal cone.) This property is a key to an application in intersection theory: given a pair of closed subschemes in some ambient space, while the scheme-theoretic intersection has irreducible components of various dimensions, depending delicately on the positions of , the normal cone to is of pure dimension.
Suppose is an embedding. This can be deformed to the embedding of inside the normal cone (as the zero section) in the following sense:: 6 there is a flat family
with generic fiber and special fiber such that there exists a family of closed embeddings
over such that
Over any point the associated embeddings are an embedding
The fiber over is the embedding of 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 with a cycle in can be given as the pushforward of an intersection of with the pullback of in .
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 CWV in CXY.
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 CWV of a vector bundle CXY with the zero section X. However this intersection product is just given by applying the Gysin isomorphism to CWV.
Concretely, the deformation to the normal cone can be constructed by means of blowup. Precisely, let
be the blow-up of along . The exceptional divisor is , the projective completion of the normal cone; for the notation used here see Cone (algebraic geometry) § Properties. The normal cone is an open subscheme of and is embedded as a zero-section into .
Now, we note:
The map , the followed by projection, is flat.
There is an induced closed embedding
that is a morphism over .
M is trivial away from zero; i.e., and restricts to the trivial embedding
as the divisor is the sum
where is the blow-up of Y along X and is viewed as an effective Cartier divisor.
As divisors and intersect at , where sits at infinity in .
Item 1 is clear (check torsion-free-ness). In general, given , we have . Since is already an effective Cartier divisor on , we get
yielding . Item 3 follows from the fact the blowdown map π is an isomorphism away from the center . 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 in M does not intersect . Thus, one gets the deformation of i to the zero-section embedding of X into the normal cone.
More concretely, suppose there is an étale morphism from an affine finite-type -scheme together with a locally closed immersion into a smooth affine finite-type -scheme . Then one can show
meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence
to be exact on the right hand side. Moreover, for special cases discussed below, we are now considering the quotient as a continuation of the previous sequence as a triangle in some triangulated category. This is because the local stack quotient can be interpreted as
The intrinsic normal cone to , denoted as ,: 29 is then defined by replacing the normal bundle with the normal cone ; i.e.,
Example: One has that is a local complete intersection if and only if . In particular, if is smooth, then is the classifying stack of the tangent bundle , which is a commutative group scheme over .
More generally, let is a Deligne-Mumford Type (DM-type) morphism of Artin Stacks which is locally of finite type. Then is characterised as the closed substack such that, for any étale map for which factors through some smooth map (e.g., ), the pullback is: