# Normal cone

For normal cones in real vector spaces, see convex cone.

In algebraic geometry, the normal cone CXY of a subscheme X of a scheme Y is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.

## Definition

The normal cone CXY of an embedding i: X Y, defined by some sheaf of ideals I is defined as the relative Spec

When the embedding i is regular the normal cone is the normal bundle, the vector bundle on X corresponding to the dual of the sheaf I/I2.

If X is a point, then the normal cone and the normal bundle to it are also called the tangent cone and the tangent space (Zariski tangent space) to the point. When Y = Spec R is affine, the definition means that the normal cone to X = Spec R/I is the Spec of the associated graded ring of R with respect to I.

If Y is the diagonal X × X and the embedding i is the diagonal embedding, then the normal bundle to X in Y is the tangent bundle to X.

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 ×k D, flat over the ring D of dual numbers and having X as the special fiber, and H0(X, NX Y).[1]

## Deformation to the normal cone

Suppose i: X Y is an embedding. This can be deformed to the embedding of X in the normal cone CXY in the following sense: there is a family of embeddings parameterized by an element t of the projective or affine line, such that if t=0 the embedding is the embedding into the normal cone, and for other t is it isomorphic to the given embedding i. (See #Construction of the deformation to the normal cone below for 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 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.

## Construction of the deformation to the normal cone

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#Properties. The normal cone is an open subscheme of and is embedded as a zero-section into .

Now, we note:

1. The map , the followed by projection, is flat.
2. There is an induced closed embedding
that is a morphism over .
3. M is trivial away from zero; i.e., and restricts to the trivial embedding
.
4. as the divisor is the sum
where is the blow-up of Y along X and is viewed as an effective Cartier divisor.
5. 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.

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.

## References

1. Hartshorne, Ch. III, Exercise 9.7.
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