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).

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.