Normal cone

Definition

The normal cone C_XY 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 E = π − 1 ( X ) ; which is the projective cone of ⊕ 0 ∞ I n ⊗ O Y O X = ⊕ 0 ∞ I n / I n + 1 . 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, N_X 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 C_XY 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 C_Y(X) and C_W(V), so that we want to find the product of X and C_WV in C_XY. 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_WV of a vector bundle C_XY with the zero section X. However this intersection product is just given by applying the Gysin isomorphism to C_WV.

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 Y × P 1 along X × 0 . The exceptional divisor is C X Y ¯ = P ( C X Y ⊕ 1 ) , the projective completion of the normal cone; for the notation used here see cone#Properties. The normal cone C X Y is an open subscheme of C X Y ¯ and X is embedded as a zero-section into C X Y .

Now, we note:

Item 1. is clear (check torsion-free-ness). In general, given X ⊂ Y , we have Bl V ⁡ X ⊂ Bl V ⁡ Y . Since X × 0 is already an effective Cartier divisor on X × P 1 , we get

yielding i ~ . Item 3. follows from the fact the blowdown map π is an isomorphism away from the center X × 0 . The last two items are seen from explicit local computation. ◻

Now, the last item in the previous paragraph implies that the image of X × 0 in M does not intersect Y ~ . Thus, one gets the deformation of i to the zero-section embedding of X into the normal cone.