Normal cone

In algebraic geometry, the normal cone C_XY 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 C_XY of an embedding i: X → Y, defined by some sheaf of ideals I is defined as the relative Spec

$\operatorname{Spec}_X (\oplus_{n = 0}^{\infty} I^n / I^{n+1}).$

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.

The normal cone (or rather its projective cousin) appears as a result of blow-up. Precisely, let

$\pi: \operatorname{Bl}_X Y = \operatorname{Proj}_Y(\oplus_0^{\infty} I^n) \to Y$

be the blow-up of Y along X. Then, by definition, the exceptional divisor is the pre-image $E = \pi^{-1}(X)$; which is the projective cone of $\oplus_0^{\infty} I^n \otimes_{\mathcal{O}_Y} \mathcal{O}_X = \oplus_0^{\infty} I^n/I^{n+1}$. Thus,

$E = \mathbb{P}(C_X Y)$.

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 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

$\pi: M \to Y \times \mathbb{P}^1$

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 cone#Properties. 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:

1. The map $\rho: M \to \mathbb{P}^1$, the $\pi$ followed by projection, is flat.
2. There is an induced closed embedding

   $\widetilde{i}: X \times \mathbb{P}^1 \hookrightarrow M$

   that is a morphism over $\mathbb{P}^1$.
3. 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

   $X \times (\mathbb{P}^1 - 0) \hookrightarrow Y \times (\mathbb{P}^1 - 0)$.

4. $\rho^{-1}(0)$ 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.
5. 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. $\square$

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.

References

• William Fulton. (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323