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

${\displaystyle \operatorname {Spec} _{X}(\oplus _{n=0}^{\infty }I^{n}/I^{n+1}).}$

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.

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

${\displaystyle \pi :\operatorname {Bl} _{X}Y=\operatorname {Proj} _{Y}(\oplus _{n=0}^{\infty }I^{n})\to Y}$

be the blow-up of Y along X. Then, by definition, the exceptional divisor is the pre-image ${\displaystyle E=\pi ^{-1}(X)}$; which is the projective cone of ${\displaystyle \oplus _{0}^{\infty }I^{n}\otimes _{{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}=\oplus _{0}^{\infty }I^{n}/I^{n+1}}$. Thus,

${\displaystyle E=\mathbb {P} (C_{X}Y)}$.

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).^[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 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

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

be the blow-up of ${\displaystyle Y\times \mathbb {P} ^{1}}$ along ${\displaystyle X\times 0}$. The exceptional divisor is ${\displaystyle {\overline {C_{X}Y}}=\mathbb {P} (C_{X}Y\oplus 1)}$, the projective completion of the normal cone; for the notation used here see cone#Properties. The normal cone ${\displaystyle C_{X}Y}$ is an open subscheme of ${\displaystyle {\overline {C_{X}Y}}}$ and $X$ is embedded as a zero-section into ${\displaystyle C_{X}Y}$.

Now, we note:

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

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

   that is a morphism over $\mathbb {P} ^{1}$.
3. M is trivial away from zero; i.e., ${\displaystyle \rho ^{-1}(\mathbb {P} ^{1}-0)=Y\times (\mathbb {P} ^{1}-0)}$ and ${\displaystyle {\widetilde {i}}}$ restricts to the trivial embedding

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

4. ${\displaystyle \rho ^{-1}(0)}$ as the divisor is the sum

   ${\displaystyle {\overline {C_{X}Y}}+{\widetilde {Y}}}$

   where ${\displaystyle {\widetilde {Y}}}$ is the blow-up of Y along X and is viewed as an effective Cartier divisor.
5. As divisors ${\displaystyle {\overline {C_{X}Y}}}$ and ${\displaystyle {\widetilde {Y}}}$ intersect at ${\displaystyle \mathbb {P} (C)}$, where ${\displaystyle \mathbb {P} (C)}$ sits at infinity in ${\displaystyle {\overline {C_{X}Y}}}$.

Item 1. is clear (check torsion-free-ness). In general, given $X\subset Y$, we have ${\displaystyle \operatorname {Bl} _{V}X\subset \operatorname {Bl} _{V}Y}$. Since ${\displaystyle X\times 0}$ is already an effective Cartier divisor on ${\displaystyle X\times \mathbb {P} ^{1}}$, we get

${\displaystyle X\times \mathbb {P} ^{1}=\operatorname {Bl} _{X\times 0}X\times \mathbb {P} ^{1}\hookrightarrow M}$,

yielding ${\displaystyle {\widetilde {i}}}$. Item 3. follows from the fact the blowdown map π is an isomorphism away from the center ${\displaystyle X\times 0}$. The last two items are seen from explicit local computation. $\square$

Now, the last item in the previous paragraph implies that the image of ${\displaystyle X\times 0}$ in M does not intersect ${\displaystyle {\widetilde {Y}}}$. 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.

• 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
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157