# Normal 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

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

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 product 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, NX Y).[1]

## Properties

If ${\displaystyle i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z}$ are regular embedding, then ${\displaystyle j\circ i}$ is a regular embedding and there is a natural exact sequence of vector bundles on X:[2]

${\displaystyle 0\to N_{X/Y}\to N_{X/Z}\to i^{*}N_{Y/Z}\to 0}$.

If ${\displaystyle Y_{i}\hookrightarrow X}$ are regular embeddings of codimensions ${\displaystyle c_{i}}$ and if ${\displaystyle W:=\bigcap _{i}Y_{i}\hookrightarrow X}$ is a regular embedding of codimension ${\displaystyle \sum c_{i}}$, then[3]

${\displaystyle N_{W/X}=\bigoplus _{i}N_{Y_{i}/X}|_{W}}$.

In particular, if ${\displaystyle X\to S}$ is a smooth morphism, then the normal bundle to the diagonal embedding${\displaystyle \Delta :X\hookrightarrow X\times _{S}\cdots \times _{S}X}$ (r-fold) is the direct sum of r - 1 copies of the relative tangent bundle ${\displaystyle T_{X/S}}$.

If ${\displaystyle X\hookrightarrow X'}$ is a closed immersion and if ${\displaystyle Y'\to Y}$ is a flat morphism such that ${\displaystyle X'=X\times _{Y}Y'}$, then[4]

${\displaystyle C_{X'/Y'}=C_{X/Y}\times _{X}X'.}$

If ${\displaystyle X\to S}$ is a smooth morphism and ${\displaystyle X\hookrightarrow Y}$ is a regular embedding, then there is a natural exact sequence of vector bundles on X:[5]

${\displaystyle 0\to T_{X/S}\to T_{Y/S}|_{X}\to N_{X/Y}\to 0}$,

(which is a special case of an exact sequence for cotangent sheaves.)

Let ${\displaystyle X}$ be a scheme of finite type over a field and ${\displaystyle W\subset X}$ a closed subscheme. If ${\displaystyle X}$ is of pure dimension r; i.e., every irreducible component has dimension r, then ${\displaystyle C_{W/X}}$ is also of pure dimension r.[6] (This can be seen as a consequence of #Deformation to the normal cone.) This property is a key to an application in intersection theory: given a pair of closed subschemes ${\displaystyle V,X}$ in some ambient space, while the scheme-theoretic intersection ${\displaystyle V\cap X}$ has irreducible components of various dimensions, depending delicately on the positions of ${\displaystyle V,X}$, the normal cone to ${\displaystyle V\cap X}$ is of pure dimension.

## Examples

• Let ${\displaystyle D\hookrightarrow X}$ be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is[7]
${\displaystyle N_{D/X}={\mathcal {O}}_{D}(D):={\mathcal {O}}_{X}(D)|_{D}}$.

### Non-regular Embedding

Consider the non-regular embedding

${\displaystyle X={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xz,yz)}}\right)\to \mathbb {A} ^{3}}$

then, we can compute the normal cone by first observing

{\displaystyle {\begin{aligned}I&=(xz,yz)\\I^{2}&=(x^{2}z^{2},zyz^{2},y^{2}z^{2})\\\end{aligned}}}

If we make the auxillary variables ${\displaystyle a=xz}$ and ${\displaystyle b=yz}$ then observe that

${\displaystyle I^{2}=(a^{2},ab,b^{2})}$

giving the relation

${\displaystyle a^{2}b^{2}-ab=0}$

We can use this to give a presentation of the normal cone:

${\displaystyle C_{X}\mathbb {A} ^{3}={\text{Spec}}_{X}\left({\frac {{\mathcal {O}}_{X}[a,b]}{(a^{2}b^{2}-ab)}}\right)}$

## 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 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.

Concretely, 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 ${\displaystyle X}$ 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 ${\displaystyle \pi }$ 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 ${\displaystyle \mathbb {P} ^{1}}$.
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 ${\displaystyle X\subset Y}$, 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. ${\displaystyle \square }$

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.

## Intrinsic normal cone

Let X be a Deligne–Mumford stack locally of finite type over a field k. If ${\displaystyle {\textbf {L}}_{X}}$ denotes the cotangent complex of X relative to k, then the intrinsic normal bundle to X is the quotient stack

${\displaystyle {\mathfrak {R}}_{X}:=h^{1}/h^{0}({\textbf {L}}_{X,{\text{fppf}}}^{\vee })}$

which is the stack of fppf ${\displaystyle {\textbf {L}}_{X}^{\vee ,0}}$-torsors on ${\displaystyle {\textbf {L}}_{X}^{\vee ,1}}$. More concretely, suppose there is an étale morphism ${\displaystyle U\to X}$ from an affine finite-type k-scheme U together with a locally closed immersion ${\displaystyle f:U\to M}$ into a smooth affine finite-type k-scheme M. Then one can show

${\displaystyle {\mathfrak {R}}_{X}|_{U}=[N_{U/M}/f^{*}T_{M}].}$

The intrinsic normal cone to X, denoted as ${\displaystyle {\mathfrak {C}}_{X}}$, is then defined by replacing the normal bundle ${\displaystyle N_{U/M}}$ with the normal cone ${\displaystyle C_{U/M}}$; i.e.,

${\displaystyle {\mathfrak {C}}_{X}|_{U}=[C_{U/M}/f^{*}T_{M}].}$

Example: One has that ${\displaystyle X}$ is a local complete intersection if and only if ${\displaystyle {\mathfrak {C}}_{X}={\mathfrak {R}}_{X}}$. In particular, if X is smooth, then ${\displaystyle {\mathfrak {C}}_{X}={\mathfrak {R}}_{X}=BT_{X}}$ is the classifying stack of the tangent bundle ${\displaystyle T_{X}}$, which is a commutative group scheme over X.

## Notes

1. Hartshorne, Ch. III, Exercise 9.7.
2. Fulton, Appendix B.7.4.
3. Fulton, Appendix B.7.4.
4. Fulton, The first part of the proof of Theorem 6.5.
5. Fulton, Appendix B 7.1.
6. Fulton, Appendix B. 6.6.
7. Fulton, Appendix B.6.2.

## References

• Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910.
• 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