Normal cone

For normal cones in real vector spaces, see convex cone.

In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.

Definition

The normal cone C_XY or $C_{X/Y}$ of an embedding i: X → Y, defined by some sheaf of ideals I is defined as the relative Spec

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

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

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

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 ${\textstyle \bigoplus _{0}^{\infty }I^{n}\otimes _{{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}=\bigoplus _{0}^{\infty }I^{n}/I^{n+1}}$. Thus,

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

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).

Properties

Compositions of regular embeddings

If $i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z$ are regular embeddings, then $j\circ i$ is a regular embedding and there is a natural exact sequence of vector bundles on X:

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

If $Y_{i}\hookrightarrow X$ are regular embeddings of codimensions $c_{i}$ and if ${\textstyle W:=\bigcap _{i}Y_{i}\hookrightarrow X}$ is a regular embedding of codimension

$$\sum c_{i}$$

then

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

In particular, if $X\to S$ is a smooth morphism, then the normal bundle to the diagonal embedding

$$\Delta :X\hookrightarrow X\times _{S}\cdots \times _{S}X$$

(r-fold) is the direct sum of r − 1 copies of the relative tangent bundle $T_{X/S}$.

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

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

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

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

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

Cartesian square

For a Cartesian square of schemes

$${\begin{matrix}X'&\to &Y'\\\downarrow &&\downarrow \\X&\to &Y\end{matrix}}$$

with $f:X'\to X$ the vertical map, there is a closed embedding

$$C_{X'/Y'}\hookrightarrow f^{*}C_{X/Y}$$

of normal cones.

Dimension of components

Let $X$ be a scheme of finite type over a field and $W\subset X$ a closed subscheme. If $X$ is of pure dimension r; i.e., every irreducible component has dimension r, then $C_{W/X}$ is also of pure dimension r. (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 $V,X$ in some ambient space, while the scheme-theoretic intersection $V\cap X$ has irreducible components of various dimensions, depending delicately on the positions of $V,X$, the normal cone to $V\cap X$ is of pure dimension.

Examples

Let $D\hookrightarrow X$ be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is

$$N_{D/X}={\mathcal {O}}_{D}(D):={\mathcal {O}}_{X}(D)|_{D}.$$

Non-regular Embedding

Consider the non-regular embedding

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

then, we can compute the normal cone by first observing

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

If we make the auxiliary variables $a=xz$ and $b=yz$ we get the relation

$$ya-xb=0.$$

We can use this to give a presentation of the normal cone as the relative spectrum

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

Since $\mathbb {A} ^{3}$ is affine, we can just write out the relative spectrum as the affine scheme

$$C_{X}\mathbb {A} ^{3}={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z][a,b]}{(xz,yz,ya-xb)}}\right)$$

giving us the normal cone.

Geometry of this normal cone

The normal cone's geometry can be further explored by looking at the fibers for various closed points of $X$. Note that geometrically $X$ is the union of the $xy$-plane $H$ with the $z$-axis $L$,

$$X=H\cup L$$

so the points of interest are smooth points on the plane, smooth points on the axis, and the point on their intersection. Any smooth point on the plane is given by a map

$${\begin{matrix}x\mapsto z_{1}&y\mapsto z_{2}&z\mapsto 0\end{matrix}}$$

for $z_{1},z_{2}\in \mathbb {C}$ and either $z_{1}\neq 0$ or $z_{2}\neq 0$. Since it's arbitrary which point we take, for convenience let's assume $z_{1}\neq 0,z_{2}=0$. Hence the fiber of $C_{X}\mathbb {A} ^{3}$ at the point $p=(z_{1},0,0)$ is isomorphic to

$$C_{X}\mathbb {A} ^{3}|_{p}\cong {\frac {\mathbb {C} [a,b]}{(z_{1}b)}}\cong \mathbb {C} [a]$$

giving the normal cone as a one dimensional line, as expected. For a point $q$ on the axis, this is given by a map

$${\begin{matrix}x\mapsto 0&y\mapsto 0&z\mapsto z_{3}\end{matrix}}$$

hence the fiber at the point $q=(0,0,z_{3})$ is

$$C_{X}\mathbb {A} ^{3}|_{q}\cong {\frac {\mathbb {C} [a,b]}{(0)}}\cong \mathbb {C} [a,b]$$

which gives a plane. At the origin $r=(0,0,0)$, the normal cone over that point is again isomorphic to $\mathbb {C} [a,b]$.

Nodal cubic

For the nodal cubic curve $Y$ given by the polynomial $y^{2}+x^{2}(x-1)$ over $\mathbb {C}$, and $X$ the point at the node, the cone has the isomorphism

$$C_{X/Y}\cong {\text{Spec}}\left(\mathbb {C} [x,y]/\left(y^{2}-x^{2}\right)\right)$$

showing the normal cone has more components than the scheme it lies over.

Deformation to the normal cone

Suppose $i:X\to Y$ is an embedding. This can be deformed to the embedding of $X$ inside the normal cone $C_{X/Y}$ (as the zero section) in the following sense: there is a flat family

$$\pi :M_{X/Y}^{o}\to \mathbb {P} ^{1}$$

with generic fiber $Y$ and special fiber $C_{X/Y}$ such that there exists a family of closed embeddings

$$X\times \mathbb {P} ^{1}\hookrightarrow M_{X/Y}^{o}$$

over $\mathbb {P} ^{1}$ such that

1. Over any point $t\in \mathbb {P} ^{1}-\{0\}$ the associated embeddings are an embedding $X\times \{t\}\hookrightarrow Y$
2. The fiber over $0\in \mathbb {P} ^{1}$is the embedding of $X\hookrightarrow C_{X/Y}$ given by the zero section.

This construction defines a tool analogous to differential topology where non-transverse intersections are performed in a tubular neighborhood of the intersection. Now, the intersection of $X$ with a cycle $Z$ in $Y$ can be given as the pushforward of an intersection of $X$ with the pullback of $Z$ in $C_{X/Y}$.

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.

Concretely, 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 (algebraic geometry) § 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. Q.E.D.

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.

Intrinsic normal cone

Intrinsic normal bundle

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

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

which is the stack of fppf ${\textbf {L}}_{X}^{\vee ,0}$-torsors on ${\textbf {L}}_{X}^{\vee ,1}$. A concrete interpretation of this stack quotient can be given by looking at its behavior locally in the etale topos of the stack $X$.

Properties of intrinsic normal bundle

More concretely, suppose there is an étale morphism $U\to X$ from an affine finite-type $k$-scheme $U$ together with a locally closed immersion $f:U\to M$ into a smooth affine finite-type $k$-scheme $M$. Then one can show

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

meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence

$${\mathcal {T}}_{U}\to {\mathcal {T}}_{M}|_{U}\to {\mathcal {N}}_{U/M}$$

to be exact on the right hand side. Moreover, for special cases discussed below, we are now considering the quotient as a continuation of the previous sequence as a triangle in some triangulated category. This is because the local stack quotient $[N_{U/M}/f^{*}T_{M}]$ can be interpreted as

$$B{\mathcal {T}}_{U}={\mathcal {T}}_{U}[+1]$$

in certain cases.

Normal cone

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

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

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

More generally, let $X\to Y$ is a Deligne-Mumford Type (DM-type) morphism of Artin Stacks which is locally of finite type. Then ${\mathfrak {C}}_{X/Y}\subseteq {\mathfrak {N}}_{X/Y}$ is characterised as the closed substack such that, for any étale map $U\to X$ for which $U\to X\to Y$ factors through some smooth map $M\to Y$ (e.g., $\mathbb {A} _{Y}^{n}\to Y$), the pullback is:

$${\mathfrak {C}}_{X/Y}|_{U}=[C_{U/M}/T_{M/Y}|_{U}].$$

See also

• Abelian cone
• Segre class
• Residual intersection
• Virtual fundamental class

References

• Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. arXiv:alg-geom/9601010. doi:10.1007/s002220050136. ISSN 0020-9910. S2CID 18533009.
• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 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, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

External links

• Fibers of the normal cone