Normal cone

If ${\displaystyle i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z}$ are regular embeddings, 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 ${\textstyle W:=\bigcap _{i}Y_{i}\hookrightarrow X}$ is a regular embedding of codimension

$${\displaystyle \sum c_{i}}$$

then[2]

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

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[3]

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

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:[4]

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

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

For a Cartesian square of schemes

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

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

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

of normal cones.

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.[5] (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.

Consider the non-regular embedding[7]^: 4–5

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

then, we can compute the normal cone by first observing

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

If we make the auxiliary variables ${\displaystyle a=xz}$ and ${\displaystyle b=yz}$ we get the relation

$${\displaystyle ya-xb=0.}$$

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

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

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

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

giving us the normal cone.

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

$${\displaystyle X=H\cup L}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 (algebraic geometry) § 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. Q.E.D.

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.

Let ${\displaystyle X}$ be a Deligne–Mumford stack locally of finite type over a field ${\displaystyle k}$. If ${\displaystyle {\textbf {L}}_{X}}$ denotes the cotangent complex of X relative to ${\displaystyle k}$, then the intrinsic normal bundle[8]^: 27 to ${\displaystyle X}$ is the quotient stack

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

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

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

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

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

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

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 ${\displaystyle [N_{U/M}/f^{*}T_{M}]}$ can be interpreted as

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

in certain cases.

The intrinsic normal cone to ${\displaystyle X}$, denoted as ${\displaystyle {\mathfrak {C}}_{X}}$,[8]^: 29 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 ${\displaystyle {\mathfrak {C}}_{X}={\mathfrak {N}}_{X}}$. In particular, if ${\displaystyle X}$ is smooth, then ${\displaystyle {\mathfrak {C}}_{X}={\mathfrak {N}}_{X}=BT_{X}}$ is the classifying stack of the tangent bundle ${\displaystyle T_{X}}$, which is a commutative group scheme over ${\displaystyle X}$.

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

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