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 $${\mathrm{Spec}}_X(\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{I}^n/I^{n+1}).$$ 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 $$\pi :{\mathrm{Bl}}_{X}Y={\mathrm{Proj}}_Y\left(\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{I}^n\right)\to Y$$ 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 ${⨁}_0^{\mathrm{\infty }}{I}^n{\otimes }_{{{𝒪}}_Y}{{𝒪}}_X={⨁}_0^{\mathrm{\infty }}{I}^n/I^{n+1}$. Thus, $$E=\mathbb{\mathbb{P}}(C_{X}Y).$$ 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 $W:=\bigcap _i{Y}_i\hookrightarrow X$ is a regular embedding of codimension $$\sum c_i$$ then $$N_{W/X}=\underset{i}{⨁}{N}_{Y_i/X}{|}_W.$$ In particular, if $X\to S$ is a smooth morphism, then the normal bundle to the diagonal embedding $$\mathrm{\Delta }:X\hookrightarrow X{\times }_S\cdots {\times }_{S}X$$ (r-fold) is the direct sum of r − 1 copies of the relative tangent bundle $T_{X/S}$. If $X\hookrightarrow X^{\prime }$ is a closed immersion and if $Y^{\prime }\to Y$ is a flat morphism such that $X^{\prime }=X{\times }_Y{Y}^{\prime }$, then $$C_{X^{\prime }/Y^{\prime }}=C_{X/Y}{\times }_X{X}^{\prime }.$$ 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 $$\begin{array}{ccc}{X}^{\prime }& \to & Y^{\prime }\\ \downarrow & & \downarrow \\ X& \to & Y\end{array}$$ with $f:X^{\prime }\to X$ the vertical map, there is a closed embedding $$C_{X^{\prime }/Y^{\prime }}\hookrightarrow f^{*}{C}_{X/Y}$$ 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 $$N_{D/X}={{𝒪}}_D(D):={{𝒪}}_X(D){|}_D.$$

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 $$\begin{array}{rl}I& =(xz,yz)\\ I^2& =(x^2{z}^2,xy{z}^2,y^2{z}^2)\\ \end{array}$$ 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 $$C_X{\mathbb{𝔸}}^3={\text{Spec}}_X\left(\frac{{{𝒪}}_X[a,b]}{(ya-xb)}\right)$$ 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 $$\begin{array}{ccc}x\mapsto z_1& y\mapsto z_2& z\mapsto 0\end{array}$$ for $z_1,z_2\in \mathbb{ℂ}$ and either $z_1\ne 0$ or $z_2\ne 0$. Since it is arbitrary which point we take, for convenience let us assume $z_1\ne 0,z_2=0$. 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 $$\begin{array}{ccc}x\mapsto 0& y\mapsto 0& z\mapsto z_3\end{array}$$ 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 $$C_{X/Y}\cong \text{Spec}(\mathbb{ℂ}[x,y]/(y^2-x^2))$$ 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 $$\pi :M_{X/Y}^o\to {\mathbb{\mathbb{P}}}^1$$ with generic fiber $Y$ and special fiber $C_{X/Y}$ such that there exists a family of closed embeddings $$X\times {\mathbb{\mathbb{P}}}^1\hookrightarrow M_{X/Y}^o$$ over ${\mathbb{\mathbb{P}}}^1$ such that

1. Over any point $t\in {\mathbb{\mathbb{P}}}^1-\{0\}$ the associated embeddings are an embedding $X\times \{t\}\hookrightarrow Y$
2. The fiber over $0\in {\mathbb{\mathbb{P}}}^1$ 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 $$\pi :M\to Y\times {\mathbb{\mathbb{P}}}^1$$ be the blow-up of $Y\times {\mathbb{\mathbb{P}}}^1$ along $X\times 0$. The exceptional divisor is $\overline{C_{X}Y}=\mathbb{\mathbb{P}}(C_{X}Y\oplus 1)$, 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 $\rho :M\to {\mathbb{\mathbb{P}}}^1$, the $\pi$ followed by projection, is flat.
2. There is an induced closed embedding $$\tilde{i}:X\times {\mathbb{\mathbb{P}}}^1\hookrightarrow M$$ that is a morphism over ${\mathbb{\mathbb{P}}}^1$.
3. M is trivial away from zero; i.e., ${\rho }^{-1}({\mathbb{\mathbb{P}}}^1-0)=Y\times ({\mathbb{\mathbb{P}}}^1-0)$ and $\tilde{i}$ restricts to the trivial embedding $$X\times ({\mathbb{\mathbb{P}}}^1-0)\hookrightarrow Y\times ({\mathbb{\mathbb{P}}}^1-0).$$
4. ${\rho }^{-1}(0)$ as the divisor is the sum $$\overline{C_{X}Y}+\tilde{Y}$$ where $\tilde{Y}$ is the blow-up of Y along X and is viewed as an effective Cartier divisor.
5. As divisors $\overline{C_{X}Y}$ and $\tilde{Y}$ intersect at $\mathbb{\mathbb{P}}(C)$, where $\mathbb{\mathbb{P}}(C)$ sits at infinity in $\overline{C_{X}Y}$.

Item 1 is clear (check torsion-free-ness). In general, given $X\subset Y$, we have ${\mathrm{Bl}}_{V}X\subset {\mathrm{Bl}}_{V}Y$. Since $X\times 0$ is already an effective Cartier divisor on $X\times {\mathbb{\mathbb{P}}}^1$, we get $$X\times {\mathbb{\mathbb{P}}}^1={\mathrm{Bl}}_{X\times 0}X\times {\mathbb{\mathbb{P}}}^1\hookrightarrow M,$$ yielding $\tilde{i}$. 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 $\tilde{Y}$. 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 ${\text{L}}_X$ denotes the cotangent complex of X relative to $k$, then the intrinsic normal bundle to $X$ is the quotient stack $${\mathfrak{𝔑}}_X:=h^1/h^0({\text{L}}_{X,\text{fppf}}^{\vee })$$ which is the stack of fppf ${\text{L}}_X^{\vee ,0}$-torsors on ${\text{L}}_X^{\vee ,1}$. 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 $${{𝒯}}_U\to {{𝒯}}_M{|}_U\to {{𝒩}}_{U/M}$$ 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 $$B{{𝒯}}_U={{𝒯}}_U[+1]$$ 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 ${\mathfrak{ℭ}}_X={\mathfrak{𝔑}}_X=B{T}_X$ 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