Normal 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.

The normal cone C_XY or ${\displaystyle C_{X/Y}}$ of an embedding i: X → Y, defined by some sheaf of ideals I is defined as the relative Spec $${\displaystyle \operatorname {Spec} _{X}\left(\bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}\right).}$$

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 $${\displaystyle \pi :\operatorname {Bl} _{X}Y=\operatorname {Proj} _{Y}\left(\bigoplus _{n=0}^{\infty }I^{n}\right)\to Y}$$ be the blow-up of Y along X. Then, by definition, the exceptional divisor is the pre-image ${\displaystyle E=\pi ^{-1}(X)}$; 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, $${\displaystyle E=\mathbb {P} (C_{X}Y).}$$

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]

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][citation needed]} $${\displaystyle C_{X'/Y'}=C_{X/Y}\times _{X}X'.}$$

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.

Let ${\displaystyle D\hookrightarrow X}$ be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is^[6] $${\displaystyle N_{D/X}={\mathcal {O}}_{D}(D):={\mathcal {O}}_{X}(D)|_{D}.}$$

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 is arbitrary which point we take, for convenience let us 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.

Suppose ${\displaystyle i:X\to Y}$ is an embedding. This can be deformed to the embedding of ${\displaystyle X}$ inside the normal cone ${\displaystyle C_{X/Y}}$ (as the zero section) in the following sense:^[7]: 6 there is a flat family $${\displaystyle \pi :M_{X/Y}^{o}\to \mathbb {P} ^{1}}$$ with generic fiber ${\displaystyle Y}$ and special fiber ${\displaystyle C_{X/Y}}$ such that there exists a family of closed embeddings $${\displaystyle X\times \mathbb {P} ^{1}\hookrightarrow M_{X/Y}^{o}}$$ over ${\displaystyle \mathbb {P} ^{1}}$ such that

1. Over any point ${\displaystyle t\in \mathbb {P} ^{1}-\{0\}}$ the associated embeddings are an embedding ${\displaystyle X\times \{t\}\hookrightarrow Y}$
2. The fiber over ${\displaystyle 0\in \mathbb {P} ^{1}}$ is the embedding of ${\displaystyle X\hookrightarrow C_{X/Y}}$ 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 ${\displaystyle X}$ with a cycle ${\displaystyle Z}$ in ${\displaystyle Y}$ can be given as the pushforward of an intersection of ${\displaystyle X}$ with the pullback of ${\displaystyle Z}$ in ${\displaystyle C_{X/Y}}$.

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

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

• 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

• Fibers of the normal cone