Normal cone explained

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 or

C _{X/Y} of an embedding, defined by some sheaf of ideals I is defined as the relative Spec$$\backslash operatorname\_X\; \backslash left(\backslash bigoplus\_^\; I^n\; /\; I^\backslash right).$$

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 .

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$$\backslash pi:\; \backslash operatorname\_X\; Y\; =\; \backslash operatorname\_Y\; \backslash left(\backslash bigoplus\_^\; I^n\backslash right)\; \backslash to\; Y$$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 $\backslash bigoplus\_0^\; I^n\; \backslash otimes\_\; \backslash mathcal\_X\; =\; \backslash bigoplus\_0^\; I^n/I^$. Thus,$$E\; =\; \backslash mathbb(C\_X\; Y).$$

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, 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:$$0\backslash to\; N\_\; \backslash to\; N\_\; \backslash to\; i^*\; N\_\; \backslash to\; 0.$$

If

Y _{i} \hookrightarrow X are regular embeddings of codimensions c _{i} and if $W\; :=\; \backslash bigcap\_i\; Y\_i\; \backslash hookrightarrow\; X$ is a regular embedding of codimension $$\backslash sum\; c\_i$$ then$$N\_\; =\; \backslash bigoplus\_i\; N\_|\_W.$$In particular, if X \to S is a smooth morphism, then the normal bundle to the diagonal embedding $$\backslash Delta:\; X\; \backslash hookrightarrow\; X\; \backslash times\_S\; \backslash cdots\; \backslash times\_S\; X$$ (r-fold) is the direct sum of 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 x _{Y} Y' , then$$C\_\; =\; C\_\; \backslash times\_X\; X\text{'}.$$

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:$$0\; \backslash to\; T\_\; \backslash to\; T\_|\_X\; \backslash to\; N\_\; \backslash to\; 0,$$(which is a special case of an exact sequence for cotangent sheaves.)

Cartesian square

For a Cartesian square of schemes with

f:X' \to X the vertical map, there is a closed embedding $$C\_\; \backslash hookrightarrow\; f^*C\_$$ 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 ; 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

1. 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 $$N\_\; =\; \backslash mathcal\_D(D)\; :=\; \backslash mathcal\_X(D)|\_D.$$

Non-regular Embedding

Consider the non-regular embedding^[1] $$X\; =\; \backslash text\backslash left(\backslash frac\backslash right)\; \backslash to\; \backslash mathbb^3$$then, we can compute the normal cone by first observingIf 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$$C\_X\; \backslash mathbb^3\; =\; \backslash text\_X\; \backslash left(\backslash frac\; \backslash right)$$Since A ^{3} is affine, we can just write out the relative spectrum as the affine scheme $$C\_X\; \backslash mathbb^3\; =\; \backslash text\backslash left(\backslash frac\; \backslash right)$$ 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 , $$X\; =\; H\; \backslash cup\; L$$ 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 for z _{1,z} _{2} \in C and either z _{1} ≠ 0 or z _{2 ≠} 0 . Since it's arbitrary which point we take, for convenience let's assume z _{1} ≠ 0 , z _{2} = 0 . Hence the fiber of C ^{3} _{X A} at the point p = (z _{1,0,0)} is isomorphic to $$C\_X\; \backslash mathbb^3\; |\_p\; \backslash cong\; \backslash frac\; \backslash cong\; \backslash mathbb[a]$$ giving the normal cone as a one dimensional line, as expected. For a point q on the axis, this is given by a map hence the fiber at the point q = ( 0 , 0 ,z _{3)} is $$C\_X\; \backslash mathbb^3\; |\_q\; \backslash cong\; \backslash frac\; \backslash cong\; \backslash mathbb[a,b]$$ which gives a plane. At the origin r = ( 0 , 0 , 0 ) , the normal cone over that point is again isomorphic to C [a,b] .

Nodal cubic

For the nodal cubic curve

Y given by the polynomial y ^{2} + x ^{2(x-1)} over C , and X the point at the node, the cone has the isomorphism $$C\_\; \backslash cong\; \backslash text\backslash left(\backslash mathbb[x,y]/\backslash left(y^2-x^2\backslash right)\backslash right)$$ 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 $$\backslash pi\; :\; M^o\_\; \backslash to\; \backslash mathbb^1$$ with generic fiber Y and special fiber C _{X/Y} such that there exists a family of closed embeddings $$X\; \backslash times\; \backslash mathbb^1\; \backslash hookrightarrow\; M^o\_$$ over P ^{1} such that

1. Over any point

t \in P ^{1-\{0\}} the associated embeddings are an embedding X x \{t\} \hookrightarrow Y

1. The fiber over

0 \in 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$$\backslash pi:\; M\; \backslash to\; Y\; \backslash times\; \backslash mathbb^1$$be the blow-up of

Y x P ^{1} along X x 0 . The exceptional divisor is \overline{C _{X} Y} = P (C _{X} Y ⊕ 1 ) , the projective completion of the normal cone; for the notation used here see . 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 P ^{1} , the \pi followed by projection, is flat.

1. There is an induced closed embedding $$\backslash widetilde:\; X\; \backslash times\; \backslash mathbb^1\; \backslash hookrightarrow\; M$$ that is a morphism over

P ^{1} .

1. M is trivial away from zero; i.e.,

\rho ^{-1} ( P ^{1} - 0 ) = Y x ( P ^{1} - 0 ) and \widetilde{i} restricts to the trivial embedding $$X\; \backslash times\; (\backslash mathbb^1\; -\; 0)\; \backslash hookrightarrow\; Y\; \backslash times\; (\backslash mathbb^1\; -\; 0).$$\rho ^{-1} ( 0 ) as the divisor is the sum $$\backslash overline\; +\; \backslash widetilde$$ where \widetilde{Y} is the blow-up of Y along X and is viewed as an effective Cartier divisor.

1. As divisors

\overline{C _{X} Y} and \widetilde{Y} intersect at P (C) , where 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 x 0 is already an effective Cartier divisor on X x P ^{1} , we get$$X\; \backslash times\; \backslash mathbb^1\; =\; \backslash operatorname\_\; X\; \backslash times\; \backslash mathbb^1\; \backslash hookrightarrow\; M,$$yielding \widetilde{i} . Item 3 follows from the fact the blowdown map π is an isomorphism away from the center X x 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 x 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 bf{L} _{X} denotes the cotangent complex of X relative to k , then the intrinsic normal bundle^[2] to X is the quotient stack $$\backslash mathfrak\_X\; :=\; h^1\; /\; h^0(\backslash textbf\_^)$$ which is the stack of fppf bf{L} ^{\vee, 0} _{X} -torsors on bf{L} ^{\vee, 1} _{X} . 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 $$\backslash mathfrak\_X\; |\_U\; =\; [N\_\{U/M\}/f^*\; T\_M]$$ meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence $$\backslash mathcal\_U\; \backslash to\; \backslash mathcal\_M\; |\_U\; \backslash to\; \backslash mathcal\_$$ 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 $$B\; \backslash mathcal\_U\; =\; \backslash mathcal\_U[+1]$$ in certain cases.

Normal cone

The intrinsic normal cone to

X , denoted as ak{C} _{X} , is then defined by replacing the normal bundle N _{U/M} with the normal cone C _{U/M} ; i.e.,$$\backslash mathfrak\_X|\_U\; =\; [C\_\{U/M\}\; /\; f^*\; T\_M].$$

Example: One has that

X is a local complete intersection if and only if ak{C} _{X} = ak{N} _{X} . In particular, if X is smooth, then ak{C} _{X} = ak{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 ak{C} _{X/Y} \subseteq ak{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., A ^{n} _{Y} \to Y ), the pullback is:$$\backslash mathfrak\_|\_U\; =\; [C\_\{U/M\}\; /\; T\_\{M/Y\}|\_U].$$

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. en. 128. 1. 45–88. 10.1007/s002220050136. alg-geom/9601010 . 18533009 . 0020-9910.

External links

• Fibers of the normal cone

Notes and References

1. Battistella. Luca. Carocci. Francesca. Manolache. Cristina. 2020-04-09. Virtual classes for the working mathematician. Symmetry, Integrability and Geometry: Methods and Applications. 10.3842/SIGMA.2020.026. 1804.06048 . free.
2. Behrend. K.. Fantechi. B.. 1997-03-19. The intrinsic normal cone. Inventiones Mathematicae. 128. 1. 45–88. 10.1007/s002220050136. alg-geom/9601010 . 18533009 . 0020-9910.