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.

Definition

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

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 [math]\displaystyle{ \pi: \operatorname{Bl}_X Y = \operatorname{Proj}_Y \left(\bigoplus_{n=0}^{\infty} I^n\right) \to Y }[/math] be the blow-up of Y along X. Then, by definition, the exceptional divisor is the pre-image [math]\displaystyle{ E = \pi^{-1}(X) }[/math]; which is the projective cone of [math]\displaystyle{ \bigoplus_0^{\infty} I^n \otimes_{\mathcal{O}_Y} \mathcal{O}_X = \bigoplus_0^{\infty} I^n/I^{n+1} }[/math]. Thus, [math]\displaystyle{ E = \mathbb{P}(C_X Y). }[/math]

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 [math]\displaystyle{ i: X \hookrightarrow Y, \, j: Y \hookrightarrow Z }[/math] are regular embeddings, then [math]\displaystyle{ j \circ i }[/math] is a regular embedding and there is a natural exact sequence of vector bundles on X: [math]\displaystyle{ 0\to N_{X/Y} \to N_{X/Z} \to i^* N_{Y/Z} \to 0. }[/math]

If [math]\displaystyle{ Y_i \hookrightarrow X }[/math] are regular embeddings of codimensions [math]\displaystyle{ c_i }[/math] and if [math]\displaystyle{ W := \bigcap_i Y_i \hookrightarrow X }[/math] is a regular embedding of codimension [math]\displaystyle{ \sum c_i }[/math] then [math]\displaystyle{ N_{W/X} = \bigoplus_i N_{Y_i/X}|_W. }[/math] In particular, if [math]\displaystyle{ X \to S }[/math] is a smooth morphism, then the normal bundle to the diagonal embedding [math]\displaystyle{ \Delta: X \hookrightarrow X \times_S \cdots \times_S X }[/math] (r-fold) is the direct sum of r − 1 copies of the relative tangent bundle [math]\displaystyle{ T_{X/S} }[/math].

If [math]\displaystyle{ X \hookrightarrow X' }[/math] is a closed immersion and if [math]\displaystyle{ Y' \to Y }[/math] is a flat morphism such that [math]\displaystyle{ X' = X \times_Y Y' }[/math], then [math]\displaystyle{ C_{X'/Y'} = C_{X/Y} \times_X X'. }[/math]

If [math]\displaystyle{ X \to S }[/math] is a smooth morphism and [math]\displaystyle{ X \hookrightarrow Y }[/math] is a regular embedding, then there is a natural exact sequence of vector bundles on X: [math]\displaystyle{ 0 \to T_{X/S} \to T_{Y/S}|_X \to N_{X/Y} \to 0, }[/math] (which is a special case of an exact sequence for cotangent sheaves.)

Cartesian square

For a Cartesian square of schemes [math]\displaystyle{ \begin{matrix} X' & \to & Y' \\ \downarrow & & \downarrow \\ X & \to & Y \end{matrix} }[/math] with [math]\displaystyle{ f:X' \to X }[/math] the vertical map, there is a closed embedding [math]\displaystyle{ C_{X'/Y'} \hookrightarrow f^*C_{X/Y} }[/math] of normal cones.

Dimension of components

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

Examples

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

Non-regular Embedding

Consider the non-regular embedding [math]\displaystyle{ X = \text{Spec}\left( \frac{\mathbb{C}[x,y,z]}{(xz,yz)}\right) \to \mathbb{A}^3 }[/math] then, we can compute the normal cone by first observing [math]\displaystyle{ \begin{align} I &= (xz, yz) \\ I^2 &= (x^2z^2, xyz^2, y^2z^2) \\ \end{align} }[/math] If we make the auxiliary variables [math]\displaystyle{ a = xz }[/math] and [math]\displaystyle{ b = yz }[/math] we get the relation [math]\displaystyle{ ya - xb = 0. }[/math] We can use this to give a presentation of the normal cone as the relative spectrum [math]\displaystyle{ C_X \mathbb{A}^3 = \text{Spec}_X \left( \frac{\mathcal{O}_X[a,b]}{(ya-xb)} \right) }[/math] Since [math]\displaystyle{ \mathbb{A}^3 }[/math] is affine, we can just write out the relative spectrum as the affine scheme [math]\displaystyle{ C_X \mathbb{A}^3 = \text{Spec}\left( \frac{\mathbb{C}[x,y,z][a,b]}{(xz,yz,ya-xb)} \right) }[/math] 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 [math]\displaystyle{ X }[/math]. Note that geometrically [math]\displaystyle{ X }[/math] is the union of the [math]\displaystyle{ xy }[/math]-plane [math]\displaystyle{ H }[/math] with the [math]\displaystyle{ z }[/math]-axis [math]\displaystyle{ L }[/math], [math]\displaystyle{ X = H \cup L }[/math] 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 [math]\displaystyle{ \begin{matrix} x \mapsto z_1 & y \mapsto z_2 & z \mapsto 0 \end{matrix} }[/math] for [math]\displaystyle{ z_1,z_2 \in \mathbb{C} }[/math] and either [math]\displaystyle{ z_1 \neq 0 }[/math] or [math]\displaystyle{ z_2\neq 0 }[/math]. Since it's arbitrary which point we take, for convenience let's assume [math]\displaystyle{ z_1 \neq 0, z_2 = 0 }[/math]. Hence the fiber of [math]\displaystyle{ C_X\mathbb{A}^3 }[/math] at the point [math]\displaystyle{ p=(z_1,0,0) }[/math] is isomorphic to [math]\displaystyle{ C_X \mathbb{A}^3 |_p \cong \frac{\mathbb{C}[a,b]}{(z_1b)} \cong \mathbb{C}[a] }[/math] giving the normal cone as a one dimensional line, as expected. For a point [math]\displaystyle{ q }[/math] on the axis, this is given by a map [math]\displaystyle{ \begin{matrix} x \mapsto 0 & y \mapsto 0 & z \mapsto z_3 \end{matrix} }[/math] hence the fiber at the point [math]\displaystyle{ q = (0,0,z_3) }[/math] is [math]\displaystyle{ C_X \mathbb{A}^3 |_q \cong \frac{\mathbb{C}[a,b]}{(0)} \cong \mathbb{C}[a,b] }[/math] which gives a plane. At the origin [math]\displaystyle{ r = (0,0,0) }[/math], the normal cone over that point is again isomorphic to [math]\displaystyle{ \mathbb{C}[a,b] }[/math].

Nodal cubic

For the nodal cubic curve [math]\displaystyle{ Y }[/math] given by the polynomial [math]\displaystyle{ y^2 + x^2(x-1) }[/math] over [math]\displaystyle{ \mathbb{C} }[/math], and [math]\displaystyle{ X }[/math] the point at the node, the cone has the isomorphism [math]\displaystyle{ C_{X/Y} \cong \text{Spec}\left(\mathbb{C}[x,y]/\left(y^2-x^2\right)\right) }[/math] showing the normal cone has more components than the scheme it lies over.

Deformation to the normal cone

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

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

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 [math]\displaystyle{ \pi: M \to Y \times \mathbb{P}^1 }[/math] be the blow-up of [math]\displaystyle{ Y \times \mathbb{P}^1 }[/math] along [math]\displaystyle{ X \times 0 }[/math]. The exceptional divisor is [math]\displaystyle{ \overline{C_X Y} = \mathbb{P}(C_X Y \oplus 1) }[/math], the projective completion of the normal cone; for the notation used here see Cone (algebraic geometry) § Properties. The normal cone [math]\displaystyle{ C_X Y }[/math] is an open subscheme of [math]\displaystyle{ \overline{C_X Y} }[/math] and [math]\displaystyle{ X }[/math] is embedded as a zero-section into [math]\displaystyle{ C_X Y }[/math].

Now, we note:

1. The map [math]\displaystyle{ \rho: M \to \mathbb{P}^1 }[/math], the [math]\displaystyle{ \pi }[/math] followed by projection, is flat.
2. There is an induced closed embedding [math]\displaystyle{ \widetilde{i}: X \times \mathbb{P}^1 \hookrightarrow M }[/math] that is a morphism over [math]\displaystyle{ \mathbb{P}^1 }[/math].
3. M is trivial away from zero; i.e., [math]\displaystyle{ \rho^{-1}(\mathbb{P}^1 - 0)= Y \times (\mathbb{P}^1 - 0) }[/math] and [math]\displaystyle{ \widetilde{i} }[/math] restricts to the trivial embedding [math]\displaystyle{ X \times (\mathbb{P}^1 - 0) \hookrightarrow Y \times (\mathbb{P}^1 - 0). }[/math]
4. [math]\displaystyle{ \rho^{-1}(0) }[/math] as the divisor is the sum [math]\displaystyle{ \overline{C_X Y} + \widetilde{Y} }[/math] where [math]\displaystyle{ \widetilde{Y} }[/math] is the blow-up of Y along X and is viewed as an effective Cartier divisor.
5. As divisors [math]\displaystyle{ \overline{C_X Y} }[/math] and [math]\displaystyle{ \widetilde{Y} }[/math] intersect at [math]\displaystyle{ \mathbb{P}(C) }[/math], where [math]\displaystyle{ \mathbb{P}(C) }[/math] sits at infinity in [math]\displaystyle{ \overline{C_X Y} }[/math].

Item 1 is clear (check torsion-free-ness). In general, given [math]\displaystyle{ X \subset Y }[/math], we have [math]\displaystyle{ \operatorname{Bl}_V X \subset \operatorname{Bl}_V Y }[/math]. Since [math]\displaystyle{ X \times 0 }[/math] is already an effective Cartier divisor on [math]\displaystyle{ X \times \mathbb{P}^1 }[/math], we get [math]\displaystyle{ X \times \mathbb{P}^1 = \operatorname{Bl}_{X \times 0} X \times \mathbb{P}^1 \hookrightarrow M, }[/math] yielding [math]\displaystyle{ \widetilde{i} }[/math]. Item 3 follows from the fact the blowdown map π is an isomorphism away from the center [math]\displaystyle{ X \times 0 }[/math]. 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 [math]\displaystyle{ X \times 0 }[/math] in M does not intersect [math]\displaystyle{ \widetilde{Y} }[/math]. 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 [math]\displaystyle{ X }[/math] be a Deligne–Mumford stack locally of finite type over a field [math]\displaystyle{ k }[/math]. If [math]\displaystyle{ \textbf{L}_X }[/math] denotes the cotangent complex of X relative to [math]\displaystyle{ k }[/math], then the intrinsic normal bundle to [math]\displaystyle{ X }[/math] is the quotient stack [math]\displaystyle{ \mathfrak{N}_X := h^1 / h^0(\textbf{L}_{X, \text{fppf}}^{\vee}) }[/math] which is the stack of fppf [math]\displaystyle{ \textbf{L}_X^{\vee, 0} }[/math]-torsors on [math]\displaystyle{ \textbf{L}_X^{\vee, 1} }[/math]. A concrete interpretation of this stack quotient can be given by looking at its behavior locally in the etale topos of the stack [math]\displaystyle{ X }[/math].

Properties of intrinsic normal bundle

More concretely, suppose there is an étale morphism [math]\displaystyle{ U \to X }[/math] from an affine finite-type [math]\displaystyle{ k }[/math]-scheme [math]\displaystyle{ U }[/math] together with a locally closed immersion [math]\displaystyle{ f: U \to M }[/math] into a smooth affine finite-type [math]\displaystyle{ k }[/math]-scheme [math]\displaystyle{ M }[/math]. Then one can show [math]\displaystyle{ \mathfrak{N}_X |_U = [N_{U/M}/f^* T_M] }[/math] meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence [math]\displaystyle{ \mathcal{T}_U \to \mathcal{T}_M |_U \to \mathcal{N}_{U/M} }[/math] 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 [math]\displaystyle{ [N_{U/M}/f^* T_M] }[/math] can be interpreted as [math]\displaystyle{ B \mathcal{T}_U = \mathcal{T}_U[+1] }[/math] in certain cases.

Normal cone

The intrinsic normal cone to [math]\displaystyle{ X }[/math], denoted as [math]\displaystyle{ \mathfrak{C}_X }[/math], is then defined by replacing the normal bundle [math]\displaystyle{ N_{U/M} }[/math] with the normal cone [math]\displaystyle{ C_{U/M} }[/math]; i.e., [math]\displaystyle{ \mathfrak{C}_X|_U = [C_{U/M} / f^* T_M]. }[/math]

Example: One has that [math]\displaystyle{ X }[/math] is a local complete intersection if and only if [math]\displaystyle{ \mathfrak{C}_X = \mathfrak{N}_X }[/math]. In particular, if [math]\displaystyle{ X }[/math] is smooth, then [math]\displaystyle{ \mathfrak{C}_X = \mathfrak{N}_X = B T_X }[/math] is the classifying stack of the tangent bundle [math]\displaystyle{ T_X }[/math], which is a commutative group scheme over [math]\displaystyle{ X }[/math].

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

See also

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

References

• Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone" (in en). Inventiones Mathematicae 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910. 
• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9 

External linkes

• Fibers of the normal cone