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 C X / Y {\displaystyle C_{X/Y}} of an embedding i: X → Y, defined by some sheaf of ideals I, is defined as the relative Spec Spec X ⁡ ( ⨁ n = 0 ∞ I n / I n + 1 ) . {\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 π : Bl X ⁡ Y = Proj Y ⁡ ( ⨁ n = 0 ∞ I n ) → Y {\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 E = π − 1 ( X ) {\displaystyle E=\pi ^{-1}(X)} ; which is the projective cone of ⨁ 0 ∞ I n ⊗ O Y O X = ⨁ 0 ∞ I n / I n + 1 {\textstyle \bigoplus _{0}^{\infty }I^{n}\otimes _{{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}=\bigoplus _{0}^{\infty }I^{n}/I^{n+1}} . Thus, E = P ( C X Y ) . {\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 i : X ↪ Y , j : Y ↪ Z {\displaystyle i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z} are regular embeddings, then j ∘ i {\displaystyle j\circ i} is a regular embedding and there is a natural exact sequence of vector bundles on X:^[2] 0 → N X / Y → N X / Z → i ∗ N Y / Z → 0. {\displaystyle 0\to N_{X/Y}\to N_{X/Z}\to i^{*}N_{Y/Z}\to 0.}

If Y i ↪ X {\displaystyle Y_{i}\hookrightarrow X} are regular embeddings of codimensions c i {\displaystyle c_{i}} and if W := ⋂ i Y i ↪ X {\textstyle W:=\bigcap _{i}Y_{i}\hookrightarrow X} is a regular embedding of codimension ∑ c i {\displaystyle \sum c_{i}} then^[2] N W / X = ⨁ i N Y i / X | W . {\displaystyle N_{W/X}=\bigoplus _{i}N_{Y_{i}/X}|_{W}.} In particular, if X → S {\displaystyle X\to S} is a smooth morphism, then the normal bundle to the diagonal embedding Δ : X ↪ X × S ⋯ × S X {\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 T X / S {\displaystyle T_{X/S}} .

If X ↪ Y {\displaystyle X\hookrightarrow Y} is a closed immersion and if Y ′ → Y {\displaystyle Y'\to Y} is a flat morphism such that X ′ = X × Y Y ′ {\displaystyle X'=X\times _{Y}Y'} , then^{[3][citation needed]} C X ′ / Y ′ = C X / Y × X X ′ . {\displaystyle C_{X'/Y'}=C_{X/Y}\times _{X}X'.}

If X → S {\displaystyle X\to S} is a smooth morphism and X ↪ Y {\displaystyle X\hookrightarrow Y} is a regular embedding, then there is a natural exact sequence of vector bundles on X:^[4] 0 → T X / S → T Y / S | X → N X / Y → 0 {\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 X ′ → Y ′ ↓ ↓ X → Y {\displaystyle {\begin{matrix}X'&\to &Y'\\\downarrow &&\downarrow \\X&\to &Y\end{matrix}}} with f : X ′ → X {\displaystyle f:X'\to X} the vertical map, there is a closed embedding C X ′ / Y ′ ↪ f ∗ C X / Y {\displaystyle C_{X'/Y'}\hookrightarrow f^{*}C_{X/Y}} of normal cones.

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

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

Consider the non-regular embedding^[7]: 4–5 X = Spec ( C [ x , y , z ] ( x z , y z ) ) → A 3 {\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 I = ( x z , y z ) I 2 = ( x 2 z 2 , x y z 2 , y 2 z 2 ) {\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 a = x z {\displaystyle a=xz} and b = y z {\displaystyle b=yz} we get the relation y a − x b = 0. {\displaystyle ya-xb=0.} We can use this to give a presentation of the normal cone as the relative spectrum C X A 3 = Spec X ( O X [ a , b ] ( y a − x b ) ) {\displaystyle C_{X}\mathbb {A} ^{3}={\text{Spec}}_{X}\left({\frac {{\mathcal {O}}_{X}[a,b]}{(ya-xb)}}\right)} Since A 3 {\displaystyle \mathbb {A} ^{3}} is affine, we can just write out the relative spectrum as the affine scheme C X A 3 = Spec ( C [ x , y , z ] [ a , b ] ( x z , y z , y a − x b ) ) {\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 X {\displaystyle X} . Note that geometrically X {\displaystyle X} is the union of the x y {\displaystyle xy} -plane H {\displaystyle H} with the z {\displaystyle z} -axis L {\displaystyle L} , X = H ∪ 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 x ↦ z 1 y ↦ z 2 z ↦ 0 {\displaystyle {\begin{matrix}x\mapsto z_{1}&y\mapsto z_{2}&z\mapsto 0\end{matrix}}} for z 1 , z 2 ∈ C {\displaystyle z_{1},z_{2}\in \mathbb {C} } and either z 1 ≠ 0 {\displaystyle z_{1}\neq 0} or z 2 ≠ 0 {\displaystyle z_{2}\neq 0} . Since it is arbitrary which point we take, for convenience let us assume z 1 ≠ 0 , z 2 = 0 {\displaystyle z_{1}\neq 0,z_{2}=0} . Hence the fiber of C X A 3 {\displaystyle C_{X}\mathbb {A} ^{3}} at the point p = ( z 1 , 0 , 0 ) {\displaystyle p=(z_{1},0,0)} is isomorphic to C X A 3 | p ≅ C [ a , b ] ( z 1 b ) ≅ C [ a ] {\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 q {\displaystyle q} on the axis, this is given by a map x ↦ 0 y ↦ 0 z ↦ z 3 {\displaystyle {\begin{matrix}x\mapsto 0&y\mapsto 0&z\mapsto z_{3}\end{matrix}}} hence the fiber at the point q = ( 0 , 0 , z 3 ) {\displaystyle q=(0,0,z_{3})} is C X A 3 | q ≅ C [ a , b ] ( 0 ) ≅ C [ a , b ] {\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 r = ( 0 , 0 , 0 ) {\displaystyle r=(0,0,0)} , the normal cone over that point is again isomorphic to C [ a , b ] {\displaystyle \mathbb {C} [a,b]} .

For the nodal cubic curve Y {\displaystyle Y} given by the polynomial y 2 + x 2 ( x − 1 ) {\displaystyle y^{2}+x^{2}(x-1)} over C {\displaystyle \mathbb {C} } , and X {\displaystyle X} the point at the node, the cone has the isomorphism C X / Y ≅ Spec ( C [ x , y ] / ( y 2 − x 2 ) ) {\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 i : X → Y {\displaystyle i:X\to Y} is an embedding. This can be deformed to the embedding of X {\displaystyle X} inside the normal cone C X / Y {\displaystyle C_{X/Y}} (as the zero section) in the following sense:^[7]: 6 there is a flat family π : M X / Y o → P 1 {\displaystyle \pi :M_{X/Y}^{o}\to \mathbb {P} ^{1}} with generic fiber Y {\displaystyle Y} and special fiber C X / Y {\displaystyle C_{X/Y}} such that there exists a family of closed embeddings X × P 1 ↪ M X / Y o {\displaystyle X\times \mathbb {P} ^{1}\hookrightarrow M_{X/Y}^{o}} over P 1 {\displaystyle \mathbb {P} ^{1}} such that

1. Over any point t ∈ P 1 − { 0 } {\displaystyle t\in \mathbb {P} ^{1}-\{0\}} the associated embeddings are an embedding X × { t } ↪ Y {\displaystyle X\times \{t\}\hookrightarrow Y}
2. The fiber over 0 ∈ P 1 {\displaystyle 0\in \mathbb {P} ^{1}} is the embedding of X ↪ C X / Y {\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 X {\displaystyle X} with a cycle Z {\displaystyle Z} in Y {\displaystyle Y} can be given as the pushforward of an intersection of X {\displaystyle X} with the pullback of Z {\displaystyle Z} in C X / Y {\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 π : M → Y × P 1 {\displaystyle \pi :M\to Y\times \mathbb {P} ^{1}} be the blow-up of Y × P 1 {\displaystyle Y\times \mathbb {P} ^{1}} along X × 0 {\displaystyle X\times 0} . The exceptional divisor is C X Y ¯ = P ( C X Y ⊕ 1 ) {\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 C X Y {\displaystyle C_{X}Y} is an open subscheme of C X Y ¯ {\displaystyle {\overline {C_{X}Y}}} and X {\displaystyle X} is embedded as a zero-section into C X Y {\displaystyle C_{X}Y} .

Now, we note:

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

Item 1 is clear (check torsion-free-ness). In general, given X ⊂ Y {\displaystyle X\subset Y} , we have Bl V ⁡ X ⊂ Bl V ⁡ Y {\displaystyle \operatorname {Bl} _{V}X\subset \operatorname {Bl} _{V}Y} . Since X × 0 {\displaystyle X\times 0} is already an effective Cartier divisor on X × P 1 {\displaystyle X\times \mathbb {P} ^{1}} , we get X × P 1 = Bl X × 0 ⁡ X × P 1 ↪ M , {\displaystyle X\times \mathbb {P} ^{1}=\operatorname {Bl} _{X\times 0}X\times \mathbb {P} ^{1}\hookrightarrow M,} yielding i ~ {\displaystyle {\widetilde {i}}} . Item 3 follows from the fact the blowdown map π is an isomorphism away from the center X × 0 {\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 X × 0 {\displaystyle X\times 0} in M does not intersect Y ~ {\displaystyle {\widetilde {Y}}} . Thus, one gets the deformation of i to the zero-section embedding of X into the normal cone.

Let X {\displaystyle X} be a Deligne–Mumford stack locally of finite type over a field k {\displaystyle k} . If L X {\displaystyle {\textbf {L}}_{X}} denotes the cotangent complex of X relative to k {\displaystyle k} , then the intrinsic normal bundle^[8]: 27 to X {\displaystyle X} is the quotient stack N X := h 1 / h 0 ( L X , fppf ∨ ) {\displaystyle {\mathfrak {N}}_{X}:=h^{1}/h^{0}({\textbf {L}}_{X,{\text{fppf}}}^{\vee })} which is the stack of fppf L X ∨ , 0 {\displaystyle {\textbf {L}}_{X}^{\vee ,0}} -torsors on L X ∨ , 1 {\displaystyle {\textbf {L}}_{X}^{\vee ,1}} . A concrete interpretation of this stack quotient can be given by looking at its behavior locally in the étale topos of the stack X {\displaystyle X} .

More concretely, suppose there is an étale morphism U → X {\displaystyle U\to X} from an affine finite-type k {\displaystyle k} -scheme U {\displaystyle U} together with a locally closed immersion f : U → M {\displaystyle f:U\to M} into a smooth affine finite-type k {\displaystyle k} -scheme M {\displaystyle M} . Then one can show N X | U = [ N U / M / f ∗ T M ] {\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 T U → T M | U → N U / M {\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 [ N U / M / f ∗ T M ] {\displaystyle [N_{U/M}/f^{*}T_{M}]} can be interpreted as B T U = T U [ + 1 ] {\displaystyle B{\mathcal {T}}_{U}={\mathcal {T}}_{U}[+1]} in certain cases.

The intrinsic normal cone to X {\displaystyle X} , denoted as C X {\displaystyle {\mathfrak {C}}_{X}} ,^[8]: 29 is then defined by replacing the normal bundle N U / M {\displaystyle N_{U/M}} with the normal cone C U / M {\displaystyle C_{U/M}} ; i.e., C X | U = [ C U / M / f ∗ T M ] . {\displaystyle {\mathfrak {C}}_{X}|_{U}=[C_{U/M}/f^{*}T_{M}].}

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

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