Normal cone

In algebraic geometry, the normal cone C_XY of a subscheme X of a scheme Y is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.

For normal cones in real vector spaces, see convex cone.

Definition

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

${\displaystyle \operatorname {Spec} _{X}(\oplus _{n=0}^{\infty }I^{n}/I^{n+1}).}$

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

${\displaystyle \pi$ :\operatorname {Bl} _{X}Y=\operatorname {Proj} _{Y}(\oplus _{n=0}^{\infty }I^{n})\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 ${\displaystyle \oplus _{0}^{\infty }I^{n}\otimes _{{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}=\oplus _{0}^{\infty }I^{n}/I^{n+1}}$. 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]

Properties

If ${\displaystyle i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z}$ are regular embedding, 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 ${\displaystyle W:=\bigcap _{i}Y_{i}\hookrightarrow X}$ is a regular embedding of codimension ${\displaystyle \sum c_{i}}$, then[3]

${\displaystyle N_{W/X}=\bigoplus _{i}N_{Y_{i}/X}|_{W}}$.

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[4]

${\displaystyle C_{X'/Y'}=C_{X/Y}\times _{X}X'.}$

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:[5]

${\displaystyle 0\to T_{X/S}\to T_{Y/S}|_{X}\to N_{X/Y}\to 0}$,

(which is a special case of an exact sequence for cotangent sheaves.)

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.[6] (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.

Examples

• Let ${\displaystyle D\hookrightarrow X}$ be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is[7]

  ${\displaystyle N_{D/X}={\mathcal {O}}_{D}(D):={\mathcal {O}}_{X}(D)|_{D}}$.

Non-regular Embedding

Consider the non-regular embedding

${\displaystyle X={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xz,yz)}}\right)\to \mathbb {A} ^{3}}$

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 ${\displaystyle b=yz}$ then observe that

${\displaystyle I^{2}=(a^{2},ab,b^{2})}$

giving the relation

${\displaystyle a^{2}b^{2}-ab=0.}$

We can use this to give a presentation of the normal cone:

${\displaystyle C_{X}\mathbb {A} ^{3}={\text{Spec}}_{X}\left({\frac {{\mathcal {O}}_{X}[a,b]}{(a^{2}b^{2}-ab)}}\right)}$

Deformation to the normal cone

Suppose i: X → Y is an embedding. This can be deformed to the embedding of X in the normal cone C_XY in the following sense: there is a family of embeddings parameterized by an element t of the projective or affine line, such that if t=0 the embedding is the embedding into the normal cone, and for other t is it isomorphic to the given embedding i. (See below for 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

${\displaystyle \pi :M\to Y\times \mathbb {P} ^{1}}$

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#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. ${\displaystyle \square }$

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.

Intrinsic normal cone

Let X be a Deligne–Mumford stack locally of finite type over a field k. If ${\displaystyle {\textbf {L}}_{X}}$ denotes the cotangent complex of X relative to k, then the intrinsic normal bundle to 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 ${\displaystyle {\textbf {L}}_{X}^{\vee ,1}}$. More concretely, suppose there is an étale morphism ${\displaystyle U\to X}$ from an affine finite-type k-scheme U together with a locally closed immersion ${\displaystyle f:U\to M}$ into a smooth affine finite-type k-scheme M. Then one can show

${\displaystyle {\mathfrak {N}}_{X}|_{U}=[N_{U/M}/f^{*}T_{M}].}$

The intrinsic normal cone to X, denoted as ${\displaystyle {\mathfrak {C}}_{X}}$, 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 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 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 \'etale 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:

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

See also

• Residual intersection

Notes

1. Hartshorne, Ch. III, Exercise 9.7. harvnb error: no target: CITEREFHartshorne (help)
2. Fulton, Appendix B.7.4. harvnb error: no target: CITEREFFulton (help)
3. Fulton, Appendix B.7.4. harvnb error: no target: CITEREFFulton (help)
4. Fulton, The first part of the proof of Theorem 6.5. harvnb error: no target: CITEREFFulton (help)
5. Fulton, Appendix B 7.1. harvnb error: no target: CITEREFFulton (help)
6. Fulton, Appendix B. 6.6. harvnb error: no target: CITEREFFulton (help)
7. Fulton, Appendix B.6.2. harvnb error: no target: CITEREFFulton (help)

References

• Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910.
• William Fulton. (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, MR 1644323
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157