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.

Definition

The normal cone C_XY of an embedding i: X → Y, defined by some sheaf of ideals I is defined as the relative Spec

\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 diagonal 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

\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 $E=\pi ^{-1}(X)$ ; which is the projective cone of $\oplus _{0}^{\infty }I^{n}\otimes _{{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}=\oplus _{0}^{\infty }I^{n}/I^{n+1}$ . Thus,

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 $i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z$ are regular embedding, then $j\circ i$ is a regular embedding and there is a natural exact sequence of vector bundles on X:^[2]

0\to N_{X/Y}\to N_{X/Z}\to i^{*}N_{Y/Z}\to 0

.

If $Y_{i}\hookrightarrow X$ are regular embeddings of codimensions $c_{i}$ and if $W:=\bigcap _{i}Y_{i}\hookrightarrow X$ is a regular embedding of codimension $\sum c_{i}$ , then^[3]

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

.

In particular, if $X\to S$ is a smooth morphism, then the normal bundle to the diagonal embedding $\Delta :X\hookrightarrow X\times _{S}\cdots \times _{S}X$ (r-fold) is the direct sum of r - 1 copies of the relative tangent bundle $T_{X/S}$ .

If $X\hookrightarrow X'$ is a closed immersion and if $Y'\to Y$ is a flat morphism such that $X'=X\times _{Y}Y'$ , then^[4]

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

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:^[5]

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 $X$ be a scheme of finite type over a field and $W\subset X$ a closed subscheme. If $X$ is of pure dimension r; i.e., every irreducible component has dimension r, then $C_{W/X}$ 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 $V,X$ 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^[7]
$N_{D/X}={\mathcal {O}}_{D}(D):={\mathcal {O}}_{X}(D)|_{D}$ .

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

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

be the blow-up of $Y\times \mathbb {P} ^{1}$ along $X\times 0$ . The exceptional divisor is ${\overline {C_{X}Y}}=\mathbb {P} (C_{X}Y\oplus 1)$ , the projective completion of the normal cone; for the notation used here see cone#Properties. The normal cone $C_{X}Y$ is an open subscheme of ${\overline {C_{X}Y}}$ and $X$ is embedded as a zero-section into $C_{X}Y$ .

Now, we note:

The map $\rho :M\to \mathbb {P} ^{1}$ , the $\pi$ followed by projection, is flat.
There is an induced closed embedding
${\widetilde {i}}:X\times \mathbb {P} ^{1}\hookrightarrow M$
that is a morphism over $\mathbb {P} ^{1}$ .
M is trivial away from zero; i.e., $\rho ^{-1}(\mathbb {P} ^{1}-0)=Y\times (\mathbb {P} ^{1}-0)$ and ${\widetilde {i}}$ restricts to the trivial embedding
$X\times (\mathbb {P} ^{1}-0)\hookrightarrow Y\times (\mathbb {P} ^{1}-0)$ .
$\rho ^{-1}(0)$ as the divisor is the sum
${\overline {C_{X}Y}}+{\widetilde {Y}}$
where ${\widetilde {Y}}$ is the blow-up of Y along X and is viewed as an effective Cartier divisor.
As divisors ${\overline {C_{X}Y}}$ and ${\widetilde {Y}}$ intersect at $\mathbb {P} (C)$ , where $\mathbb {P} (C)$ 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 $X\times 0$ is already an effective Cartier divisor on $X\times \mathbb {P} ^{1}$ , we get

X\times \mathbb {P} ^{1}=\operatorname {Bl} _{X\times 0}X\times \mathbb {P} ^{1}\hookrightarrow M

,

yielding ${\widetilde {i}}$ . Item 3. follows from the fact the blowdown map π is an isomorphism away from the center $X\times 0$ . The last two items are seen from explicit local computation. $\square$

Now, the last item in the previous paragraph implies that the image of $X\times 0$ 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

Let X be a Deligne–Mumford stack locally of finite type over a field k. If ${\textbf {L}}_{X}$ denotes the cotangent complex of X relative to k, then the intrinsic normal bundle to X is the quotient stack

{\mathfrak {R}}_{X}:=h^{1}/h^{0}({\textbf {L}}_{X,{\text{fppf}}}^{\vee })

which is the stack of fppf ${\textbf {L}}_{X}^{\vee ,0}$ -torsors on ${\textbf {L}}_{X}^{\vee ,1}$ . More concretely, supoose 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

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

The intrinsic normal cone to X, denoted as ${\mathfrak {C}}_{X}$ , is then defined by replacing the normal bundle $N_{U/M}$ with the normal cone $C_{U/M}$ ; i.e.,

{\mathfrak {C}}_{X}|_{U}=[C_{U/M}/f^{*}T_{M}].

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

Notes

↑ Hartshorne, Ch. III, Exercise 9.7.
↑ Fulton, Appendix B.7.4.
↑ Fulton, Appendix B.7.4.
↑ Fulton, The first part of the proof of Theorem 6.5.
↑ Fulton, Appendix B 7.1.
↑ Fulton, Appendix B. 6.6.
↑ Fulton, Appendix B.6.2.

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

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[1] Hartshorne, Ch. III, Exercise 9.7.

[2] Fulton, Appendix B.7.4.

[3] Fulton, Appendix B.7.4.

[4] Fulton, The first part of the proof of Theorem 6.5.

[5] Fulton, Appendix B 7.1.

[6] Fulton, Appendix B. 6.6.

[7] Fulton, Appendix B.6.2.