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 CXY 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, NX Y).
Properties
Compositions of regular embeddings 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:
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
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