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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.
Article
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.
Contents
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
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.
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. (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}
Examples
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
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 CY(X) and CW(V), so that we want to find the product of X and CWV in CXY.
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 CWV of a vector bundle CXY with the zero section X. However this intersection product is just given by applying the Gysin isomorphism to CWV.
Concretely, the deformation to the normal cone can be constructed by means of blowup. Precisely, let
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).
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
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
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