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.
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.
The normal cone (or rather its projective cousin) appears as a result of blow-up.
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).
is a regular embedding and there is a natural exact sequence of vector bundles on X:[2]
is a smooth morphism, then the normal bundle to the diagonal embedding
(r-fold) is the direct sum of r − 1 copies of the relative tangent bundle
is a regular embedding, then there is a natural exact sequence of vector bundles on X:[4]
(which is a special case of an exact sequence for cotangent sheaves.)
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
has irreducible components of various dimensions, depending delicately on the positions of
We can use this to give a presentation of the normal cone as the relative spectrum
The normal cone's geometry can be further explored by looking at the fibers for various closed points of
giving the normal cone as a one dimensional line, as expected.
showing the normal cone has more components than the scheme it lies over.
such that This construction defines a tool analogous to differential topology where non-transverse intersections are performed in a tubular neighborhood of the intersection.
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.
Now, we note: Item 1 is clear (check torsion-free-ness).
Item 3 follows from the fact the blowdown map π is an isomorphism away from the center
Now, the last item in the previous paragraph implies that the image of
Thus, one gets the deformation of i to the zero-section embedding of X into the normal cone.
be a Deligne–Mumford stack locally of finite type over a field
meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence
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.
is characterised as the closed substack such that, for any étale map