In algebraic geometry,[] the normal cone CXY 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 CXY or of an embedding i: X → Y, defined by some sheaf of idealsI 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 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 appears as a result of blow-up. Precisely, let be the blow-up of Y along X. Then, by definition, the exceptional divisor is the pre-image ; which is the projective cone of. Thus, 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 ×kD, flat over the ringD of dual numbers and having X as the special fiber, and H0.
Properties
If are regular embedding, then is a regular embedding and there is a natural exact sequence of vector bundles on X: If are regular embeddings of codimensions and if is a regular embedding of codimension, then In particular, if is a smooth morphism, then the normal bundle to the diagonal embedding is the direct sum of r - 1 copies of the relative tangent bundle. If is a closed immersion and if is a flat morphism such that, then If is a smooth morphism and is a regular embedding, then there is a natural exact sequence of vector bundles on X: Let be a scheme of finite type over a field and a closed subscheme. If is of ; i.e., every irreducible component has dimension r, then is also of pure dimension r. This property is a key to an application in intersection theory: given a pair of closed subschemes in some ambient space, while the scheme-theoretic intersection has irreducible components of various dimensions, depending delicately on the positions of, the normal cone to is of pure dimension.
Examples
Let be an effective Cartier divisor. Then the normal bundle to it is
:.
Non-regular Embedding
Consider the non-regular embedding then, we can compute the normal cone by first observing If we make the auxiliary variables and then observe that giving the relation We can use this to give a presentation of the normal cone:
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 CXY 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. 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 and CW, 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 be the blow-up of along. The exceptional divisor is, the projective completion of the normal cone; for the notation used here see cone#Properties. The normal cone is an open subscheme of and is embedded as a zero-section into. Now, we note: Item 1. is clear. In general, given, we have. Since is already an effective Cartier divisor on, we get yielding. Item 3. follows from the fact the blowdown map π is an isomorphism away from the center. The last two items are seen from explicit local computation. Now, the last item in the previous paragraph implies that the image of in M does not intersect. 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 denotes the cotangent complex of X relative to k, then the intrinsic normal bundle to X is the quotient stack which is the stack of fppf -torsors on. More concretely, suppose there is an étale morphism from an affine finite-type k-scheme U together with a locally closed immersion into a smooth affine finite-type k-scheme M. Then one can show The intrinsic normal cone to X, denoted as, is then defined by replacing the normal bundle with the normal cone ; i.e., Example: One has that is a local complete intersectionif and only if. In particular, if X is smooth, then is the classifying stack of the tangent bundle, which is a commutative group scheme over X. More generally, let is a Deligne-Mumford Type morphism of Artin Stacks which is locally of finite type. Then is characterised as the closed substack such that, for any \'etale map for which factors through some smooth map , the pullback is:
