Normal cone

In algebraic geometry, the normal cone C"X""Y" 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"X""Y" or formula_1 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"/"I"2.
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
be the blow-up of "Y" along "X". Then, by definition, the exceptional divisor is the pre-image formula_4; which is the projective cone of formula_5. 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" ×"k" "D", flat over the ring "D" of dual numbers and having "X" as the special fiber, and "H"0("X", "N""X" "Y").
Properties.
If formula_7 are regular embedding, then formula_8 is a regular embedding and there is a natural exact sequence of vector bundles on "X":
If formula_10 are regular embeddings of codimensions formula_11 and if formula_12 is a regular embedding of codimension formula_13, then
In particular, if formula_15 is a smooth morphism, then the normal bundle to the diagonal embeddingformula_16 ("r"-fold) is the direct sum of "r" - 1 copies of the relative tangent bundle formula_17.
If formula_18 is a closed immersion and if formula_19 is a flat morphism such that formula_20, then
If formula_15 is a smooth morphism and formula_23 is a regular embedding, then there is a natural exact sequence of vector bundles on "X":
Let formula_25 be a scheme of finite type over a field and formula_26 a closed subscheme. If formula_25 is of ; i.e., every irreducible component has dimension "r", then formula_28 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 formula_29 in some ambient space, while the scheme-theoretic intersection formula_30 has irreducible components of various dimensions, depending delicately on the positions of formula_29, the normal cone to formula_30 is of pure dimension.
Examples.
Non-regular Embedding.
Consider the non-regular embedding
then, we can compute the normal cone by first observing
If we make the auxiliary variables formula_37 and formula_38 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 C"X""Y" 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""W""V" in "C""X""Y".
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""W""V" of a vector bundle "C""X""Y" with the zero section "X". However this intersection product is just given by applying the Gysin isomorphism to "C""W""V".
Concretely, the deformation to the normal cone can be constructed by means of blowup. Precisely, let
be the blow-up of formula_43 along formula_44. The exceptional divisor is formula_45, the projective completion of the normal cone; for the notation used here see cone#Properties. The normal cone formula_46 is an open subscheme of formula_47 and formula_25 is embedded as a zero-section into formula_46.
Now, we note:
Item 1. is clear (check torsion-free-ness). In general, given formula_50, we have formula_51. Since formula_44 is already an effective Cartier divisor on formula_53, we get
yielding formula_55. Item 3. follows from the fact the blowdown map π is an isomorphism away from the center formula_44. The last two items are seen from explicit local computation. formula_57
Now, the last item in the previous paragraph implies that the image of formula_44 in "M" does not intersect formula_59. 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 formula_60 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 formula_62-torsors on formula_63. More concretely, suppose there is an étale morphism formula_64 from an affine finite-type "k"-scheme "U" together with a locally closed immersion formula_65 into a smooth affine finite-type "k"-scheme "M". Then one can show
The intrinsic normal cone to "X", denoted as formula_67, is then defined by replacing the normal bundle formula_68 with the normal cone formula_69; i.e.,
Example: One has that formula_25 is a local complete intersection if and only if formula_72. In particular, if "X" is smooth, then formula_73 is the classifying stack of the tangent bundle formula_74, which is a commutative group scheme over "X".
More generally, let formula_75 is a Deligne-Mumford Type (DM-type) morphism of Artin Stacks which is locally of finite type. Then formula_76 is characterised as the closed substack such that, for any étale map formula_64 for which formula_78 factors through some smooth map formula_79 (e.g., formula_80), the pullback is: