Normal cone
In algebraic geometry, the normal cone formula_1 of a subscheme formula_2 of a scheme formula_3 is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry. Definition. The normal cone C"Y" or formula_4 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". 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_7; which is the projective cone of formula_8. 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" × "D", flat over the ring "D" of dual numbers and having "X" as the special fiber, and "H"("X", "N" "Y"). Properties. Compositions of regular embeddings. If formula_10 are regular embeddings, then formula_11 is a regular embedding and there is a natural exact sequence of vector bundles on "X": If  formula_13 are regular embeddings of codimensions formula_14 and if formula_15 is a regular embedding of codimensionformula_16then In particular, if formula_18 is a smooth morphism, then the normal bundle to the diagonal embeddingformula_19 ("r"-fold)is the direct sum of "r" - 1 copies of the relative tangent bundle formula_20. If formula_21 is a closed immersion and if formula_22 is a flat morphism such that formula_23, then If formula_18 is a smooth morphism and formula_26 is a regular embedding, then there is a natural exact sequence of vector bundles on "X": Cartesian square. For a Cartesian square of schemesformula_28with formula_29 the vertical map, there  is a closed embeddingformula_30of normal cones. Dimension of components. Let formula_2 be a scheme of finite type over a field and formula_32 a closed subscheme. If formula_2 is of ; i.e., every irreducible component has dimension "r", then formula_34 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_35 in some ambient space, while the scheme-theoretic intersection formula_36 has irreducible components of various dimensions, depending delicately on the positions of formula_35, the normal  cone to formula_36 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_43 and formula_44 we get the relation We can use this to give a presentation of the normal cone as the relative spectrum Since formula_47 is affine, we can just write out the relative spectrum as the affine schemeformula_48giving us the normal cone. Geometry of this normal cone. The normal cone's geometry can be further explored by looking at the fibers for various closed points of formula_2 . Note that geometrically formula_2 is the union of the formula_51-plane formula_52 with the formula_53-axis formula_54,formula_55so 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 mapformula_56for formula_57and either formula_58 or formula_59. Since it's arbitrary which point we take, for convenience let's assume formula_60, hence the fiber of formula_61 at the point formula_62 is isomorphic toformula_63giving the normal cone as a one dimensional line, as expected. For a point formula_64 on the axis, this is given  by a mapformula_65hence the fiber at the point formula_66 isformula_67which gives a plane. At the origin formula_68, the normal cone over that point is again isomorphic to formula_69. Nodal cubic. For the nodal cubic curve formula_3 given by the polynomial formula_71 over formula_72, and formula_2 the point at the node, the cone has the isomorphismformula_74showing the normal cone has more components than the scheme it lies over. Deformation to the normal cone. Suppose formula_75 is an embedding. This can be deformed to the embedding of formula_2 inside the normal cone formula_4 (as the zero section) in the following sense: there  is a flat familyformula_78with generic fiber formula_3 and special fiber formula_4 such that there exists a family of closed embeddingsformula_81over formula_82 such that 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 formula_2 with a cycle formula_88 in formula_3 can be given as the pushforward of an intersection of formula_2 with the pullback of formula_88 in formula_4. 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"("X") and "C"("V"), so that we want to find the product of "X" and "C""V" in "C""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""V  of a vector bundle C""Y" with the zero section "X". However this intersection product is just given by applying the Gysin isomorphism to "C""V". Concretely, the deformation to the normal cone can be constructed by means of blowup. Precisely, let be the blow-up of formula_94 along formula_95. The exceptional divisor is formula_96, the projective completion of the normal cone; for the notation used here see cone#Properties. The normal cone formula_97 is an open subscheme of formula_98 and formula_2 is embedded as a zero-section into formula_97. Now, we note: Item 1. is clear (check torsion-free-ness). In general , given formula_101, we have formula_102. Since formula_95 is already an effective Cartier divisor on formula_104, we get yielding formula_106. Item 3. follows from the fact the blowdown map π is an isomorphism away from the center formula_95. The last two items are seen from explicit local computation. formula_108 Now, the last item in the previous paragraph implies that the image of formula_95 in "M" does not intersect formula_110. Thus, one gets the deformation of "i" to the zero-section embedding of "X" into the normal cone. Intrinsic normal cone. Intrinsic normal bundle. Let formula_2 be a Deligne–Mumford stack locally  of finite type over a field formula_112. If formula_113 denotes the cotangent complex of "X" relative to formula_112, then the intrinsic normal bundle to formula_2 is the quotient stackformula_116which is the stack of fppf formula_117-torsors on formula_118. A concrete interpretation of this stack quotient can be given by looking at its behavior locally in the etale topos of the stack formula_2. Properties of intrinsic normal bundle. More concretely, suppose there is an étale morphism formula_120 from an affine finite-type formula_112-scheme formula_122 together with a locally closed immersion formula_123 into a smooth affine finite-type formula_112-scheme formula_125 . Then one can showformula_126meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequenceformula_127to 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 formula_128 can be interpreted asformula_129in certain cases. Normal cone. The intrinsic normal cone to formula_2, denoted as formula_131, is then defined by replacing the normal bundle formula_132 with the normal cone formula_133; i.e., Example: One has that  formula_2 is a local complete intersection if and only if formula_136. In particular, if formula_2 is smooth, then formula_138 is the classifying stack of the tangent bundle formula_139, which is a commutative group scheme over formula_2. More generally, let formula_141 is a Deligne-Mumford Type (DM-type) morphism of Artin Stacks which is locally of finite type. Then formula_142 is characterised as the closed substack such that, for any étale map formula_120 for which formula_144 factors through some smooth map formula_145 (e.g., formula_146), the pullback is: