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.

DefinitionEdit

The normal cone CXY or   of an embedding i: XY, 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.

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  ; 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 ×k D, flat over the ring D of dual numbers and having X as the special fiber, and H0(X, NX Y).[1]

PropertiesEdit

Compositions of regular embeddingsEdit

If   are regular embeddings, then   is a regular embedding and there is a natural exact sequence of vector bundles on X:[2]

 

If   are regular embeddings of codimensions   and if   is a regular embedding of codimension

 
then[2]
 
In particular, if   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  .

If   is a closed immersion and if   is a flat morphism such that  , then[3][citation needed]

 

If   is a smooth morphism and   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.)

Cartesian squareEdit

For a Cartesian square of schemes

 
with   the vertical map, there is a closed embedding
 
of normal cones.

Dimension of componentsEdit

Let   be a scheme of finite type over a field and   a closed subscheme. If   is of pure dimension r; i.e., every irreducible component has dimension r, then   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   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.

ExamplesEdit

Let   be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is[6]

 

Non-regular EmbeddingEdit

Consider the non-regular embedding[7]: 4–5 

 
then, we can compute the normal cone by first observing
 
If we make the auxiliary variables   and   we get the relation
 
We can use this to give a presentation of the normal cone as the relative spectrum
 
Since   is affine, we can just write out the relative spectrum as the affine scheme
 
giving us the normal cone.

Geometry of this normal coneEdit

The normal cone's geometry can be further explored by looking at the fibers for various closed points of  . Note that geometrically   is the union of the  -plane   with the  -axis  ,

 
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
 
for  and either   or  . Since it's arbitrary which point we take, for convenience let's assume  , hence the fiber of   at the point   is isomorphic to
 
giving the normal cone as a one dimensional line, as expected. For a point   on the axis, this is given by a map
 
hence the fiber at the point   is
 
which gives a plane. At the origin  , the normal cone over that point is again isomorphic to  .

Nodal cubicEdit

For the nodal cubic curve   given by the polynomial   over  , and   the point at the node, the cone has the isomorphism

 
showing the normal cone has more components than the scheme it lies over.

Deformation to the normal coneEdit

Suppose   is an embedding. This can be deformed to the embedding of   inside the normal cone   (as the zero section) in the following sense:[7]: 6  there is a flat family

 
with generic fiber   and special fiber   such that there exists a family of closed embeddings
 
over   such that
  1. Over any point   the associated embeddings are an embedding  
  2. The fiber over  is the embedding of   given by the zero section.

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   with a cycle   in   can be given as the pushforward of an intersection of   with the pullback of   in  .

ConstructionEdit

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

 
be the blow-up of   along  . The exceptional divisor is  , the projective completion of the normal cone; for the notation used here see Cone (algebraic geometry) § Properties. The normal cone   is an open subscheme of   and   is embedded as a zero-section into  .

Now, we note:

  1. The map  , the   followed by projection, is flat.
  2. There is an induced closed embedding
     
    that is a morphism over  .
  3. M is trivial away from zero; i.e.,   and   restricts to the trivial embedding
     
  4.   as the divisor is the sum
     
    where   is the blow-up of Y along X and is viewed as an effective Cartier divisor.
  5. As divisors   and   intersect at  , where   sits at infinity in  .

Item 1 is clear (check torsion-free-ness). 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. Q.E.D.

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 coneEdit

Intrinsic normal bundleEdit

Let   be a Deligne–Mumford stack locally of finite type over a field  . If   denotes the cotangent complex of X relative to  , then the intrinsic normal bundle[8]: 27  to   is the quotient stack

 
which is the stack of fppf  -torsors on  . A concrete interpretation of this stack quotient can be given by looking at its behavior locally in the etale topos of the stack  .

Properties of intrinsic normal bundleEdit

More concretely, suppose there is an étale morphism   from an affine finite-type  -scheme   together with a locally closed immersion   into a smooth affine finite-type  -scheme  . Then one can show

 
meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence
 
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   can be interpreted as
 
in certain cases.

Normal coneEdit

The intrinsic normal cone to  , denoted as  ,[8]: 29  is then defined by replacing the normal bundle   with the normal cone  ; i.e.,

 

Example: One has that   is a local complete intersection if and only if  . In particular, if   is smooth, then   is the classifying stack of the tangent bundle  , which is a commutative group scheme over  .

More generally, let   is a Deligne-Mumford Type (DM-type) morphism of Artin Stacks which is locally of finite type. Then   is characterised as the closed substack such that, for any étale map   for which   factors through some smooth map   (e.g.,  ), the pullback is:

 

See alsoEdit

NotesEdit

  1. ^ Hartshorne 1977, p. Ch. III, Exercise 9.7..
  2. ^ a b Fulton 1998, p. Appendix B.7.4..
  3. ^ Fulton 1998, p. The first part of the proof of Theorem 6.5..
  4. ^ Fulton 1998, p. Appendix B 7.1..
  5. ^ Fulton 1998, p. Appendix B. 6.6..
  6. ^ Fulton 1998, p. Appendix B.6.2..
  7. ^ a b Battistella, Luca; Carocci, Francesca; Manolache, Cristina (2020-04-09). "Virtual classes for the working mathematician". Symmetry, Integrability and Geometry: Methods and Applications. doi:10.3842/SIGMA.2020.026.
  8. ^ a b Behrend, K.; Fantechi, B. (1997-03-19). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910.

ReferencesEdit

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