# 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* 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 diagonal *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 *H*^{0}(*X*, *N*_{X} *Y*).^{[1]}

## 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 #Construction of the deformation to the normal cone 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*.

## Construction of the deformation to the normal cone

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:

- The map , the followed by projection, is flat.
- There is an induced closed embedding
*M*is trivial away from zero; i.e., and restricts to the trivial embedding- .

- as the divisor is the sum
*Y*along*X*and is viewed as an effective Cartier divisor. - 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.

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.

## References

- ↑ Hartshorne, Ch. III, Exercise 9.7.

- William Fulton. (1998),
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.,**2**(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323 - Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157