Normal cone

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 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/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]

Properties

If are regular embedding, 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[3]

.

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[4]

If is a smooth morphism and is a regular embedding, then there is a natural exact sequence of vector bundles on X:[5]

,

(which is a special case of an exact sequence for cotangent sheaves.)

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.[6] (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.

Examples

  • Let be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is[7]
    .

Non-regular Embedding

Consider the non-regular embedding

then, we can compute the normal cone by first observing

If we make the auxillary 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. (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 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#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.

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 intersection if 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.

See also

Notes

  1. Hartshorne, Ch. III, Exercise 9.7.
  2. Fulton, Appendix B.7.4.
  3. Fulton, Appendix B.7.4.
  4. Fulton, The first part of the proof of Theorem 6.5.
  5. Fulton, Appendix B 7.1.
  6. Fulton, Appendix B. 6.6.
  7. Fulton, Appendix B.6.2.

References

  • Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910.
  • 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
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