Segre class

In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by Beniamino Segre (1953).

Definition

Suppose is a cone over , is the projection from the projective completion of to , and is the anti-tautological line bundle on . Viewing the Chern class as a group endomorphism of the Chow group of , the total Segre class of is given by:

The th Segre class is simply the th graded piece of . If is of pure dimension over then this is given by:

The reason for using rather than is that this makes the total Segre class stable under addition of the trivial bundle .

If Z is a closed subscheme of an algebraic scheme X, then denote the Segre class of the normal cone to .

Relation to Chern classes for vector bundles

For a holomorphic vector bundle over a complex manifold a total Segre class is the inverse to the total Chern class , see e.g.[1]

Explicitly, for a total Chern class

one gets the total Segre class

where

Let be Chern roots, i.e. formal eigenvalues of where is a curvature of a connection on .

While the Chern class c(E) is written as

where is an elementary symmetric polynomial of degree in variables

the Segre for the dual bundle which has Chern roots is written as

Expanding the above expression in powers of one can see that is represented by a complete homogeneous symmetric polynomial of

Properties

Here are some basic properties.

  • For any cone C (e.g., a vector bundle), .[2]
  • For a cone C and a vector bundle E,
    [3]
  • If E is a vector bundle, then[4]
    for .
    is the identity operator.
    for another vector bundle F.
  • If L is a line bundle, then , minus the first Chern class of L.[4]
  • If E is a vector bundle of rank , then, for a line bundle L,
    [5]

A key property of a Segre class is birational invariance: this is contained in the following. Let be a proper morphism between algebraic schemes such that is irreducible and each irreducible component of maps onto . Then, for each closed subscheme , and the restriction of ,

[6]

Similarly, if is a flat morphism of constant relative dimension between pure-dimensional algebraic schemes, then, for each closed subscheme , and the restriction of ,

[7]

A basic example of binational invariance is provided by a blow-up. Let be a blow-up along some closed subscheme Z. Since the exceptional divisor is an effective Cartier divisor and the normal cone (or normal bundle) to it is ,

where we used the notation .[8] Thus,

where is given by .

Examples

Example 1

Let Z be a smooth curve that is a complete intersection of effective Cartier divisors on a variety X. Assume the dimension of X is n + 1. Then the Segre class of the normal cone to is:[9]

Indeed, for example, if Z is regularly embedded into X, then, since is the normal bundle and (see Normal cone#Properties), we have:

Example 2

The following is Example 3.2.22. of (Fulton 1998). It recovers some classical results from Schubert's book on enumerative geometry.

Viewing the dual projective space as the Grassmann bundle parametrizing the 2-planes in , consider the tautological exact sequence

where are the tautological sub and quotient bundles. With , the projective bundle is the variety of conics in . With , we have and so, using Chern class#Computation formulae,

and thus

where The coefficients in have the enumerative geometric meanings; for example, 92 is the number of conics meeting 8 general lines.

See also: Residual intersection#Example: conics tangent to given five conics.

Example 3

Let X be a surface and effective Cartier divisors on it. Let be the scheme-theoretic intersection of and (viewing those divisors as closed subschemes). For simplicity, suppose meet only at a single point P with the same multiplicity m and that P is a smooth point of X. Then[10]

To see this, consider the blow-up of X along P and let , the strict transform of Z. By the formula at #Properties,

Since where , the formula above results.

Multiplicity along a subvariety

Let be the local ring of a variety X at a closed subvariety V codimension n (for example, V can be a closed point). Then is a polynomial of degree n in t for large t; i.e., it can be written as the lower-degree terms and the integer is called the multiplicity of A.

The Segre class of encodes this multiplicity: the coefficient of in is .[11]

References

  1. Fulton W. (1998). Intersection theory, p.50. Springer, 1998.
  2. Fulton, Example 4.1.1.
  3. Fulton, Example 4.1.5.
  4. Fulton, Proposition 3.1.
  5. Fulton, Example 3.1.1.
  6. Fulton, Proposition 4.2. (a)
  7. Fulton, Proposition 4.2. (b)
  8. Fulton, § 2.5.
  9. Fulton, Example 9.1.1.
  10. Fulton, Example 4.2.2.
  11. Fulton, Example 4.3.1.
  • Segre, Beniamino (1953), "Nuovi metodi e resultati nella geometria sulle varietà algebriche", Ann. Mat. Pura Appl. (in Italian), 35 (4): 1–127, MR 0061420
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.