In mathematics, the Segre class is a characteristic class used in the study of coness, 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). In the modern treatment of intersection theory in algebraic geometry, as developed e.g. in the definitive book of Fulton (1998), Segre classes play a fundamental role.

Definition

is a cone is the projection from the projective completion is the anti-tautological line bundle. Viewing the Chern class as a group endomorphism of the Chow group is given by:

then this is given by:

.

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

Relation to Chern classes for vector bundles

For a holomorphic vector bundle over a complex manifold is the inverse to the total Chern class, see e.g. Fulton (1998).

Explicitly, for a total Chern class

one gets the total Segre class

where

is a curvature of a connection.

While the Chern class c(E) is written as

is an elementary symmetric polynomial,

the Segre for the dual bundle is written as

is represented by a complete homogeneous symmetric polynomial.

Properties

Here are some basic properties.

• For any cone C.

• For a cone C and a vector bundle E,

• If E is a vector bundle, then

  1. .

  2. is the identity operator.

  3. for another vector bundle F.

  If L, minus the first Chern class of L.

• If E, then, for a line bundle L,

be a proper morphism between algebraic schemes is irreducible and each irreducible component,

is a flat morphism,

be a blow-up along some closed subscheme Z. Since the exceptional divisor,

. Thus,

.

Examples

Example 1

Let Z on a variety X. Assume the dimension of X is n + 1. Then the Segre class of the normal cone is:

Indeed, for example, if Z is regularly embedded into X (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, consider the tautological exact sequence

, the projective bundle and so, using Chern class#Computation formulae,

and thus

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

Example 3

Let X be the scheme-theoretic intersection meet only at a single point P with the same multiplicity m and that P is a smooth point of X. Then

of X along P, the strict transform of Z. By the formula at #Properties,

, the formula above results.

Multiplicity along a subvariety

be the local ring of a variety X at a closed subvariety V codimension n (for example, V can be a closed point is a polynomial of degree n in t for large t is called the multiplicity of A.

.

References

Bibliography