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.

[1] 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.

Viewing the Chern class

, the total Segre class of

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

a total Segre class

is the inverse to the total Chern class

[3] Explicitly, for a total Chern class one gets the total Segre class where Let

be Chern roots, i.e. formal eigenvalues of

While the Chern class c(E) is written as where

is an elementary symmetric polynomial of degree

is written as Expanding the above expression in powers of

is represented by a complete homogeneous symmetric polynomial of

A key property of a Segre class is birational invariance: this is contained in the following.

be a proper morphism between algebraic schemes such that

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

, A basic example of birational invariance is provided by a blow-up.

Let Z be a smooth curve that is a complete intersection of effective Cartier divisors

Then the Segre class of the normal cone

(see Normal cone#Properties), we have: The following is Example 3.2.22. of Fulton (1998).

[2] It recovers some classical results from Schubert's book on enumerative geometry.

Viewing the dual projective space

and so, using Chern class#Computation formulae, and thus where

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

(viewing those divisors as closed subschemes).

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; i.e., it can be written as