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