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 .
In the modern treatment of intersection theory in algebraic geometry, as developed e.g. in the definitive book of Fulton, Segre classes play a fundamental role.
Definition.
Suppose formula_1 is a cone over formula_2, formula_3 is the projection from the projective completion formula_4 of formula_1 to formula_6, and formula_7 is the anti-tautological line bundle on formula_4. Viewing the Chern class formula_9 as a group endomorphism of the Chow group of formula_4, the total Segre class of formula_1 is given by:
The formula_13th Segre class formula_14 is simply the formula_13th graded piece of formula_16. If formula_1 is of pure dimension formula_18 over formula_6 then this is given by:
The reason for using formula_4 rather than formula_22 is that this makes the total Segre class stable under addition of the trivial bundle formula_23.
If "Z" is a closed subscheme of an algebraic scheme "X", then formula_24 denote the Segre class of the normal cone to formula_25.
Relation to Chern classes for vector bundles.
For a holomorphic vector bundle formula_26 over a complex manifold formula_27 a total Segre class formula_28 is the inverse to the total Chern class formula_29, see e.g.
Explicitly, for a total Chern class
one gets the total Segre class
where
Let formula_33 be Chern roots, i.e. formal eigenvalues of formula_34 where formula_35 is a curvature of a connection on formula_36.
While the Chern class c(E) is written as
where formula_38 is an elementary symmetric polynomial of degree formula_39 in variables formula_33
the Segre for the dual bundle formula_41 which has Chern roots formula_42 is written as
Expanding the above expression in powers of formula_44 one can see that formula_45 is represented by
a complete homogeneous symmetric polynomial of formula_44
Properties.
Here are some basic properties.
A key property of a Segre class is birational invariance: this is contained in the following. Let formula_56 be a proper morphism between algebraic schemes such that formula_57 is irreducible and each irreducible component of formula_6 maps onto formula_57. Then, for each closed subscheme formula_60, formula_61 and formula_62 the restriction of formula_63,
Similarly, if formula_65 is a flat morphism of constant relative dimension between pure-dimensional algebraic schemes, then, for each closed subscheme formula_60, formula_67 and formula_68 the restriction of formula_69,
A basic example of binational invariance is provided by a blow-up. Let formula_71 be a blow-up along some closed subscheme "Z". Since the exceptional divisor formula_72 is an effective Cartier divisor and the normal cone (or normal bundle) to it is formula_73,
where we used the notation formula_75. Thus,
where formula_77 is given by formula_78.
Examples.
Example 1.
Let "Z" be a smooth curve that is a complete intersection of effective Cartier divisors formula_79 on a variety "X". Assume the dimension of "X" is "n" + 1. Then the Segre class of the normal cone formula_80 to formula_25 is:
Indeed, for example, if "Z" is regularly embedded into "X", then, since formula_83 is the normal bundle and formula_84 (see Normal cone#Properties), we have:
Example 2.
The following is Example 3.2.22. of . It recovers some classical results from Schubert's book on enumerative geometry.
Viewing the dual projective space formula_86 as the Grassmann bundle formula_87 parametrizing the 2-planes in formula_88, consider the tautological exact sequence
where formula_90 are the tautological sub and quotient bundles. With formula_91, the projective bundle formula_92 is the variety of conics in formula_88. With formula_94, we have formula_95 and so, using Chern class#Computation formulae,
and thus
where formula_98 The coefficients in formula_28 have the enumerative geometric meanings; for example, 92 is the number of conics meeting 8 general lines.
See also: .
Example 3.
Let "X" be a surface and formula_100 effective Cartier divisors on it. Let formula_101 be the scheme-theoretic intersection of formula_102 and formula_103 (viewing those divisors as closed subschemes). For simplicity, suppose formula_104 meet only at a single point "P" with the same multiplicity "m" and that "P" is a smooth point of "X". Then
To see this, consider the blow-up formula_71 of "X" along "P" and let formula_107, the strict transform of "Z". By the formula at #Properties,
Since formula_109 where formula_110, the formula above results.
Multiplicity along a subvariety.
Let formula_111 be the local ring of a variety "X" at a closed subvariety "V" codimension "n" (for example, "V" can be a closed point). Then formula_112 is a polynomial of degree "n" in "t" for large "t"; i.e., it can be written as formula_113 the lower-degree terms and the integer formula_114 is called the multiplicity of "A".
The Segre class formula_115 of formula_116 encodes this multiplicity: the coefficient of formula_117 in formula_115 is formula_114.