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 Segre (1953).[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.[2]

Definition

edit

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

edit

For a holomorphic vector bundle   over a complex manifold   a total Segre class   is the inverse to the total Chern class  , see e.g. Fulton (1998).[3]

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

edit

Here are some basic properties.

  • For any cone C (e.g., a vector bundle),  .[4]
  • For a cone C and a vector bundle E,
     [5]
  • If E is a vector bundle, then[6]
      for  .
      is the identity operator.
      for another vector bundle F.
  • If L is a line bundle, then  , minus the first Chern class of L.[6]
  • If E is a vector bundle of rank  , then, for a line bundle L,
     [7]

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  ,

 [8]

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

 [9]

A basic example of birational 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  .[10] Thus,

 

where   is given by  .

Examples

edit

Example 1

edit

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:[11]

 

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

edit

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   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.

Example 3

edit

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[12]

 

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

edit

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  .[13]

References

edit
  1. ^ Segre 1953
  2. ^ a b Fulton 1998
  3. ^ Fulton 1998, p.50.
  4. ^ Fulton 1998, Example 4.1.1.
  5. ^ Fulton 1998, Example 4.1.5.
  6. ^ a b Fulton 1998, Proposition 3.1.
  7. ^ Fulton 1998, Example 3.1.1.
  8. ^ Fulton 1998, Proposition 4.2. (a)
  9. ^ Fulton 1998, Proposition 4.2. (b)
  10. ^ Fulton 1998, § 2.5.
  11. ^ Fulton 1998, Example 9.1.1.
  12. ^ Fulton 1998, Example 4.2.2.
  13. ^ Fulton 1998, Example 4.3.1.

Bibliography

edit
  • Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
  • Segre, Beniamino (1953), "Nuovi metodi e resultati nella geometria sulle varietà algebriche", Ann. Mat. Pura Appl. (in Italian), 35 (4): 1–127, MR 0061420