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 Beniamino Segre (1953).{{harvnb|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.{{harvnb|Fulton|1998}}

Definition

Suppose C is a cone over X

, q is the projection from the projective completion \mathbb{P}(C \oplus 1) of C to X, and \mathcal{O}(1) is the anti-tautological line bundle on \mathbb{P}(C \oplus 1). Viewing the Chern class c_1(\mathcal{O}(1)) as a group endomorphism of the Chow group of \mathbb{P}(C \oplus 1), the total Segre class of C is given by:

:s(C) = q_* \left( \sum_{i \geq 0} c_1(\mathcal{O}(1))^{i} [\mathbb{P}(C \oplus 1)] \right).

The ith Segre class s_i(C) is simply the ith graded piece of s(C). If C is of pure dimension r over X then this is given by:

:s_i(C) = q_* \left( c_1(\mathcal{O}(1))^{r+i} [\mathbb{P}(C \oplus 1)] \right).

The reason for using \mathbb{P}(C \oplus 1) rather than \mathbb{P}(C) is that this makes the total Segre class stable under addition of the trivial bundle \mathcal{O}.

If Z is a closed subscheme of an algebraic scheme X, then s(Z, X) denote the Segre class of the normal cone to Z \hookrightarrow X.

=Relation to Chern classes for vector bundles=

For a holomorphic vector bundle E over a complex manifold M a total Segre class s(E) is the inverse to the total Chern class c(E), see e.g. Fulton (1998).{{harvnb|Fulton|1998|loc=p.50.}}

Explicitly, for a total Chern class

:

c(E) = 1+c_1(E) + c_2(E) + \cdots \,

one gets the total Segre class

:

s(E) = 1 + s_1 (E) + s_2 (E) + \cdots \,

where

:

c_1(E) = -s_1(E), \quad c_2(E) = s_1(E)^2 - s_2(E), \quad \dots, \quad c_n(E) = -s_1(E)c_{n-1}(E) - s_2(E) c_{n-2}(E) - \cdots - s_n(E)

Let x_1, \dots, x_k be Chern roots, i.e. formal eigenvalues of \frac{ i \Omega }{ 2\pi} where \Omega is a curvature of a connection on E .

While the Chern class c(E) is written as

: c(E) = \prod_{i=1}^{k} (1+x_i) = c_0 + c_1 + \cdots + c_k \,

where c_i is an elementary symmetric polynomial of degree i in variables x_1, \dots, x_k

the Segre for the dual bundle E^\vee which has Chern roots -x_1, \dots, -x_k is written as

: s(E^\vee) = \prod_{i=1}^{k} \frac {1} { 1 - x_i } = s_0 + s_1 + \cdots

Expanding the above expression in powers of x_1, \dots x_k one can see that s_i (E^\vee) is represented by

a complete homogeneous symmetric polynomial of x_1, \dots x_k

Properties

Here are some basic properties.

  • For any cone C (e.g., a vector bundle), s(C \oplus 1) = s(C).{{harvnb|Fulton|1998|loc=Example 4.1.1.}}
  • For a cone C and a vector bundle E,
  • :c(E)s(C \oplus E) = s(C).

{{harvnb|Fulton|1998|loc=Example 4.1.5.}}

  • If E is a vector bundle, then{{harvnb|Fulton|1998|loc=Proposition 3.1.}}
  • :s_i(E) = 0 for i < 0.
  • :s_0(E) is the identity operator.
  • :s_i(E) \circ s_j(F) = s_j(F) \circ s_i(E) for another vector bundle F.
  • If L is a line bundle, then s_1(L) = -c_1(L), minus the first Chern class of L.
  • If E is a vector bundle of rank e + 1, then, for a line bundle L,
  • :s_p(E \otimes L) = \sum_{i=0}^p (-1)^{p-i} \binom{e+p}{e+i} s_i(E) c_1(L)^{p-i}.{{harvnb|Fulton|1998|loc=Example 3.1.1.}}

A key property of a Segre class is birational invariance: this is contained in the following. Let p: X \to Y be a proper morphism between algebraic schemes such that Y is irreducible and each irreducible component of X maps onto Y. Then, for each closed subscheme W \subset Y, V = p^{-1}(W) and p_V: V \to W the restriction of p,

:{p_V}_*(s(V, X)) = \operatorname{deg}(p) \, s(W, Y).{{harvnb|Fulton|1998|loc=Proposition 4.2. (a)}}

Similarly, if f: X \to Y is a flat morphism of constant relative dimension between pure-dimensional algebraic schemes, then, for each closed subscheme W \subset Y, V = f^{-1}(W) and f_V: V \to W the restriction of f,

:{f_V}^*(s(W, Y)) = s(V, X).{{harvnb|Fulton|1998|loc=Proposition 4.2. (b)}}

A basic example of birational invariance is provided by a blow-up. Let \pi: \widetilde{X} \to X be a blow-up along some closed subscheme Z. Since the exceptional divisor E := \pi^{-1}(Z) \hookrightarrow \widetilde{X} is an effective Cartier divisor and the normal cone (or normal bundle) to it is \mathcal{O}_E(E) := \mathcal{O}_X(E)|_E,

:\begin{align}

s(E, \widetilde{X}) &= c(\mathcal{O}_E(E))^{-1} [E] \\

&= [E] - E \cdot [E] + E \cdot (E \cdot [E]) + \cdots,

\end{align}

where we used the notation D \cdot \alpha = c_1(\mathcal{O}(D))\alpha.{{harvnb|Fulton|1998|loc=§ 2.5.}} Thus,

:s(Z, X) = g_* \left( \sum_{k=1}^{\infty} (-1)^{k-1} E^k \right)

where g: E = \pi^{-1}(Z) \to Z is given by \pi.

Examples

= Example 1 =

Let Z be a smooth curve that is a complete intersection of effective Cartier divisors D_1, \dots, D_n on a variety X. Assume the dimension of X is n + 1. Then the Segre class of the normal cone C_{Z/X} to Z \hookrightarrow X is:{{harvnb|Fulton|1998|loc=Example 9.1.1.}}

:s(C_{Z/X}) = [Z] - \sum_{i=1}^n D_i \cdot [Z].

Indeed, for example, if Z is regularly embedded into X, then, since C_{Z/X} = N_{Z/X} is the normal bundle and N_{Z/X} = \bigoplus_{i=1}^n N_{D_i/X}|_Z (see Normal cone#Properties), we have:

:s(C_{Z/X}) = c(N_{Z/X})^{-1}[Z] = \prod_{i=1}^d (1-c_1(\mathcal{O}_X(D_i))) [Z] = [Z] - \sum_{i=1}^n D_i \cdot [Z].

= 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 \breve{\mathbb{P}^3} as the Grassmann bundle p: \breve{\mathbb{P}^3} \to * parametrizing the 2-planes in \mathbb{P}^3, consider the tautological exact sequence

:0 \to S \to p^* \mathbb{C}^3 \to Q \to 0

where S, Q are the tautological sub and quotient bundles. With E = \operatorname{Sym}^2(S^* \otimes Q^*), the projective bundle q: X = \mathbb{P}(E) \to \breve{\mathbb{P}^3} is the variety of conics in \mathbb{P}^3. With \beta = c_1(Q^*), we have c(S^* \otimes Q^*) = 2 \beta + 2\beta^2 and so, using Chern class#Computation formulae,

:c(E) = 1 + 8 \beta + 30 \beta^2 + 60 \beta^3

and thus

:s(E) = 1 + 8 h + 34 h^2 + 92 h^3

where h = -\beta = c_1(Q). The coefficients in s(E) have the enumerative geometric meanings; for example, 92 is the number of conics meeting 8 general lines.

{{See also|Residual intersection#Example: conics tangent to given five conics}}

= Example 3 =

Let X be a surface and A, B, D effective Cartier divisors on it. Let Z \subset X be the scheme-theoretic intersection of A + D and B + D (viewing those divisors as closed subschemes). For simplicity, suppose A, B meet only at a single point P with the same multiplicity m and that P is a smooth point of X. Then{{harvnb|Fulton|1998|loc=Example 4.2.2.}}

:s(Z, X) = [D] + (m^2[P] - D \cdot [D]).

To see this, consider the blow-up \pi: \widetilde{X} \to X of X along P and let g: \widetilde{Z} = \pi^{-1}Z \to Z, the strict transform of Z. By the formula at #Properties,

:s(Z, X) = g_* ([\widetilde{Z}]) - g_*(\widetilde{Z} \cdot [\widetilde{Z}]).

Since \widetilde{Z} = \pi^* D + mE where E = \pi^{-1} P, the formula above results.

Multiplicity along a subvariety

Let (A, \mathfrak{m}) be the local ring of a variety X at a closed subvariety V codimension n (for example, V can be a closed point). Then \operatorname{length}_A(A/\mathfrak{m}^t) is a polynomial of degree n in t for large t; i.e., it can be written as { e(A)^n \over n!} t^n + the lower-degree terms and the integer e(A) is called the multiplicity of A.

The Segre class s(V, X) of V \subset X encodes this multiplicity: the coefficient of [V] in s(V, X) is e(A).{{harvnb|Fulton|1998|loc=Example 4.3.1.}}

References

{{reflist}}

Bibliography

|last=Segre|first= Beniamino

|title=Nuovi metodi e resultati nella geometria sulle varietà algebriche|language=Italian

|journal=Ann. Mat. Pura Appl. |issue=4|volume= 35|year=1953|pages=1–127 |authorlink=Beniamino Segre}}

Category:Intersection theory

Category:Characteristic classes