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 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.

Definition

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

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

The [math]\displaystyle{ i }[/math]th Segre class [math]\displaystyle{ s_i(C) }[/math] is simply the [math]\displaystyle{ i }[/math]th graded piece of [math]\displaystyle{ s(C) }[/math]. If [math]\displaystyle{ C }[/math] is of pure dimension [math]\displaystyle{ r }[/math] over [math]\displaystyle{ X }[/math] then this is given by:

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

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

If Z is a closed subscheme of an algebraic scheme X, then [math]\displaystyle{ s(Z, X) }[/math] denote the Segre class of the normal cone to [math]\displaystyle{ Z \hookrightarrow X }[/math].

Relation to Chern classes for vector bundles

For a holomorphic vector bundle [math]\displaystyle{ E }[/math] over a complex manifold [math]\displaystyle{ M }[/math] a total Segre class [math]\displaystyle{ s(E) }[/math] is the inverse to the total Chern class [math]\displaystyle{ c(E) }[/math], see e.g. Fulton (1998).

Explicitly, for a total Chern class

[math]\displaystyle{ c(E) = 1+c_1(E) + c_2(E) + \cdots \, }[/math]

one gets the total Segre class

[math]\displaystyle{ s(E) = 1 + s_1 (E) + s_2 (E) + \cdots \, }[/math]

where

[math]\displaystyle{ 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) }[/math]

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

While the Chern class c(E) is written as

[math]\displaystyle{ c(E) = \prod_{i=1}^{k} (1+x_i) = c_0 + c_1 + \cdots + c_k \, }[/math]

where [math]\displaystyle{ c_i }[/math] is an elementary symmetric polynomial of degree [math]\displaystyle{ i }[/math] in variables [math]\displaystyle{ x_1, \dots, x_k }[/math]

the Segre for the dual bundle [math]\displaystyle{ E^\vee }[/math] which has Chern roots [math]\displaystyle{ -x_1, \dots, -x_k }[/math] is written as

[math]\displaystyle{ s(E^\vee) = \prod_{i=1}^{k} \frac {1} { 1 - x_i } = s_0 + s_1 + \cdots }[/math]

Expanding the above expression in powers of [math]\displaystyle{ x_1, \dots x_k }[/math] one can see that [math]\displaystyle{ s_i (E^\vee) }[/math] is represented by a complete homogeneous symmetric polynomial of [math]\displaystyle{ x_1, \dots x_k }[/math]

Properties

Here are some basic properties.

• For any cone C (e.g., a vector bundle), [math]\displaystyle{ s(C \oplus 1) = s(C) }[/math].
• For a cone C and a vector bundle E,

  [math]\displaystyle{ c(E)s(C \oplus E) = s(C). }[/math]

• If E is a vector bundle, then

  [math]\displaystyle{ s_i(E) = 0 }[/math] for [math]\displaystyle{ i \lt 0 }[/math].

  [math]\displaystyle{ s_0(E) }[/math] is the identity operator.

  [math]\displaystyle{ s_i(E) \circ s_j(F) = s_j(F) \circ s_i(E) }[/math] for another vector bundle F.

• If L is a line bundle, then [math]\displaystyle{ s_1(L) = -c_1(L) }[/math], minus the first Chern class of L.
• If E is a vector bundle of rank [math]\displaystyle{ e + 1 }[/math], then, for a line bundle L,

  [math]\displaystyle{ 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}. }[/math]

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

[math]\displaystyle{ {p_V}_*(s(V, X)) = \operatorname{deg}(p) \, s(W, Y). }[/math]

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

[math]\displaystyle{ {f_V}^*(s(W, Y)) = s(V, X). }[/math]

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

[math]\displaystyle{ \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} }[/math]

where we used the notation [math]\displaystyle{ D \cdot \alpha = c_1(\mathcal{O}(D))\alpha }[/math]. Thus,

[math]\displaystyle{ s(Z, X) = g_* \left( \sum_{k=1}^{\infty} (-1)^{k-1} E^k \right) }[/math]

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

Examples

Example 1

Let Z be a smooth curve that is a complete intersection of effective Cartier divisors [math]\displaystyle{ D_1, \dots, D_n }[/math] on a variety X. Assume the dimension of X is n + 1. Then the Segre class of the normal cone [math]\displaystyle{ C_{Z/X} }[/math] to [math]\displaystyle{ Z \hookrightarrow X }[/math] is:

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

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

[math]\displaystyle{ 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]. }[/math]

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

[math]\displaystyle{ 0 \to S \to p^* \mathbb{C}^3 \to Q \to 0 }[/math]

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

[math]\displaystyle{ c(E) = 1 + 8 \beta + 30 \beta^2 + 60 \beta^3 }[/math]

and thus

[math]\displaystyle{ s(E) = 1 + 8 h + 34 h^2 + 92 h^3 }[/math]

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

See also: Residual intersection.

Example 3

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

[math]\displaystyle{ s(Z, X) = [D] + (m^2[P] - D \cdot [D]). }[/math]

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

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

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

Multiplicity along a subvariety

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

The Segre class [math]\displaystyle{ s(V, X) }[/math] of [math]\displaystyle{ V \subset X }[/math] encodes this multiplicity: the coefficient of [math]\displaystyle{ [V] }[/math] in [math]\displaystyle{ s(V, X) }[/math] is [math]\displaystyle{ e(A) }[/math].

Bibliography

• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4 
• {{citation|mr=0061420

|last=Segre|first= Beniamino |title=Nuovi metodi e resultati nella geometria sulle varietà algebriche|language=Italian