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

s (C) = q_{*} (\sum_{i \geq 0} c_{1} (𝒪 (1))^{i} [ℙ (C \oplus 1)]) .

The $i$ th Segre class $s_{i} (C)$ is simply the $i$ th graded piece of $s (C)$ . If $C$ is of pure dimension $r$ over $X$ then this is given by:

s_{i} (C) = q_{*} (c_{1} (𝒪 (1))^{r + i} [ℙ (C \oplus 1)]) .

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

If Z is a closed subscheme of an algebraic scheme X, then $s (Z, X)$ denotes the Segre class of the normal cone to $Z ↪ X$ .

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).

Explicitly, for a total Chern class

c (E) = 1 + c_{1} (E) + c_{2} (E) + \dots

one gets the total Segre class

s (E) = 1 + s_{1} (E) + s_{2} (E) + \dots

where

c_{1} (E) = - s_{1} (E), c_{2} (E) = s_{1} (E)^{2} - s_{2} (E), \dots, c_{n} (E) = - s_{1} (E) c_{n - 1} (E) - s_{2} (E) c_{n - 2} (E) - \dots - s_{n} (E)

Let $x_{1}, \dots, x_{k}$ be Chern roots, i.e. formal eigenvalues of $\frac{i Ω}{2 π}$ where $Ω$ 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} + \dots + 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 $E^{\lor}$ which has Chern roots $- x_{1}, \dots, - x_{k}$ is written as

s (E^{\lor}) = \prod_{i = 1}^{k} \frac{1}{1 - x_{i}} = s_{0} + s_{1} + \dots

Expanding the above expression in powers of $x_{1}, \dots x_{k}$ one can see that $s_{i} (E^{\lor})$ 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), $s (C \oplus 1) = s (C)$ .
For a cone C and a vector bundle E,
$c (E) s (C \oplus E) = s (C) .$
If E is a vector bundle, then
$s_{i} (E) = 0$ 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,
$s_{p} (E \otimes L) = \sum_{i = 0}^{p} (- 1)^{p - i} (\binom{e + p}{e + i}) s_{i} (E) c_{1} (L)^{p - i} .$

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)) = \deg (p) s (W, Y) .

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) .

A basic example of birational invariance is provided by a blow-up. Let $π : \tilde{X} \to X$ be a blow-up along some closed subscheme Z. Since the exceptional divisor $E := π^{- 1} (Z) ↪ \tilde{X}$ is an effective Cartier divisor and the normal cone (or normal bundle) to it is $𝒪_{E} (E) := 𝒪_{X} (E) |_{E}$ ,

\begin{matrix} s (E, \tilde{X}) & = c (𝒪_{E} (E))^{- 1} [E] \\ = [E] - E \cdot [E] + E \cdot (E \cdot [E]) + \dots, \end{matrix}

where we used the notation $D \cdot α = c_{1} (𝒪 (D)) α$ . Thus,

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

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

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 $Z ↪ X$ is:

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 $N_{Z / X} = ⨁_{i = 1}^{n} N_{D_{i} / X} |_{Z}$ (see Normal cone#Properties), we have:

s (C_{Z / X}) = c (N_{Z / X})^{- 1} [Z] = \prod_{i = 1}^{d} (1 - c_{1} (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 $\overset{˘}{ℙ^{3}}$ as the Grassmann bundle $p : \overset{˘}{ℙ^{3}} \to *$ parametrizing the 2-planes in $ℙ^{3}$ , consider the tautological exact sequence

0 \to S \to p^{*} ℂ^{3} \to Q \to 0

where $S, Q$ are the tautological sub and quotient bundles. With $E = {Sym}^{2} (S^{*} \otimes Q^{*})$ , the projective bundle $q : X = ℙ (E) \to \overset{˘}{ℙ^{3}}$ is the variety of conics in $ℙ^{3}$ . With $β = c_{1} (Q^{*})$ , we have $c (S^{*} \otimes Q^{*}) = 2 β + 2 β^{2}$ and so, using Chern class#Computation formulae,

c (E) = 1 + 8 β + 30 β^{2} + 60 β^{3}

and thus

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

where $h = - β = c_{1} (Q) .$ The coefficients in $s (E)$ have the enumerative geometric meanings; for example, 92 is the number of conics meeting 8 general lines.

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

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

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

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

Since $\tilde{Z} = π^{*} D + m E$ where $E = π^{- 1} P$ , the formula above results.

Multiplicity along a subvariety

Let $(A, 𝔪)$ be the local ring of a variety X at a closed subvariety V codimension n (for example, V can be a closed point). Then ${length}_{A} (A / 𝔪^{t})$ is a polynomial of degree n in t for large t; i.e., it can be written as $\frac{e (A)^{n}}{n!} t^{n} +$ 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 $e (A)$ .

References

↑ Segre 1953
1 2 Fulton 1998
↑ Fulton 1998, p.50.
↑ Fulton 1998, Example 4.1.1.
↑ Fulton 1998, Example 4.1.5.
1 2 Fulton 1998, Proposition 3.1.
↑ Fulton 1998, Example 3.1.1.
↑ Fulton 1998, Proposition 4.2. (a)
↑ Fulton 1998, Proposition 4.2. (b)
↑ Fulton 1998, § 2.5.
↑ Fulton 1998, Example 9.1.1.
↑ Fulton 1998, Example 4.2.2.
↑ Fulton 1998, Example 4.3.1.

Bibliography

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

[1] Segre 1953

[Fulton_1998-2] 1 2 Fulton 1998

[3] Fulton 1998, p.50.

[4] Fulton 1998, Example 4.1.1.

[5] Fulton 1998, Example 4.1.5.

[Fulton-6] 1 2 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.