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 $C$ is a cone over $X$, $q$ is the projection from the projective completion $\mathbb(C \oplus 1)$ of $C$ to $X$, and $\mathcal(1)$ is the anti-tautological line bundle on $\mathbb(C \oplus 1)$. Viewing the Chern class $c_1(\mathcal(1))$ as a group endomorphism of the Chow group of $\mathbb(C \oplus 1)$, the total Segre class of $C$ is given by:
:s(C) = q_* \left( \sum_ c_1(\mathcal(1))^ mathbb(C \oplus 1)\right).
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_* \left( c_1(\mathcal(1))^ mathbb(C \oplus 1)\right).

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

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).

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

: $c_1(E) = -s_1(E), \quad c_2(E) = s_1(E)^2 - s_2(E), \quad \dots, \quad c_n(E) = -s_1(E)c_(E) - s_2(E) c_(E) - \cdots - s_n(E)$

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

While the Chern class c(E) is written as

:$c(E) = \prod_^ (1+x_i) = c_0 + c_1 + \cdots + c_k \,$

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

:$s(E^\vee) = \prod_^ \frac = s_0 + s_1 + \cdots$

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), $s(C \oplus 1) = s(C)$.
*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'',
*:$s_p(E \otimes L) = \sum_^p (-1)^ \binom s_i(E) c_1(L)^.$

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 $W \subset Y$, $V = p^(W)$ and $p_V: V \to W$ the restriction of $p$,
:$_*(s(V, X)) = \operatorname(p) \, s(W, Y).$
Similarly, if $f: X \to Y$ is a flat morphism of constant relative dimension between pure-dimensional algebraic schemes, then, for each closed subscheme $W \subset Y$, $V = f^(W)$ and $f_V: V \to W$ the restriction of $f$,
:$^*(s(W, Y)) = s(V, X).$

A basic example of birational invariance is provided by a blow-up. Let $\pi: \widetilde \to X$ be a blow-up along some closed subscheme ''Z''. Since the exceptional divisor $E := \pi^(Z) \hookrightarrow \widetilde$ is an effective Cartier divisor and the normal cone (or normal bundle) to it is $\mathcal_E(E) := \mathcal_X(E), _E$,
:\begin
s(E, \widetilde) &= c(\mathcal_E(E))^ \\
&= - E \cdot + E \cdot (E \cdot + \cdots,
\end
where we used the notation $D \cdot \alpha = c_1(\mathcal(D))\alpha$. Thus,
:$s(Z, X) = g_* \left( \sum_^ (-1)^ E^k \right)$
where $g: E = \pi^(Z) \to Z$ 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 $C_$ to $Z \hookrightarrow X$ is:
:s(C_) = - \sum_^n D_i \cdot
Indeed, for example, if ''Z'' is regularly embedded into ''X'', then, since $C_ = N_$ is the normal bundle and $N_ = \bigoplus_^n N_, _Z$ (see Normal cone#Properties), we have:
:s(C_) = c(N_)^ = \prod_^d (1-c_1(\mathcal_X(D_i))) = - \sum_^n D_i \cdot

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 $\breve$ as the Grassmann bundle $p: \breve \to *$ parametrizing the 2-planes in $\mathbb^3$, consider the tautological exact sequence
:$0 \to S \to p^* \mathbb^3 \to Q \to 0$
where $S, Q$ are the tautological sub and quotient bundles. With $E = \operatorname^2(S^* \otimes Q^*)$, the projective bundle $q: X = \mathbb(E) \to \breve$ is the variety of conics in $\mathbb^3$. 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
:s(Z, X) = + (m^2 - D \cdot .
To see this, consider the blow-up $\pi: \widetilde \to X$ of ''X'' along ''P'' and let $g: \widetilde = \pi^Z \to Z$, the strict transform of ''Z''. By the formula at #Properties,
:s(Z, X) = g_* ( widetilde - g_*(\widetilde \cdot widetilde.
Since $\widetilde = \pi^* D + mE$ where $E = \pi^ P$, the formula above results.

Multiplicity along a subvariety

Let $(A, \mathfrak)$ be the local ring of a variety ''X'' at a closed subvariety ''V'' codimension ''n'' (for example, ''V'' can be a closed point). Then $\operatorname_A(A/\mathfrak^t)$ is a polynomial of degree ''n'' in ''t'' for large ''t''; i.e., it can be written as $t^n +$ 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 /math> in $s(V, X)$ is $e(A)$.

References

Bibliography

*
*{{citation, mr=0061420
, 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

Intersection theory
Characteristic classes