# 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).

## Definition

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

${\displaystyle s(C)=q_{*}\left(\sum _{i\geq 0}c_{1}({\mathcal {O}}(1))^{i}[\mathbb {P} (C\oplus 1)]\right).}$

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

${\displaystyle s_{i}(C)=q_{*}\left(c_{1}({\mathcal {O}}(1))^{r+i}[\mathbb {P} (C\oplus 1)]\right).}$

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

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

### Relation to Chern classes for vector bundles

For a holomorphic vector bundle ${\displaystyle E}$ over a complex manifold ${\displaystyle M}$ a total Segre class ${\displaystyle s(E)}$ is the inverse to the total Chern class ${\displaystyle c(E)}$, see e.g.[1]

Explicitly, for a total Chern class

${\displaystyle c(E)=1+c_{1}(E)+c_{2}(E)+\cdots \,}$

one gets the total Segre class

${\displaystyle s(E)=1+s_{1}(E)+s_{2}(E)+\cdots \,}$

where

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

Let ${\displaystyle x_{1},\dots ,x_{k}}$ be Chern roots, i.e. formal eigenvalues of ${\displaystyle {\frac {i\Omega }{2\pi }}}$ where ${\displaystyle \Omega }$ is a curvature of a connection on ${\displaystyle E}$.

While the Chern class c(E) is written as

${\displaystyle c(E)=\prod _{i=1}^{k}(1+x_{i})=c_{0}+c_{1}+\cdots +c_{k}\,}$

where ${\displaystyle c_{i}}$ is an elementary symmetric polynomial of degree ${\displaystyle i}$ in variables ${\displaystyle x_{1},\dots ,x_{k}}$

the Segre for the dual bundle ${\displaystyle E^{\vee }}$ which has Chern roots ${\displaystyle -x_{1},\dots ,-x_{k}}$ is written as

${\displaystyle s(E^{\vee })=\prod _{i=1}^{k}{\frac {1}{1-x_{i}}}=s_{0}+s_{1}+\cdots }$

Expanding the above expression in powers of ${\displaystyle x_{1},\dots x_{k}}$ one can see that ${\displaystyle s_{i}(E^{\vee })}$ is represented by a complete homogeneous symmetric polynomial of ${\displaystyle x_{1},\dots x_{k}}$

## Properties

Here are some basic properties.

• For any cone C (e.g., a vector bundle), ${\displaystyle s(C\oplus 1)=s(C)}$.[2]
• For a cone C and a vector bundle E,
${\displaystyle c(E)s(C\oplus E)=s(C).}$[3]
• If E is a vector bundle, then[4]
${\displaystyle s_{i}(E)=0}$ for ${\displaystyle i<0}$.
${\displaystyle s_{0}(E)}$ is the identity operator.
${\displaystyle s_{i}(E)\circ s_{j}(F)=s_{j}(F)\circ s_{i}(E)}$ for another vector bundle F.
• If L is a line bundle, then ${\displaystyle s_{1}(L)=-c_{1}(L)}$, minus the first Chern class of L.[4]
• If E is a vector bundle of rank ${\displaystyle e+1}$, then, for a line bundle L,
${\displaystyle s_{p}(E\otimes L)=\sum _{i=0}^{p}(-1)^{p-i}{\binom {e+p}{e+i}}s_{i}(E)c_{1}(L)^{p-i}.}$[5]

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

${\displaystyle {p_{V}}_{*}(s(V,X))=\operatorname {deg} (p)\,s(W,Y).}$[6]

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

${\displaystyle {f_{V}}^{*}(s(W,Y))=s(V,X).}$[7]

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

{\displaystyle {\begin{aligned}s(E,{\widetilde {X}})&=c({\mathcal {O}}_{E}(E))^{-1}[E]\\&=[E]-E\cdot [E]+E\cdot (E\cdot [E])+\cdots ,\end{aligned}}}

where we used the notation ${\displaystyle D\cdot \alpha =c_{1}({\mathcal {O}}(D))\alpha }$.[8] Thus,

${\displaystyle s(Z,X)=g_{*}\left(\sum _{k=1}^{\infty }(-1)^{k-1}E^{k}\right)}$

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

## Examples

### Example 1

Let Z be a smooth curve that is a complete intersection of effective Cartier divisors ${\displaystyle D_{1},\dots ,D_{n}}$ on a variety X. Assume the dimension of X is n + 1. Then the Segre class of the normal cone ${\displaystyle C_{Z/X}}$ to ${\displaystyle Z\hookrightarrow X}$ is:[9]

${\displaystyle s(C_{Z/X})=[Z]-\sum _{i=1}^{n}D_{i}\cdot [Z].}$

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

${\displaystyle s(C_{Z/X})=c(N_{Z/X})^{-1}[Z]=\prod _{i=1}^{d}(1-c_{1}({\mathcal {O}}_{X}(D_{i})))[Z]=[Z]-\sum _{i=1}^{n}D_{i}\cdot [Z].}$

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

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

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

${\displaystyle c(E)=1+8\beta +30\beta ^{2}+60\beta ^{3}}$

and thus

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

where ${\displaystyle h=-\beta =c_{1}(Q).}$ The coefficients in ${\displaystyle s(E)}$ 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 ${\displaystyle A,B,D}$ effective Cartier divisors on it. Let ${\displaystyle Z\subset X}$ be the scheme-theoretic intersection of ${\displaystyle A+D}$ and ${\displaystyle B+D}$ (viewing those divisors as closed subschemes). For simplicity, suppose ${\displaystyle A,B}$ meet only at a single point P with the same multiplicity m and that P is a smooth point of X. Then[10]

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

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

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

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

## Multiplicity along a subvariety

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

The Segre class ${\displaystyle s(V,X)}$ of ${\displaystyle V\subset X}$ encodes this multiplicity: the coefficient of ${\displaystyle [V]}$ in ${\displaystyle s(V,X)}$ is ${\displaystyle e(A)}$.[11]

## References

1. Fulton W. (1998). Intersection theory, p.50. Springer, 1998.
2. Fulton, Example 4.1.1.
3. Fulton, Example 4.1.5.
4. Fulton, Proposition 3.1.
5. Fulton, Example 3.1.1.
6. Fulton, Proposition 4.2. (a)
7. Fulton, Proposition 4.2. (b)
8. Fulton, § 2.5.
9. Fulton, Example 9.1.1.
10. Fulton, Example 4.2.2.
11. Fulton, Example 4.3.1.
• Segre, Beniamino (1953), "Nuovi metodi e resultati nella geometria sulle varietà algebriche", Ann. Mat. Pura Appl. (in Italian), 35 (4): 1–127, MR 0061420
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