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 $\mathbb {P} (C\oplus 1)$ 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 $i$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:

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

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).

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), $s(C\oplus 1)=s(C)$.
• 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 $p$,

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

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 $\pi :{\widetilde {X}}\to X$ 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 ${\mathcal {O}}_{E}(E):={\mathcal {O}}_{X}(E)|_{E}$,

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

where we used the notation $D\cdot \alpha =c_{1}({\mathcal {O}}(D))\alpha$. 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:

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

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

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

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

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

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 $e(A)$.

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