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).¹ 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.²

Suppose that ${\displaystyle C}$ is a cone over ${\displaystyle X}$, that ${\displaystyle q}$ is the projection from the projective completion ${\displaystyle \mathbb {P} (C\oplus 1)}$ of ${\displaystyle C}$ to ${\displaystyle X}$, and that ${\displaystyle {\mathcal {O}}(1)}$ 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 ${\displaystyle Z\hookrightarrow X}$.

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. Fulton (1998).³

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 ${\displaystyle x_{1},\dots x_{k}}$.

Here are some basic properties.

• For any cone C (e.g., a vector bundle), ${\displaystyle s(C\oplus 1)=s(C)}$.⁴
• For a cone C and a vector bundle E,

  ${\displaystyle c(E)s(C\oplus E)=s(C).}$⁵

• If E is a vector bundle, then⁶

  ${\displaystyle s_{i}(E)=0}$ 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.⁶
• If E is a vector bundle of rank ${\displaystyle e+1}$, then, for a line bundle L,

  ${\displaystyle s_{p}(E\otimes L)=\sum _{i=0}^{p}(-1)^{p-i}{\binom {e+p}{e+i}}s_{i}(E)c_{1}(L)^{p-i}.}$⁷

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 ${\displaystyle p}$,

${\displaystyle {p_{V}}_{*}(s(V,X))=\operatorname {deg} (p)\,s(W,Y).}$⁸

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 ${\displaystyle f}$,

${\displaystyle {f_{V}}^{*}(s(W,Y))=s(V,X).}$⁹

A basic example of birational 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 ${\displaystyle D\cdot \alpha =c_{1}({\mathcal {O}}(D))\alpha }$.¹⁰ 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 ${\displaystyle \pi }$.

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:¹¹

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

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:

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

The following is Example 3.2.22. of Fulton (1998).² 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.

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. Then¹²

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

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.

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 ${\displaystyle e(A)}$.¹³