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.

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 ${\displaystyle Z\hookrightarrow X}$.

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).

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

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

• If E is a vector bundle, then

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

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

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 ${\displaystyle p}$,

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

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 ${\displaystyle f}$,

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

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 ${\displaystyle D\cdot \alpha =c_{1}({\mathcal {O}}(D))\alpha }$. 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:

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

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). 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. Then

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

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 ${\displaystyle e(A)}$.

• 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