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 $𝒪(1)$ is the anti-tautological line bundle on $\mathbb{P}(C\oplus 1)$. Viewing the Chern class $c_1(𝒪(1))$ as a group endomorphism of the Chow group of $\mathbb{P}(C\oplus 1)$, the total Segre class of $C$ is given by:

$s(C)=q_{\ast }(\sum _{i\ge 0}{c}_1(𝒪(1){)}^i[\mathbb{P}(C\oplus 1)]).$

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:

$s_i(C)=q_{\ast }(c_1(𝒪(1){)}^{r+i}[\mathbb{P}(C\oplus 1)]).$

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 $𝒪$.

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

$c_1(E)=-s_1(E),c_2(E)=s_1(E{)}^2-s_2(E),\dots ,c_n(E)=-s_1(E)c_{n-1}(E)-s_2(E)c_{n-2}(E)-\cdots -s_n(E)$

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,

  $s_p(E\otimes L)=\sum _{i=0}^p(-1{)}^{p-i}(\genfrac{}{}{0ex}{}{e+p}{e+i})s_i(E)c_1(L{)}^{p-i}.$

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 $p$,

${p_V}_{\ast }(s(V,X))=\mathrm{deg}(p)s(W,Y).$

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 $f$,

${f_V}^{\ast }(s(W,Y))=s(V,X).$

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

$\begin{array}{cc}s(E,\tilde{X})& =c({𝒪}_E(E){)}^{-1}[E]\\ & =[E]-E\cdot [E]+E\cdot (E\cdot [E])+\cdots ,\end{array}$

where we used the notation $D\cdot \alpha =c_1(𝒪(D))\alpha$. Thus,

$s(Z,X)=g_{\ast }(\sum _{k=1}^{\mathrm{\infty }}(-1{)}^{k-1}{E}^k)$

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 $N_{Z/X}=\underset{i=1}{\overset{n}{⨁}}{N}_{D_i/X}{|}_Z$ (see Normal cone#Properties), we have:

$s(C_{Z/X})=c(N_{Z/X}{)}^{-1}[Z]=\prod _{i=1}^d(1-c_1({𝒪}_X(D_i)))[Z]=[Z]-\sum _{i=1}^n{D}_i\cdot [Z].$

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 $p:\breve{{\mathbb{P}}^3}\to \ast$ parametrizing the 2-planes in ${\mathbb{P}}^3$, consider the tautological exact sequence

$0\to S\to p^{\ast }{ℂ}^3\to Q\to 0$

where $S,Q$ are the tautological sub and quotient bundles. With $E={\mathrm{Sym}}^2(S^{\ast }\otimes Q^{\ast })$, the projective bundle $q:X=\mathbb{P}(E)\to \breve{{\mathbb{P}}^3}$ is the variety of conics in ${\mathbb{P}}^3$. With $\beta =c_1(Q^{\ast })$, we have $c(S^{\ast }\otimes Q^{\ast })=2\beta +2{\beta }^2$ 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.

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 $\pi :\tilde{X}\to X$ of X along P and let $g:\tilde{Z}={\pi }^{-1}Z\to Z$, the strict transform of Z. By the formula at #Properties,

$s(Z,X)=g_{\ast }([\tilde{Z}])-g_{\ast }(\tilde{Z}\cdot [\tilde{Z}]).$

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

Multiplicity along a subvariety

Let $(A,𝔪)$ be the local ring of a variety X at a closed subvariety V codimension n (for example, V can be a closed point). Then ${\mathrm{length}}_A(A/{𝔪}^t)$ is a polynomial of degree n in t for large t; i.e., it can be written as $\frac{e(A{)}^n}{n!}{t}^n+$ 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

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