Segre class

Suppose $C$ is a cone over $X$

, $q$ is the projection from the projective completion $\backslash mathbb\{P\}(C\; \backslash oplus\; 1)$ of $C$ to $X$, and $\backslash mathcal\{O\}(1)$ is the anti-tautological line bundle on $\backslash mathbb\{P\}(C\; \backslash oplus\; 1)$. Viewing the Chern class $c\_1(\backslash mathcal\{O\}(1))$ as a group endomorphism of the Chow group of $\backslash mathbb\{P\}(C\; \backslash oplus\; 1)$, the total Segre class of $C$ is given by:

:$s(C)\; =\; q\_*\; \backslash left(\; \backslash sum\_\{i\; \backslash geq\; 0\}\; c\_1(\backslash mathcal\{O\}(1))^\{i\}\; [\backslash mathbb\{P\}(C\; \backslash oplus\; 1)]\; \backslash right).$

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:

:$s\_i(C)\; =\; q\_*\; \backslash left(\; c\_1(\backslash mathcal\{O\}(1))^\{r+i\}\; [\backslash mathbb\{P\}(C\; \backslash oplus\; 1)]\; \backslash right).$

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

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

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).{{harvnb|Fulton|1998|loc=p.50.}}

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 $x\_1,\; \backslash dots,\; x\_k$ be Chern roots, i.e. formal eigenvalues of $\backslash frac\{\; i\; \backslash Omega\; \}\{\; 2\backslash pi\}$ where $\backslash Omega$ is a curvature of a connection on $E$.

While the Chern class c(E) is written as

:$c(E)\; =\; \backslash prod\_\{i=1\}^\{k\}\; (1+x\_i)\; =\; c\_0\; +\; c\_1\; +\; \backslash cdots\; +\; c\_k\; \backslash ,$

where $c\_i$ is an elementary symmetric polynomial of degree $i$ in variables $x\_1,\; \backslash dots,\; x\_k$

the Segre for the dual bundle $E^\backslash vee$ which has Chern roots $-x\_1,\; \backslash dots,\; -x\_k$ is written as

:$s(E^\backslash vee)\; =\; \backslash prod\_\{i=1\}^\{k\}\; \backslash frac\; \{1\}\; \{\; 1\; -\; x\_i\; \}\; =\; s\_0\; +\; s\_1\; +\; \backslash cdots$

Expanding the above expression in powers of $x\_1,\; \backslash dots\; x\_k$ one can see that $s\_i\; (E^\backslash vee)$ is represented by

a complete homogeneous symmetric polynomial of $x\_1,\; \backslash dots\; x\_k$

Here are some basic properties.

• For any cone C (e.g., a vector bundle), $s(C\; \backslash oplus\; 1)\; =\; s(C)$.{{harvnb|Fulton|1998|loc=Example 4.1.1.}}
• For a cone C and a vector bundle E,
• :$c(E)s(C\; \backslash oplus\; E)\; =\; s(C).$

{{harvnb|Fulton|1998|loc=Example 4.1.5.}}

• If E is a vector bundle, then{{harvnb|Fulton|1998|loc=Proposition 3.1.}}
• :$s\_i(E)\; =\; 0$ for .
• :$s\_0(E)$ is the identity operator.
• :$s\_i(E)\; \backslash circ\; s\_j(F)\; =\; s\_j(F)\; \backslash circ\; s\_i(E)$ 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,
• :$s\_p(E\; \backslash otimes\; L)\; =\; \backslash sum\_\{i=0\}^p\; (-1)^\{p-i\}\; \backslash binom\{e+p\}\{e+i\}\; s\_i(E)\; c\_1(L)^\{p-i\}.${{harvnb|Fulton|1998|loc=Example 3.1.1.}}

A key property of a Segre class is birational invariance: this is contained in the following. Let $p:\; X\; \backslash to\; Y$ 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 $W\; \backslash subset\; Y$, $V\; =\; p^\{-1\}(W)$ and $p\_V:\; V\; \backslash to\; W$ the restriction of $p$,

:$\{p\_V\}\_*(s(V,\; X))\; =\; \backslash operatorname\{deg\}(p)\; \backslash ,\; s(W,\; Y).${{harvnb|Fulton|1998|loc=Proposition 4.2. (a)}}

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

:$\{f\_V\}^*(s(W,\; Y))\; =\; s(V,\; X).${{harvnb|Fulton|1998|loc=Proposition 4.2. (b)}}

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

:$\backslash begin\{align\}$

s(E, \widetilde{X}) &= c(\mathcal{O}_E(E))^{-1} [E] \\

&= [E] - E \cdot [E] + E \cdot (E \cdot [E]) + \cdots,

\end{align}

where we used the notation $D\; \backslash cdot\; \backslash alpha\; =\; c\_1(\backslash mathcal\{O\}(D))\backslash alpha$.{{harvnb|Fulton|1998|loc=§ 2.5.}} Thus,

:$s(Z,\; X)\; =\; g\_*\; \backslash left(\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; (-1)^\{k-1\}\; E^k\; \backslash right)$

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

Let Z be a smooth curve that is a complete intersection of effective Cartier divisors $D\_1,\; \backslash dots,\; D\_n$ on a variety X. Assume the dimension of X is n + 1. Then the Segre class of the normal cone $C\_\{Z/X\}$ to $Z\; \backslash hookrightarrow\; X$ is:{{harvnb|Fulton|1998|loc=Example 9.1.1.}}

:$s(C\_\{Z/X\})\; =\; [Z]\; -\; \backslash sum\_\{i=1\}^n\; D\_i\; \backslash cdot\; [Z].$

Indeed, for example, if Z is regularly embedded into X, then, since $C\_\{Z/X\}\; =\; N\_\{Z/X\}$ is the normal bundle and $N\_\{Z/X\}\; =\; \backslash bigoplus\_\{i=1\}^n\; N\_\{D\_i/X\}|\_Z$ (see Normal cone#Properties), we have:

:$s(C\_\{Z/X\})\; =\; c(N\_\{Z/X\})^\{-1\}[Z]\; =\; \backslash prod\_\{i=1\}^d\; (1-c\_1(\backslash mathcal\{O\}\_X(D\_i)))\; [Z]\; =\; [Z]\; -\; \backslash sum\_\{i=1\}^n\; D\_i\; \backslash cdot\; [Z].$

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

:$0\; \backslash to\; S\; \backslash to\; p^*\; \backslash mathbb\{C\}^3\; \backslash to\; Q\; \backslash to\; 0$

where $S,\; Q$ are the tautological sub and quotient bundles. With $E\; =\; \backslash operatorname\{Sym\}^2(S^*\; \backslash otimes\; Q^*)$, the projective bundle $q:\; X\; =\; \backslash mathbb\{P\}(E)\; \backslash to\; \backslash breve\{\backslash mathbb\{P\}^3\}$ is the variety of conics in $\backslash mathbb\{P\}^3$. With $\backslash beta\; =\; c\_1(Q^*)$, we have $c(S^*\; \backslash otimes\; Q^*)\; =\; 2\; \backslash beta\; +\; 2\backslash beta^2$ and so, using Chern class#Computation formulae,

:$c(E)\; =\; 1\; +\; 8\; \backslash beta\; +\; 30\; \backslash beta^2\; +\; 60\; \backslash beta^3$

and thus

:$s(E)\; =\; 1\; +\; 8\; h\; +\; 34\; h^2\; +\; 92\; h^3$

where $h\; =\; -\backslash beta\; =\; c\_1(Q).$ 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}}

Let X be a surface and $A,\; B,\; D$ effective Cartier divisors on it. Let $Z\; \backslash subset\; X$ 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{{harvnb|Fulton|1998|loc=Example 4.2.2.}}

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

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

:$s(Z,\; X)\; =\; g\_*\; ([\backslash widetilde\{Z\}])\; -\; g\_*(\backslash widetilde\{Z\}\; \backslash cdot\; [\backslash widetilde\{Z\}]).$

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

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

The Segre class $s(V,\; X)$ of $V\; \backslash subset\; X$ encodes this multiplicity: the coefficient of $[V]$ in $s(V,\; X)$ is $e(A)$.{{harvnb|Fulton|1998|loc=Example 4.3.1.}}

{{reflist}}

• {{Citation | title=Intersection theory | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd | first=William |last=Fulton}}
• {{citation|mr=0061420

|last=Segre|first= Beniamino

|title=Nuovi metodi e resultati nella geometria sulle varietà algebriche|language=Italian

|journal=Ann. Mat. Pura Appl. |issue=4|volume= 35|year=1953|pages=1–127 |authorlink=Beniamino Segre}}

Category:Intersection theory

Category:Characteristic classes