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{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 \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 $$\pi$$.

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), 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,
 * $$c(E) = 1 + 8 \beta + 30 \beta^2 + 60 \beta^3$$

and thus
 * $$s(E) = 1 + 8 h + 34 h^2 + 92 h^3$$

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 $$\pi: \widetilde{X} \to X$$ of X along P and let $$g: \widetilde{Z} = \pi^{-1}Z \to Z$$, the strict transform of Z. By the formula at ,
 * $$s(Z, X) = g_* ([\widetilde{Z}]) - g_*(\widetilde{Z} \cdot [\widetilde{Z}]).$$

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 $$e(A)$$.