Computation of Segre numbers: an application to Whitney stratifications

Rüdiger Achilles (Bologna, Italy)

achilles@dm.unibo.it http://www.dm.unibo.it/˜ achilles

Commutative and Non-Commutative Algebraic Geometry Chisinau, June 6–11, 2004 with Mirella Manaresi (Bologna) and Davide Aliffi (Bologna)

1. R. Achilles and D. Aliffi, Segre: a script for the REDUCE package CALI. Bologna, 1999-2001. Available at

http://www.dm.unibo.it/~achilles/segre/.

2. R. Achilles and M. Manaresi, Self-intersections of surfaces and Whitney stratifications. Proc. Edinburgh Math. Soc.(2) 46 (2003), 545–559.

1. Stückrad-Vogel intersection cycle Stückrad - Vogel, 1982; Flenner - O’Carroll - Vogel, Joins and Intersections, Springer, 1999.

2. Generalized Samuel multiplicities and Segre numbers Achilles - Manaresi, Math. Ann. 309 (1997), 573–591; Gaffney - Gassler, J. Algeb. Geom. 8 (1999), 695–736.

3. Polar multiplicities and Whitney stratifications Teissier, SLN 961 (1982), 314–491.

4. Result and examples Stückrad-Vogel intersection cycle X, Y ⊂ P^{n}, equidimensional

v(X, Y) = v_{0} + v_{1} + … if v_{k} ≠ 0 then v_{k} is of dimension k (k = 0, 1, …)

Example 1

X : x^{2} − yz = 0  cone
Y : x = y = 0  line on the cone

cut out Y by generic planes

u_{11} x + u_{12} y = 0,
u_{21} x + u_{22} y = 0;

and intersect X step by step with these generic planes components lying in X ∩ Y are collected in the cycle v(X, Y) and the rest is intersected with the next generic plane:

First step

(x^{2} − yz, u_{11} x + u_{12} y) = (x, y) ∩ (u_{12} x + u_{11} z, u_{11} x + u_{12} y)

Second step

(u_{12} x + u_{11} z, u_{11} x + u_{12} y , u_{21} x + u_{22} y) = (x, y, z).

⇒ v(X, Y) = Y + O.

Example 2 In P_{K}^{3}

X : xy − zt = 0  non-singular quadric
Y : x = z = 0  line on it

First step

(xy − zt, u_{11} x + u_{12} z) = (x, z) ∩ (u_{12} y + u_{11} t, u_{11} x + u_{12} z) Second step

(u_{12} y + u_{11} t, u_{11} x + u_{12} z , u_{21} x + u_{22} z) = (x, z, u_{12} y + u_{11} t) .

⇒ v(X, Y) = Y + P, where P is a non K-rational (or a movable) point Theorem 1 (Flenner-Manaresi, 1997) X, Y ⊂ P_{K}^{n} varieties, e := dim X + dim Y − n. Let k ∈ Z such that e − 1 ≤ k ≤ dim X ∩ Y − 1, L := K(u),

p: X_{L} ∩ Y_{L} → P_{L}^{n+e−k−1}

the generic linear projection and R(p) := ramification locus of p. Then dim R(p) ≤ k, and the associated k-cycle [R(p)]_{k} is just v_{k}(X, Y) on {x ∈ X ∩ Y | X, Y, X ∩ Y smooth at x}.

Definition 1

X, Y ⊆ Z closed subschemes of an algebraic K-scheme X, f: Z → N a morphism to an algebraic manifold with dim N ≥ dim X + dim Y − dim X ∩ Y. Then R(f) is defined to be the degeneracy locus of

f^{∗}(Ω_{N}^{1}) ⊗ O_{X∩Y} → Ω_{X∪Y}^{1} ⊗ O_{X∩Y} .

Generalized Samuel multiplicities P. Samuel, 1951 (A, 𝔪) noetherian local ring I of dimension d an 𝔪-primary ideal (that is, an ideal of finite colength)

H_{I}^{(0)} (j) := length (I^{j}/I^{j+1} )

Hilbert-Samuel function

H_{I}^{(1)} (j) := j ; ∑ ; k=0 H_{I}^{(0)} (j) = length (A/I^{j+1} ) if j ≫ 1,
= e_{0} \binom{j + d}{d} − e_{1} \binom{j + d − 1}{d − 1} + … + (−1)^{d}e_{d}

write e(I, A) for e_{0}.

Achilles-Manaresi, 1997

I not necessarily 𝔪-primary pass from the associated graded ring

G_{I}(A) := ⊕_{j≥0}I^{j}/I^{j+1}

to the bigraded ring

R = ⊕_{i,j≥0} R_{ij} = G_{𝔪}^{i}(G_{I}^{j}(A))
= ⊕_{i,j≥0} (𝔪^{i}I^{j} + I^{j+1} )/(𝔪^{i+1} I^{j} + I^{j+1} )

R_{00} = A/𝔪, have Hilbert function H^{(0,0)} (i, j) = dim R_{ij}, twofold sum transform H^{(1,1)} (i, j) := ∑_{q=0}^{j} ∑_{p=0}^{i} H^{(0,0)} (p, q), for both i, j ≫ 1 becomes a polynomial which can be written in the form

_{k+l≤d}^{∑} a_{k,l}^{(1,1)} \binom{i + k}{k}\binom{j + l}{l},

Define the generalized Samuel multiplicity to be

(a_{0,d}^{(1,1)} , a_{1,d−1}^{(1,1)} , …, a_{d,0}^{(1,1)} ) =: (c_{0}, c_{1}, …, c_{d}).

Theorem 2 Set q := dim (A/I), G := G_{I}(A), s := dim G/𝔪G. Then

1. c_{k} = 0 for k < d − s and k > q;

2. c_{d−s} = ∑_{P} e(𝔪G_{P}) · e(G/P), where P runs over all highest dimensional associated prime ideals of G/𝔪G such that dim G/P + dim G_{P} = dim G; 3. c_{q} = ∑_{𝔭} e(A_{𝔭}) · e(A/𝔭), where 𝔭 runs over all highest dimen-

sional associated prime ideals of A/𝔭 such that dim A/𝔭 + dim A_{𝔭} = dim A;

4. If A is the ring of the ruled join J := J(X, Y) ⊂ P^{2n+1} (localized at the irrelevant homogeneous maximal ideal), I the ideal of the ‘diagonal’ Δ ⊂ P^{2n+1} and π: J → XY the canonical projection onto the embedded join XY ⊂ P^{n}, then

c_{0} = deg _{Γ⊆J}^{∑} deg (Γ/XY) deg (π(Γ)),
c_{k} = deg v_{k−1}, k = 1, …, d − 1 = dim J,
c_{d} = 0.

Here the sum is taken over all irreducible components Γ of J with the induced scheme structure and π(Γ) denotes the closure of π(Γ \ Δ) ⊆ P_{K}^{n} equipped with its reduced structure. If K is algebraically closed and X, Y are irreducible (but not necessarily reduced), then

c_{0} = deg (J/XY) deg (XY).

Segre numbers of Gaffney and Gassler, 1999

A = O_{X,0}, (X, 0) ⊂ (C^{n}, 0) a reduced closed analytic space of pure dimension d and I an ideal which defines a nowhere dense subspace of (X, 0). They considered the blowup of X along I

X × P^{μ(I)−1} ⊃ Bl_{I}(X) →^{b} X

with exceptional divisor E and defined the kth Segre number as

e_{k}(I, Y) := mult _{0}(b_{∗}(H_{1} ⋯ H_{k−1} · E · Bl_{I}X)) ,

where H_{i} is a generic hyperplane on Bl_{I}X induced by one in P^{μ(I)−1} . Note that

e_{k}(I, Y) = c_{d−k}(A, I), k = 1, …, d,

that is, Segre numbers are a special case of the generalized Samuel multiplicity.

Our generalized Samuel multiplicities are also related to the degrees of Segre classes of cones and subvarieties: W ⊂ V; let W, V be nonsingular, N_{W} V the normal bundle with Chern classes c_{i}(N_{W} V) ∩ [W] and Segre classes s_{i}(N_{W} V) ∩ [W],

1 + c_{1} + c_{2} ⋯ = (1 + s_{1} + s_{2} + ⋯)^{−1}.

In general there is only a normal cone, C = C_{W} V of V along W. The total Segre class s(W, V) ∈ A_{∗}W is defined as follows: if W = V then s(W, V) = [V]. Otherwise Ṽ = Bl_{W} V, E = P(C) exceptional divisor, η: E → W projection, d := dim V = dim Ṽ. The i-fold self intersections E^{i} = E ∗ … ∗ E are well defined classes in A_{d−i}(E).

s(W, V) := _{i≥1}^{∑}(−1)^{i−1}η_{∗}(E^{i})

(B. Segre, 1953, sottovarietà covarianti)

Now N_{E}Ṽ = OṼ (E)|_{E=P(C)} = O_{C}(−1) is the dual of the canonical line bundle O_{C} on P(C). It follows

E^{i} = (−1)^{i−1}c_{1}((O_{C}(1))^{i−1}) ∩ [P(C)], hence

s = s(C) = s(W, V) = _{i≥0}^{∑} η_{∗}(c_{1}(O_{C}(1))^{i−1} ∩ [P(C)])

which makes sense for every cone C on a scheme W under the assumption that there is no irreducible component C′ of C with P(C′) = ∅. By s_{i} we denote the part of s of dimension i, and by s^{i} the part of codimension i in V. Thus, if V is equidimensional (which we assume), then s_{i} = s^{d−i}.

Connection with Samuel multiplicity: W ⊂ V irreducible and reduced, r := codim (W, V) > 0, q = d − r.

e(O_{V,W}) = e_{W} V [W] = η_{∗}(c_{1}(O_{C}(1))^{r−1} ∩ [P(C)])
= (−1)^{q−1}η_{∗}(E^{r−1}),

that is, deg s^{r}(C_{W} V) = deg s_{q}(C_{W} V) = c_{q}. In the situation of Theorem 2, (4):

c_{k} = q ; ∑ ; i=k \binom{d − k − 1}{d − i − 1} deg s_{i}(C_{Δ∩J} J)

and

deg s^{k} = deg s_{d−k} = k ; ∑ ; i=0 \binom{k − 1}{i − 1}(−1)^{k−i}c_{d−i},

k = 0, …, d − 1. Convention: \binom{m}{−1} := 0 for m ≥ 0, \binom{−1}{−1} := 1.

More general, if V is an equidimensional algebraic scheme over the base field K, L a line bundle of degree δ on V, σ_{1}, … σ_{t} ∈ H^{0}(V, L) and W := V (σ_{1}) ∩ … ∩ V (σ_{t}), then

c_{k} = q ; ∑ ; i=k \binom{d − k − 1}{d − i − 1}δ^{k−i} deg s_{i}(C_{Δ∩J} J)

and

deg s^{k} = deg s_{d−k} = k ; ∑ ; i=0 \binom{k − 1}{i − 1}(−δ)^{k−i}c_{d−i},

k = 0, …, d − 1.

Definition 2 Y ⊂ X complex projective varieties, Y non-singular. (Xreg , Y) satisfies the Whitney conditions at a point x_{0} ∈ Y if for each sequence (x_{i}) of points of Xreg and each sequence (y_{i}) of points of Y both converging to x_{0} and such that the limits lim_{x_{i}→x_{0}} T_{x_{i}}X and lim_{x_{i},y_{i}→x_{0}} x_{i}y_{i} exist in the Grassmannians G(d, n) and G(1, n) respectively, one has:

(a) lim_{x_{i}→x_{0}} T_{x_{i}}X ⊃ T_{x_{0}}Y ,
(b) lim_{x_{i}→x_{0}} T_{x_{i}}X ⊃ lim_{x_{i},y_{i}→x_{0}} x_{i}y_{i} .

We remark that (b) implies (a).

Definition 3 A Whitney stratification of X (d = dim X) is given by a filtration of X by algebraic sets F_{i}

X = F_{0} ⊇ F_{1} ⊇ … ⊇ F_{d+1} = ∅

such that

(i) F_{i} \ F_{i+1} is either empty or is a non-singular quasi-projective variety of pure codimension i. (The connected components of F_{i} \ F_{i+1} are called the strata of the stratification.)

(ii) Whenever S_{j} and S_{k} are connected components of F_{i} \ F_{i+1} and F_{l} \ F_{l+1} respectively with S_{j} ⊂ S_{k}, then the pair (S_{k}, S_{j}) satisfies the Whitney conditions (a) and (b).

Polar varieties

L_{(k)} = (n − d + k − 2)-dimensional linear subspace of P^{n}, 1 ≤ k ≤ d = dim X. The kth polar variety (or polar locus) of X associated with L_{(k)} is

P (L_{k}, X) := closure of {x ∈ Xreg | dim (T_{x}X ∩ L_{(k)}) ≥ k − 1} .

For k = 0 we set P (L_{(0)} , X) := X.

If L_{(k)} is generic, we write P_{k}(X) = P (L_{(k)}, X) and if

L_{(0)} ⊂ L_{(1)} ⊂ … ⊂ L_{(d)}

is a generic flag, then we have

X = P_{0}(X) ⊃ P_{1}(X) ⊃ … ⊃ P_{d}(X) .

Let x ∈ X. Teissier showed that the sequence of multiplicities

m_{0} = e_{x}(P_{0}(X)), …, m_{d−1} = e_{x}(P_{d−1}(X))

does not depend upon the choice of the general flag.

Theorem 3 (Teissier, 1982) The pair (Xreg , Y) satisfies the Whitney conditions in x_{0} if and only if the sequence of polar multiplicities

m_{0} = e_{y}(X), m_{1} = e_{y}(P_{1}(X)), …, m_{d−1} = e_{y}(P_{d−1}(X))

is locally constant in Y around x_{0}.

We propose the following function g to measure the singularity of X in a point x of X:

A := O_{X×X,(x,x)}, I := diagonal ideal in A,

g(x) := d ; ∑ ; i=0 c_{i}(I, A) = e(G_{I}(A)) .

Note that dim A = 2d, c_{d+1} = ⋯ = c_{2d} = 0 and that

(c_{0}(I, A), c_{1}(I, A), …, c_{d}(I, A))

is a refinement of the multiplicity

c_{d}(I, A) = e_{x}X = e(O_{X,x})

of X at x.

Theorem 4 (Achilles-Manaresi, 2003) Let X ⊂ P^{n} be a (reduced) surface and x ∈ X be a closed point. Then

X_{j} := {x ∈ X | g(x) ≥ j}, j = 0, 1, …

are closed subschemes of X or empty, and the connected components of

S_{g}(j) := g^{−1}(j) = X_{j} \ X_{j+1}

are the strata of a Whitney stratification of X (the coarsest one if n = 3).

Generalized Samuel multiplicities (c_{2}, c_{1}, c_{0}) and polar multiplicities (m_{0}, m_{1}) (both ordered by codimension) for the surface in C^{3} (or in P^{3}) defined by the equation

1. x^{4} + y^{4} − xyz = 0;

2. y^{2} − x^{3} − x^{2}z^{2} = 0. In both cases the Whitney stratification is given by

surface ⊂ z-axis ⊂ origin .

We illustrate the second example:

[figure: z; y; x; (2, 2, 0) (2, 0); (1, 0, 0) (1, 0); (2, 3, 0) (2, 1)]