Proceedings of the Edinburgh Mathematical Society Submitted Paper

Paper 24 April 2003

SELF-INTERSECTIONS OF SURFACES AND WHITNEY STRATIFICATIONS^{∗}

RÜDIGER ACHILLES AND MIRELLA MANARESI

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126 Bologna, Italy (achilles@dm.unibo.it, manaresi@dm.unibo.it)

(Received )

Abstract Let X be a surface in C^{n} or P^{n} and let C_{X}(X × X) be the normal cone to X in X × X (diagonally embedded). For a point x ∈ X, denote by g(x) := e_{x}(C_{X}(X × X)) the multiplicity of C_{X}(X × X) at x. It is a former result of the authors that g(x) is the degree at x of the Stückrad–Vogel cycle v(X, X) = ∑_{C} j(X, X; C) [C] of the self-intersection of X, that is, g(x) = ∑_{C} j(X, X; C) e_{x}(C). We prove that the stratification of X by the multiplicity g(x) is a Whitney stratification, the canonical one if n = 3. The corresponding result for hypersurfaces in A^{n} or P^{n}, diagonally embedded in a multiple product with itself, was conjectured by L. van Gastel. This is also discussed, but remains open.

Keywords: Hypersurface singularities, normal cone, Whitney stratification.

AMS 2000 Mathematics subject classification: Primary 32S15 Secondary 13H15,14C17

1. Introduction

In his article [21], H. Whitney introduced the notion of a regular stratification (later called Whitney stratification) that turned out to be a very useful tool in the study of singular complex analytic spaces. Whitney’s regularity condition had been characterized numerically by B. Teissier [19] using polar multiplicities. We will use Teissier’s numerical criterion (see Sect. 2) and the algebraic approach to intersection theory by J. Stückrad and W. Vogel (see Sect. 3) to construct the canonical (or minimal) Whitney stratification of a surface X in P^{3}. More precisely, let C_{X}(X × X) be the normal cone to X in X × X (diagonally embedded). For a point x ∈ X, denote by g(x) := e_{x}(C_{X}(X × X)) the multiplicity of C_{X}(X × X) at x. In [4] we proved that g(x) is the degree at x of the Stückrad–Vogel cycle v(X, X) = ∑_{C} j(X, X; C) [C] of the self-intersection of X, that is, g(x) = ∑_{C} j(X, X; C) e_{x}(C); see Proposition 3.6 for details. In this paper, our main result Theorem 4.2 states that the pointwise degree g(x) of the Stückrad-Vogel intersection cycle of the self-intersection of X is a stratifying function which gives the canonical Whitney stratification. Our restriction to algebraic surfaces in P^{3} or P^{n} is only

∗ Investigation supported by the MURST and the University of Bologna, Funds for Selected Research Topics. The authors are members of the G.N.S.A.G.A. of the Italian CNR.

2 R. Achilles and M. Manaresi

for simplicity. In fact, using P. Tworzewski’s extension of the Stückrad-Vogel cycle to the analytic case (see [20]), one has the analogue result for complex analytic surfaces in C^{n}. Both in the projective and in the complex analytic case, the stratifying function g(x) can be calculated by computer algebra systems: see our examples in Sect. 4.

L. van Gastel conjectured that for hypersurfaces in P^{n} or C^{n}, the pointwise degree g(x) of the intersection cycle of a suitable multiple self-intersection should also give the canonical Whitney stratification. However, we were not able to prove that the g-constant strata are smooth. Hence we conclude our paper with some examples and open problems.

Notations. We denote the n-dimensional projective space over a field K by P_{K}^{n}. If not explicitly stated to the contrary, our base field will be always the field of complex numbers and we simply write P^{n}. By a variety or subvariety of P_{K}^{n} we mean a closed reduced (but possibly reducible) equidimensional subscheme of P_{K}^{n} without embedded components. A surface is a 2-dimensional variety and a hypersurface an (n − 1)-dimensional variety. The multiplicity of a scheme X at a point x, i. e., the multiplicity of the local ring O_{X,x}, will be denoted by m_{x}(X) = e_{x}(X) = e(O_{X,x}). Finally, X_{reg} stands for the set of closed points x of X such that O_{X,x} is a regular local ring, and Sing X is the complement of X_{reg} in the set of closed points of X.

2. Polar multiplicities and Whitney stratifications

Definition 2.1. Let X ⊂ P^{n} be a d-dimensional subvariety, with 0 ≤ d ≤ n − 1, and for each 0 ≤ k ≤ d let L_{(k)} be an (n − d + k − 2)-dimensional linear subspace of P^{n}. The polar variety (or polar locus) P (L_{(k)}, X) of X associated with L_{(k)} is the closure of

{x ∈ X_{reg} | dim(T_{x}X ∩ L_{(k)}) ≥ k − 1} .

If n − d + k − 2 = −1 we set P (L_{(k)}, X) = X and L_{(k)} = ∅. This happens only in the case k = 0 and d = n − 1.

Remark 2.2. If L_{(k)} is a generic linear subspace of P^{n}, then the polar variety P (L_{(k)}, X) is either empty or equidimensional of codimension k in X, and its degree does not depend upon L_{(k)}; see for example [18], Prop. 1.2 and the Transversality Lemma 1.3.

Notation 2.3. For L_{(k)} generic we set P_{k}(X) := P (L_{(k)}, X) and call it the general k-polar variety (or k-polar locus of X). If

L: L_{(0)} ⊂ L_{(1)} ⊂ … ⊂ L_{(d)}

is a generic flag of linear subspaces of P^{n} with dim L_{(k)} = n − d + k − 2, then we have the following inclusion of the corresponding polar varieties

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

If x ∈ X, we can consider the sequence of multiplicities

M_{X,x}(L) = (m_{x}(P_{0}(X)), …, m_{x}(P_{d−1}(X))) .

Self-intersections of surfaces and Whitney stratifications 3

This sequence does not depend upon the choice of the general flag L (see [19], IV 3.1, p. 425), and is in fact constant on a Zariski open subset of the variety of flags. The number m_{x}(P_{k}(X)), for k = 0, …, d − 1, is called the general k-polar multiplicity of X at x. Note that m_{x}(P_{d}(X)) = 0 since the germ of P_{d}(X) at the point x is empty; see [19], IV 3.3.

Remark 2.4. The polar varieties P_{k}(X) arise as sets of critical points of generic linear projections, see [14], (2.2.3), p. 462–463 or [19], p. 314.

We recall the definition of a critical point of a differentiable map.

Definition 2.5. Let f: M → N be a differentiable map. A point x ∈ M is called a critical point of f if the rank of the tangent map

f_{∗x}: T_{x}M → T_{f(x)}N

is smaller than the maximal possible one, that is,

rank f_{∗x} < min(dim_{x} M, dim_{f(x)} N) .

Remark 2.6. Maintaining the notation of Definition 2.1 and Remark 2.4, we observe that the k-polar variety P (L_{(k)}, X) is the closure of the set of critical points of the restriction to X of the linear projection on P^{d−k+1} with center L_{(k)}. If dim L_{(k)} = −1, that is, if d = n − 1 and k = 0, we can consider the affine cones X̂ of X and L̂_{(0)} = {0} of L_{(0)} = ∅ in A^{n+1} and take the linear projection X̂ \ {0} → A^{n}. The closure of the set of critical points of this map is X̂; in this sense, P_{0}(X) = X can also be regarded as a set of critical points.

Definition 2.7. Let X be d-dimensional complex projective variety and let Y be a non-singular subvariety of X. We say that the pair (X_{reg}, Y) satisfies the Whitney conditions at a point x_{0} ∈ Y if for each sequence (x_{i}) of points of X_{reg} 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}} \overline{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}} \overline{x_{i}y_{i}} .

We remark that (b) implies (a).

Theorem 2.8. (Teissier [19]) With the above notation, the pair (X_{reg}, Y) satisfies the Whitney conditions in x_{0} if and only if the sequence of polar multiplicities

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

is locally constant in Y around x_{0}.

Definition 2.9. With the above notation, a Whitney stratification of X is given by a filtration of X by algebraic sets F_{i}

X = F_{0} ⊇ F_{1} ⊇ ⋯ ⊇ F_{d+1} = ∅

with the following properties:

(i) For each i = 0, …, d, the locally closed subset F_{i} \ F_{i+1} is either empty or is a nonsingular quasi-projective variety of pure codimension i. (The connected components of F_{i} \ F_{i+1} (i = 0, …, d) 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 (i, l = 0, …, d) with S_{j} ⊂ \overline{S}_{k}, then the pair (S_{k}, S_{j}) satisfies the Whitney conditions (a) and (b).

Remark 2.10. In the case of a (reduced) surface X ⊂ P^{n}, the above Theorem 2.8 of Teissier implies that a Whitney stratification of X can be obtained as follows. Set F_{0} = X, F_{1} = Sing X (the singular locus of X), and, if F_{1,1}, … F_{1,m} are the 1-dimensional irreducible components of F_{1}, set

F_{2} = Sing F_{1} ∪ {0-dimensional components of F_{1}} ∪

∪ ⋃_{j=1}^{m} {x ∈ F_{1,j} | (m_{x}(X), m_{x}(P_{1}(X))) is different from its generic value in F_{1,j}}

and F_{3} = ∅. This is the so-called canonical Whitney stratification, which is characterized by the fact that the connected components of the differences F_{i} \ F_{i+1} are the strata of the minimal or coarsest Whitney stratification, see [19], Chap. 6.

3. Intersection cycle of Stückrad and Vogel

Let X, Y be equidimensional closed subschemes of P_{K}^{n} = Proj(K[X_{0}, …, X_{n}]), where K is an arbitrary field. For indeterminates U_{ij} (0 ≤ i, j ≤ n) let L be the pure transcendental field extension K(U_{ij})_{0≤i,j≤n} and X_{L} := X ⊗_{K} L, etc. Proving a Bezout theorem for improper intersections, Stückrad and Vogel (see [7]) introduced a cycle v(X, Y) = v^{0} + ⋯ + v^{n+1} on X_{L} ∩ Y_{L}, which is obtained by an intersection algorithm on the ruled join variety

J := J(X_{L}, Y_{L}) ⊂ P_{L}^{2n+1} = Proj(L[X_{0}, …, X_{n}, Y_{0}, …, Y_{n}])

as follows:

Let Δ be the “diagonal” subspace of P_{L}^{2n+1} given by the equations

X_{0} − Y_{0} = ⋯ = X_{n} − Y_{n} = 0 ,

let H_{i} ⊆ J be the divisor given by the equation

ℓ_{i} := ∑_{j=0}^{n} U_{ij}(X_{j} − Y_{j}) = 0

and set \underline{ℓ} := (ℓ_{0}, …, ℓ_{n}). Then one defines inductively cycles β^{k} and v^{k} by setting β^{0} := [J]. If β^{k} is already defined, decompose the intersection

β^{k} ∩ H_{k} = v^{k+1} + β^{k+1} (0 ≤ k ≤ dim J) ,

where the support of v^{k+1} lies in Δ and where no component of β^{k+1} is contained in Δ. It follows that v^{k} is a (dim J − k)-cycle on X_{L} ∩ Y_{L} ≅ J ∩ Δ. The part of dimension k of the cycle v(X, Y) := v(\underline{ℓ}, J) := ∑ v^{k} will be denoted by v_{k}, so that the upper index denotes the codimension in the ruled join and the lower one the dimension of the cycle. In general, the cycle v(X, Y) is defined over L. By a result of van Gastel ([11], Prop. 3.9), a K-rational irreducible subvariety C of X_{L} ∩ Y_{L} occurs in v(X, Y) if and only if C is a distinguished variety of the intersection of X and Y in the sense of Fulton ([9], p. 95), and this is equivalent to the maximality of the analytic spread (see [2]) or the maximality of the dimension of the so-called limit of join variety (see [8]).

Definition 3.1. ([7], 2.2.1) The cycle v(X, Y) is called the v-cycle of the intersection of X and Y. An irreducible subvariety C of X_{L} ∩Y_{L} is said to be a characteristic subvariety if C occurs in v(X, Y). The coefficient of C in v(X, Y) is denoted by j(X, Y; C). Thus

v(X, Y) = ∑_{C} j(X, Y; C) [C] ,

where C runs through the characteristic subvarieties. The set of all these subvarieties is denoted by C = C(X, Y). Moreover, the set of all elements of C which are defined over K is denoted by C_{rat} = C_{rat}(X, Y), that is, C_{rat} is the set of K-rational or distinguished or fixed subvarieties and C \ C_{rat} is the set of the so-called non K-rational or movable subvarieties of the intersection of X and Y.

Remark 3.2. The Stückrad-Vogel intersection cycle can be constructed in the same way also for more than two equidimensional closed subschemes X_{1}, …, X_{r} ⊆ P_{K}^{n} by applying the intersection algorithm to the join variety J(X_{1}, …, X_{r}) ⊆ P^{N}, N := r(n + 1) − 1 and generic hyperplanes H_{0}, …, H_{N−n−1} ⊂ P^{N} whose intersection is the diagonal Δ ⊂ P^{N}; see for example [7], 2.2.14. The resulting cycle will be denoted by v(X_{1}, …, X_{r}).

We observe that for improper intersections the associativity law does not hold, that is, in general, v(X_{1}, v(X_{2}, X_{3})) ≠ v(v(X_{1}, X_{2}), X_{3}) and both of these cycles may be also different from v(X_{1}, X_{2}, X_{3}), see [7], Example 2.2.15. In Sect. 4 we will compare the cycles v(X, X, X) and v(v(X, X), X).

In the case of a self-intersection, we define inductively

X^{(1)} := X and X^{(m)} := v(X^{(m−1)}, X) ,

and we will see that on the smooth locus of X this cycle is composed of polar varieties; see Proposition 3.5 below. In order to prove this proposition, we need the following definition and result from [6].

Definition 3.3. ([6], 3.5 and 4.1) The map

p := (ℓ_{0} : …: ℓ_{r}): P_{L}^{n} −−→ P_{L}^{r}

is called the rth generic linear projection.

The set

Sm(X, Y) := {x ∈ X ∩ Y | X, Y and X ∩ Y are smooth at x}

is called the smooth locus of the pair (X, Y).

Theorem 3.4. ([6], Thm. 4.6) Let X, Y ⊆ P^{n} be algebraic varieties of dimension d and e, respectively. Let t be an integer such that 0 ≤ t ≤ dim X∩Y −1 and n ≥ d+e−t−1. Let p: X_{L} ∪ Y_{L} ⟶ P_{L}^{d+e−t−1} be the generic linear projection and R(p) = R(p, X_{L}, Y_{L}) its ramification locus. Then dim R(p) ≤ t, and the associated t-cycle [R(p)]_{t} is just v_{t}(X, Y) on Sm(X_{L}, Y_{L}).

We observe that in a projective space over an infinite field K the Stückrad–Vogel intersection cycle can be constructed also with elements u_{ij} from K instead of indeterminates U_{ij}. It is sufficient to choose these elements such that the linear forms ℓ_{i} avoid a finite number of certain prime ideals (see [3] for the precise conditions). The above results remain valid if we specialize the indeterminates U_{ij} to generic elements u_{ij} in K.

Proposition 3.5. Let X ⊂ P^{n} be a hypersurface (reduced and equidimensional without embedded components). Define inductively X^{(1)} := X and X^{(m)} := v(X^{(m−1)}, X). Then

X^{(m)} = ∑_{k=0}^{m−1} \binom{m − 1}{k}[P_{k}(X)] + w_{m}, 2 ≤ m ≤ n ,

where the support of w_{m} is contained in the singular locus of X.

Proof. For m = 2 the result follows immediately from Theorem 3.4, with t = n − 2 and d = e = n − 1, and Remark 2.4 with k = 1 (since [P_{0}(X)] = [X]). By induction assume m ≥ 3 and that

X^{(m−1)} = ∑_{k=0}^{m−2} \binom{m − 2}{k}[P_{k}(X)] + w_{m−1} .

Hence by the linearity of the Stückrad–Vogel cycle

X^{(m)} = v(X^{(m−1)}, X) = ∑_{k=0}^{m−2} \binom{m − 2}{k}v(X, P_{k}(X)) + v(X, w_{m−1}) .

Observe that the support of v(X, w_{m−1}) is contained in the singular locus of X. Now, if P_{k}(X) ≠ ∅,

v(X, P_{k}(X)) = [P_{k}(X)] + v_{n−k−2}(X, P_{k}(X)) ;

Self-intersections of surfaces and Whitney stratifications 7

see [7], Remark 2.2.7.(2). Moreover, again by Remark 2.4 and Theorem 3.4 in the nontrivial case where P_{k}(X) ≠ ∅, with t = n − k − 2, d = n − 1 and e = n − 1 − k, noting that P_{k}(X) ⊆ X,

v(X, P_{k}(X)) = [P_{k}(X)] + [P_{k+1}(X)] + w̃_{k} ,

where w̃_{k} is supported by the singular locus of X. Then we obtain

X^{(m)} = ∑_{k=0}^{m−2} \binom{m − 2}{k}([P_{k}(X)] + [P_{k+1}(X)]) + w_{m} ,

and an easy computation gives the result. □

For an arbitrary irreducible subvariety Z ⊆ X_{L} ∩ Y_{L} ⊂ P_{L}^{n} we set Z_{Δ} := J(Z, Z) ∩ Δ. By Ĵ and Ẑ_{Δ} we denote the affine cones of the ruled join J := J(X_{L}, Y_{L}) ⊂ P_{L}^{2n+1} and Z_{Δ} in the affine space A_{L}^{2n+2}. Let (A, 𝔪) be the local ring O_{Ĵ,Ẑ_{Δ}} and I ⊂ A be the ideal of the diagonal subspace Δ and let G(X, Y; Z) denote the associated graded ring G_{I}(A) = ⊕_{j=0}^{∞}I^{j}/I^{j+1}. If Z is the empty subvariety of P^{n}, then A becomes the homogeneous ring of coordinates of the ruled join J ⊂ P_{L}^{2n+1} localized at the irrelevant maximal ideal; that is, we obtain a global picture of the intersection algorithm. Let e(B) denote the multiplicity of a local ring (or graded ring) B with respect to its unique maximal (or homogeneous maximal) ideal. Finally we denote by

c_{k} := c_{k}(I, A) := c_{k}(G_{𝔪}(G_{I}(A)) (0 ≤ k ≤ dim A =: d)

the generalized Samuel multiplicities of the ideal I ⊂ A, which are defined by the leading coefficients of the Hilbert polynomial belonging to the twofold sum transform of the Hilbert function of the bigraded ring R = ⊕_{i,j=0}^{∞} R_{ij} with

R_{ij} = G_{𝔪}^{i}(G_{I}^{j}(A)) = (𝔪^{i}I^{j} + I^{j+1})/(𝔪^{i+1}I^{j} + I^{j+1}) ;

see [4], Def. 2.2.

Proposition 3.6. ([4], 2.5, 4.2 and 4.4) With the preceding notation,

e(G(X, Y; Z)) = e(G_{I}(A)) = ∑_{C} j(X, Y; C) · e(O_{C,Z}) = ∑_{k=0}^{d} c_{k}(I, A) ,

where C runs through the characteristic subvarieties of X and Y with C ⊇ Z. If Z = ∅, then d = dim A = dim J + 1,

c_{0} = j(X, Y; ∅), c_{1} = deg v_{0}, c_{2} = deg v_{1}, …, c_{d} = deg v_{d−1}

and if k > dim(X ∩ Y) + 1, then c_{k} = 0.

If Z ≠ ∅ is K-rational, then d = dim A = dim J − dim Z and

c_{k} = ∑_{C} j(X, Y; C) · e(O_{C,Z}) (0 ≤ k ≤ d) ,

where C runs through all varieties of C(X, Y) with C ⊇ Z and codim_{C} Z = k. If k > dim(X ∩ Y) − dim Z, then c_{k} = 0.

8 R. Achilles and M. Manaresi

Remark 3.7. (Analytic case) In the paper [20], P. Tworzewski has constructed an intersection cycle of complex analytic subsets X and Y of a manifold M which do not intersect necessarily properly. His construction is based on a pointwise defined intersection multiplicity g(x) = g(X × Y, Δ_{M}, x) for a point x ∈ Δ_{M}, where Δ_{M} is the diagonal of M × M and g(x) is the sum of the coordinates of the so-called extended index of intersection g̃(x); see [20], Definition (4.2), p. 185.

Let A = O_{X×Y,x} and I = I_{Δ_{M}} · O_{X×Y,x}. K. Nowak [16], [17] (see also [5]) has recently proved that g(x) = e(G_{I}(A)) and that g̃(X) is composed of the generalized Samuel multiplicities c_{0}(I, A), …, c_{dim(X∩Y)}(I, A) and of zeros.

In the analytic case, also the Segre numbers of an ideal I introduced by T. Gaffney and R. Gassler [10] are special cases of the generalized Samuel multiplicities c_{k}(I, A); see [5].

4. Self-intersection and Whitney stratification

If one wants to use the Stückrad-Vogel intersection cycle of a self-intersection of a surface X in P^{3} for the construction of a Whitney stratification of X, then it seems to be natural to consider v(X, X, X) or v(v(X, X), X) rather than v(X, X), since the latter cycle cannot have a zero-dimensional part; see [7], Remark 2.2.7.(2).

L. van Gastel proposed to use v(X, X, X) and conjectured that

v(X, X, X) = v(v(X, X), X)

(up to a field extension or up to rational equivalence). Such an associativity formula would then allow an application of Proposition 3.5. Unfortunately, the associativity law does not hold. H. Flenner suggested to us the following example from [7] (Example 2.2.15) in order to show that in general v(X, X, X) ≠ v(v(X, X), X).

Example 4.1. Consider the following configuration of two lines V_{1}, V_{2} and of a conic V_{3} in the projective plane.

[figure: V_{2}; Q; V_{3}; P; V_{1}]

Let X_{i} be the cone over V_{i} with common vertex C in P^{3} and denote by L_{P}, L_{Q} the lines P C and QC, respectively. We want to calculate the twofold self-intersections of the surface

X := X_{1} ∪ X_{2} ∪ X_{3} ⊂ P^{3} = Proj(C[x, y, z, w]) ,

Self-intersections of surfaces and Whitney stratifications 9

which is given by the equation xy(yz − x^{2}) = 0. Obviously v(X_{1}, X_{1}) = X_{1}, v(X_{1}, X_{2}) = L_{P}, v(X_{2}, X_{2}) = X_{2}, v(X_{2}, X_{3}) = L_{P} +L_{Q} and v(X_{3}, X_{3}) = X_{3} +L_{1} +L_{2}, where L_{1} and L_{2} are movable lines on the cone X_{3} passing through the vertex C. Hence v(X_{1}, L_{i}) = C, v(X_{2}, L_{i}) = C, v(X_{3}, L_{i}) = L_{i} + C. Furthermore, v(X_{1}, L_{P}) = L_{P}, v(X_{1}, L_{Q}) = C, v(X_{2}, L_{P}) = L_{P}, v(X_{2}, L_{Q}) = L_{Q}, v(X_{3}, L_{P}) = L_{P} + C, v(X_{3}, L_{Q}) = L_{Q} + C. Using the bilinearity of the Stückrad-Vogel cycle we get

v(v(X, X), X) = X_{1} + X_{2} + X_{3} + 32 · L_{P} + 6 · L_{Q} + L_{1} + ⋯ + L_{4} + 18 · C ,

where L_{3} and L_{4} are two further movable lines on X_{3} going through C. The calculation of v(X, X, X) is more difficult. Again by the bilinearity of the cycle, we have to sum up 27 cycles v_{ijk} := v(X_{i}, X_{j}, X_{k}), which by symmetry can be grouped as follows:

v_{123} = v_{132} = v_{213} = v_{231} = v_{312} = v_{321} = L_{P} + C ,
v_{112} = v_{121} = v_{211} = v_{122} = v_{212} = v_{221} = L_{P} ,
v_{113} = v_{131} = v_{311} = 2 · L_{P} ,
v_{133} = v_{313} = v_{331} = 2 · L_{P} + 2 · C ,
v_{223} = v_{232} = v_{322} = L_{P} + L_{Q} ,
v_{233} = v_{323} = v_{332} = L_{P} + L_{Q} + 2 · C
v_{111} = X_{1} and v_{222} = X_{2} ,
v_{333} = X_{3} + L_{1} + ⋯ + L_{4} + 2 · C ,

where L_{1}, … L_{4} are again movable lines as before. It follows that

v(X, X, X) = X_{1} + X_{2} + X_{3} + 30 · L_{P} + 6 · L_{Q} + L_{1} + ⋯ + L_{4} + 20 · C .

This time the contribution of the vertex C is higher by 2, but that of the line L_{P} smaller by 2 so that we have of course the same Bezout number 64.

Surprisingly it turns out that neither v(X, X, X) nor v(v(X, X), X) is needed in order to get a Whitney stratification of the surface X. As we will see in the next theorem, it is sufficient to consider the self-intersection v(X, X) and to take its local degree g(x) as a stratifying function:

Theorem 4.2. Let X ⊂ P^{n} be a (reduced) surface, x ∈ X be a closed point and g: X → N be the map defined by g(x) = e(G(X, X; x)). 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).

10 R. Achilles and M. Manaresi

Proof. At first we will show that X_{j} is closed. In fact, given a closed point x ∈ X \X_{j}, we will construct a Zariski open subset U of X containing x such that g(y) ≤ g(x) for each closed point y ∈ U. Let

C(X, X) = {C_{1}, …, C_{s}, D_{s+1}, …, D_{t}}

be the set of the characteristic subvarieties of the self-intersection of X, and assume that C_{1}, …, C_{s} pass through x but D_{s+1}, …, D_{t} do not. By [13], Lemma 2.2, the sets

D_{i} := {y ∈ C_{i} | e(O_{C_{i},y}) > e(O_{C_{i},x})}, i = 1, …, s

are closed in C_{i} and hence in X. Then U := X \ (⋃_{i=1}^{t} D_{i}) is a Zariski open subset of X containing x such that

g(y) = ∑_{i=1}^{s} j(X, X; C_{i}) · e(O_{C_{i},y}) ≤ g(x) = ∑_{i=1}^{s} j(X, X; C_{i}) · e(O_{C_{i},x})

for all y ∈ U.

Now we prove that S_{g}(j) is smooth.

At first we note that g(x) = 1 if and only if X is smooth at x. In fact, g(x) = 1 forces X to be irreducible at x. Thus g(x) = m_{x}(X) + ⋯ = 1, which implies m_{x}(X) = 1. Conversely, the regularity of O_{X,x} implies the regularity of A = O_{Ĵ,x̂_{Δ}} (see, for example, [7], 1.3.15) and of A/I ≅ O_{X̂,x̂}, where we used our notation introduced before Proposition 3.6. Hence I is generated by regular parameters, G_{I}(A) is a polynomial ring over the regular local ring A/I, and g(x) = e(G_{I}(A)) = 1.

If dim(Sing X) = 0 then, by the above consideration, the stratification is given by X and by the singular points of X. So the strata are smooth.

If dim(Sing X) = 1, then

v(X, X) = [X] + [P_{1}(X)] + ∑_{k=1}^{t} j_{k}[C_{k}] + eventual 0-dim. part ,

where C_{k} (k = 1, …, t) are all the 1-dimensional irreducible components of the singular locus of X counted by their intersection numbers j_{k}; see Proposition 3.5 and [2], 2.5. The 0-dimensional part can only occur if n ≥ 4 (see [7], Remark 2.2.7.(2)), and its fixed part (if there is any) is supported on Sing X (see [2], 2.5) and its movable part is [P_{2}(X)]. Note that for a point x ∈ X, g(x) is the degree of v(X, X) at x (Proposition 3.6):

g(x) = m_{x}(X) + m_{x}(P_{1}(X)) + ∑_{k=1}^{t} j_{k} · m_{x}(C_{k}) + j(X, X; x) .

Here we have used that m_{x}(P_{2}(X)) = 0; see the remark at the end of 2.3. Now consider a point x on a fixed C_{k}, and call it a general point of C_{k} if the following four conditions are satisfied:

(i) x is a smooth point of C_{k} (equivalently, m_{x}(C_{k}) = 1) and m_{x}(X) = e(O_{X,C_{k}}); (ii) x ∉ P_{1}(X); that is, m_{x}(P_{1}(X) = 0;

(iii) x does not belong to C_{i} with i ≠ k; that is, m_{x}(C_{i}) = 0 if i ≠ k;

(iv) x does not belong to the 0-dimensional part of v(X, X); that is, j(X, X; x) = 0.

In all general points of C_{k}, the function g takes the same value a := e(O_{X,C_{k}}) + j_{k}, but g(x) will be strictly larger than a if x ∈ C_{k} is not a general point of C_{k}. This can be easily seen from the above description of g(x). For example, if x is a singular point of C_{k}, then m_{x}(C_{k}) = e(O_{C_{k},x}) > 1. It follows that the 1-dimensional strata are formed by the general points of C_{k} (k = 1, …, t) and that they are smooth. As in the case dim(Sing X) = 0, every isolated singular point of X (which could also be in v(X, X) if n ≥ 4) itself forms a stratum. The 2-dimensional strata, being the connected components of X \ Sing X, are obviously smooth.

Now we know that S_{g}(j) := g^{−1}(j) = X_{j} \ X_{j+1} is smooth and that, by construction, for all x ∈ X_{j} \ X_{j+1} it holds that, following the notation of 3.5,

g(x) = m_{x}(X) + m_{x}(P_{1}(X)) + m_{x}(w) = j .

Here w denotes the part of v(X, X) whose support is contained in the singular locus of X, and we have used that, as noted previously, m_{x}(P_{2}(X)) = 0 (which is relevant only if n ≥ 4). By the upper-semicontinuity of the multiplicities m_{x}(…), it follows that both m_{x}(X) and m_{x}(P_{1}(X)) must be constant on the connected components of S_{g}(j) and hence, by Teissier’s result (Theorem 2.8), the connected components of S_{g}(j) are the strata of a Whitney stratification of X. Moreover, if n = 3, then we have obtained the coarsest Whitney stratification of X. In fact, to get smooth strata, ∑_{k=1}^{t} j_{k} · m_{x}(C_{k}) = m_{x}(w) (for this equality we have used that n = 3) must be constant along each stratum of an arbitrary Whitney stratification and hence, again by Theorem 2.8, g(x) must be constant, which is the condition used to construct our stratification. This finishes the proof. □

One is tempted to ask whether a Whitney stratification of a surface X can be obtained by other stratifying functions coming from self-intersections. Of course, by Remark 3.2, one could also consider the cycles v(X, X, X) and v(v(X, X), X) and take again their pointwise degree as stratifying functions to obtain what we call the g-stratifications of X by v(X, X, X) and v(v(X, X), X), respectively. Or one could stratify X by the pointwise degree of the fixed part only of v(X, X), v(X, X, X) and v(v(X, X), X), getting stratifications which we call the distinguished stratifications of X by v(X, X), v(X, X, X) and v(v(X, X), X), respectively.

The complete comparison of the g-stratifications and of the distinguished stratifications by v(X, X), v(X, X, X) and v(v(X, X), X) is still open. In particular, it is not known if the distinguished stratification by v(X, X, X) coincides with the g-stratification by v(X, X, X), but an example from Teissier ([19], p. 315) shows that for v(X, X) the two stratifications may differ and that the distinguished one needs not to be a Whitney stratification.

Example 4.3 (Teissier). Consider the hypersurface X ⊂ C^{3} with the defining equation y^{2} − x^{3} − z^{2}x^{2} = 0:

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

The singular locus of X is the z-axis, which is the only distinguished subvariety (except the whole surface X). So the distinguished stratification has only two strata and hence is not a Whitney stratification; see [19], p. 315. In order to find the g-stratification of the preceding theorem, we calculate the generalized Samuel multiplicities

(c_{0}(I, A), c_{1}(I, A), c_{2}(I, A))

for A := O_{X×X,a} (a ∈ X, and X is considered to be diagonally embedded in X × X) and I = I_{Δ} · A. This can be done by using the computer program CALI [12], a REDUCE package for commutative algebra, together with the script [1] written by D. Aliffi and the first author. We obtain (0, 3, 2) if the point a ∈ X is the origin, (0, 2, 2) for a on the z-axis but different from the origin, and (0, 0, 1) for all smooth points a of X; see the figure. Note that g(a) = c_{0}(I, A) + c_{1}(I, A) + c_{2}(I, A). Therefore we have three strata: the origin with g = 5 (which is not a stratum in the distinguished stratification), the z-axis except the origin with g = 4, and X_{reg} with g = 1.

In a similar way one finds the canonical Whitney stratification of the projective closure \overline{X} of X in P^{3} = Proj(C[t, x, y, z]). The singular locus of \overline{X} consists of the two lines L_{1} : x = y = 0 and L_{2} : t = x = 0, which intersect in the point P_{3} = (0, 0, 0, 1). To construct a Whitney stratification, the points P_{1} = (1, 0, 0, 0) and P_{2} = (0, 0, 1, 0) are also needed. The canonical Whitney stratification of \overline{X} has the following six strata: \overline{X}_{reg} with g = 1, L_{1} \ (P_{1} ∪ P_{3}) and L_{2} \ (P_{2} ∪ P_{3}) with g = 4, the point P_{1} with g = 5, and the points P_{3} and P_{2} with g = 6.

In the following example we study a surface X whose g-stratification by v(X, X) is different from its distinguished stratification by v(v(X, X), X).

Example 4.4. Let X ⊂ C^{3} be the union of the plane X_{1} with defining equation z = 0 and the nonsingular surface X_{2} with defining equation z − x^{3} + y^{2} = 0. Then the singular locus of X is the cuspidal plane curve C := X_{1} ∩ X_{2} with cusp point O. By the bilinearity of the Stückrad-Vogel cycle we get easily v(X, X) = X_{1} + X_{2} + 2 · C + P, where P is a (movable) polar curve of X_{2} that can be moved away from O. Using again

Self-intersections of surfaces and Whitney stratifications 13

the bilinearity we find v(X, v(X, X)), and since O is a smooth point both of X_{1} and X_{2} and is not lying on a general polar curve P, the point O will not appear in the cycle v(X, v(X, X)). But the point O is a singular point of C = Sing X and hence the function g(x) = m_{x}(X_{1}) + m_{x}(X_{2}) + 2 · m_{x}(C) (see v(X, X) above), being 1 on X \ C and 4 on C \ O, has in O the value 1 + 1 + 2 · m_{x}(C) = 6.

Let X be a hypersurface in P^{n} and consider the g-stratifications of X coming from X^{(n−1)} or X^{(n)}, respectively. If the g-constant strata were smooth, then one could generalize Theorem 4.2 to X ⊂ P^{n}, n ≥ 4. Unfortunately this is not the case, at least for reducible hypersurfaces. We have the following counterexample:

Example 4.5. Consider the projective closure in P^{4} of the hypersurface X ⊂ C^{3} which is defined as the union of the two smooth hypersurfaces X_{1} : t = 0 and X_{2} : t + z(z + y^{2} − x^{3}) = 0. The singular locus of X is X_{1} ∩ X_{2} and consists of the plane S_{1} : t = z = 0 and the smooth surface S_{2} : t = z + y^{2} − x^{3} = 0. Note that C := Sing(Sing X) = S_{1} ∩ S_{2} is a plane curve with a cusp at the origin. We will see that the stratifying function g is constant on C and does not jump at the cusp point. To this end we calculate the Stückrad-Vogel cycle that defines g. Obviously v(X_{1}, X_{1}) = X_{1}, v(X_{2}, X_{2}) = X_{2}, v(X_{1}, X_{2}) = S_{1} + S_{2}, and using the bilinearity of the Stückrad-Vogel cycle we get v(X, X) = X_{1}+X_{2}+2(S_{1}+S_{2}), X^{(3)} = v(v(X, X), X) = X_{1}+X_{2}+6(S_{1}+S_{2}), and X^{(4)} = v(X^{(3)}, X) = X_{1} + X_{2} + 14(S_{1} + S_{2}). This implies that g is constant on C, namely 14 if g is defined by X^{(3)} or 30 if it is defined by X^{(4)}.

We conclude with two open problems.

Problem 4.6. Let X be an irreducible (and reduced) hypersurface in P^{n} and consider the g-stratification of X by X^{(n)}. Are the g-constant strata smooth? A positive answer would allow to generalize Theorem 4.2 to such X ⊂ P^{n}: but see the preceding counterexample if X is not irreducible.

Problem 4.7. Let X be a (reduced) surface in P^{3}. Is the distinguished stratification by v(X, X, X) equal to the g-stratification by v(X, X), that is, to the canonical Whitney stratification of X?

Acknowledgements. It is a great pleasure for the authors to thank Leendert van Gastel, who generously discussed his conjectures and ideas of possible proofs with them. The authors are also grateful to the referee for useful suggestions on how to improve this paper.

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