On the index of contact and multiplicities for bigraded rings

Sławomir Rams

IMUJ PREPRINT 1999/26 KRAKÓW

Abstract We compute the index of contact p(Z, S)(c) of a set Z with a submanifold S at a point c ∈ reg(Z ∩ S) at which Z is normal pseudoflat along Z ∩ S (Thm 4.1, Cor. 4.4).

1 Introduction

In [5] so-called index of contact of a set Z with a submanifold S is applied to estimate the Łojasiewicz exponent of polynomial mappings and study a regular separation condition for projective algebraic sets (see also [10]). The main aim of this note is to prove that the index of contact and the index of intersection (introduced in [14]) coincide along a Zariski open subset of each component of Z ∩ S. The latter invariant is the Samuel multiplicity of the associated graded ring; hence, it can be computed by various computer algebra systems (see [3], [2] for details).

In analytic intersection theory, the intersection multiplicity along a proper component is defined in two steps. Having defined the intersection multiplicity at an isolated point ([6], Def. 3.4) Draper passes to the case of (proper) component of dimension greater than zero by cutting by a transversal hyperplane of complementary dimension at a regular point ([6], Def. 4.5). In this note, we prove that a similar procedure can be applied to compute the index of contact at a point c ∈ reg(Z ∩ S) at which Z is normal pseudo-flat along Z ∩ S (Thm 4.1(d), (e)). We compute the extended index of intersection of Z and S ([14], Def. 4.2) at such points and prove (Cor. 4.4, Cor. 4.5) that all the components of the complement of the normal pseudo-flateness locus (see [1] for details) appear as embedded components with positive multiplicities in so-called intersection product ([14], Def. 6.3).

[6], Def. 3.4 is generalized in [4] where the isolated intersection multiplicity of locally analytic sets X, Y at a point a is defined by intersecting some extensions of the diagonal Δ ⊂ N^{2} with the set X × Y. We prove here that the index of contact can be computed in a similar way (Thm 4.1(b), (c)). We give also a precise description of the family of the extensions of S that can be applied to compute p(Z, S)(c) in this way (Cor. 4.2).

2 Preliminaries

Let N be an n-dimensional (complex) manifold and let S be its s-dimensional closed submanifold. For an analytic cycle A = ∑_{j∈J} α_{j}C_{j} in V, where V is an open subset of N, we define the part of A supported by S as A^{S} := ∑_{j∈J,C_{j}⊂S} α_{j}C_{j}. The multiplicity of the cycle A at the point a ∈ V is defined as ν(A, a) := ∑_{j∈J} α_{j}ν(C_{j}, a), where ν(C_{j}, a) denotes the multiplicity of the analytic set C_{j} at the point a. Its support ⋃_{j∈J} C_{j} is denoted by |A|.

We fix a purely k-dimensional analytic set Z ⊂ N and a point c ∈ Z ∩ S. We put d := p − (k + s − n), where p := dim_{c}(Z ∩ S).

F_{c}^{1}(Z, S) denotes the family of locally analytic sets R ⊂ N of pure dimension n + p − k which meet Z properly and satisfy the condition S_{c} ⊂ R_{c}. Let {X_{j}}_{j=1}^{j=q} be the family of all p-dimensional components of Z ∩ S. One can see that for every R ∈ F_{c}^{1}(Z, S) the following equality holds

ν((Z · R)^{S}, c) = ∑_{j=1}^{j=q} ν(X_{j}, c) · i(Z, R; X_{j}),  (1)

where i(Z, R; X_{j}) is the multiplicity of intersection of Z and R along X_{j}. We introduce the following notation:

\begin{aligned}F_{c}^{2}(Z, S) & := & \{R ∈ F_{c}^{1}(U, Z) : Z_{c} ∩ R_{c} = (Z ∩ S)_{c}\}, \\ μ_{1}(Z, S)(c) & := & inf\{ν((Z · R)^{S}, c) : R ∈ F_{c}^{1}(Z, S)\}, \\ μ_{2}(Z, S)(c) & := & inf\{ν(Z · R, c) : R ∈ F_{c}^{2}(Z, S)\}. \end{aligned}

If c ∈ reg(Z ∩ S), we define the family T_{c}(Z, S) as the set of all (n − p)- dimensional submanifolds P ⊂ N that are transversal to Z ∩S at the point c, and meet Z properly (by Rückert’s lemma T_{c}(Z, S) ≠ ∅). Every P ∈ T_{c}(Z, S) determines the cycle P · Z in a neighbourhood of c, where ” · ” denotes the intersection product ([6], p. 190). Since c is an isolated point of Z ∩ P ∩ S, the multiplicity of isolated intersection i((Z · P), S; c) ([4], Def. 5.1) and the relative tangent cone C_{c}(Z, S) (see [4], p. 22) are well-defined.

Let G_{d}(C^{n−s}) denote the Grassmann manifold of d-dimensional subspaces of C^{n−s} and let E ⊂ C stand for the unit ball.

In section 4 we will need the following lemmas:

Lemma 2.1 Let N = E^{n}, S = E^{s} × {0}^{n−s}, a ∈ reg_{p}(Z ∩ S), P ∈ T (Z, S). Then the set of L ∈ G_{d}(C^{n−s}) which satisfy the condition

(C^{s} × L) ∩ C_{a}(Z ∩ P, S) = C^{s} × {0}^{n−s}  (2)

is an open and dense subset of G_{d}(C^{n−s}).

Proof: We can assume that d > 0. Let π : C^{s} ⊕ C^{n−s} ↦ C^{n−s} stand for the projection. By [4], Lemma 2.4 the equality C_{a}(Z ∩P, S) = C^{s}×C_{0}(π(Z ∩ P ∩ U)) holds, where U is a suficiently small neighbourhood of a and C_{0}( ·) denotes the tangent cone at the point 0. □

Lemma 2.2 Let c ∈ reg(Z ∩ S), P ∈ T_{c}(Z, S). If R ∈ F_{c}^{2}(Z, S), the germ R_{c} is smooth and satisfies the condition C_{c}(P ∩ Z, S) ∩ T_{c}R = T_{c}S, then the following equality holds

ν(Z · R, c) = μ_{2}(Z, S)(c) = i((P · Z), S; c).

Proof: Fix R̃ ∈ F_{c}^{2}(Z, S) and observe that the sets R̃, P, Z meet properly at the point c. Since P, Z and R̃, Z also meet properly, we have

i((P · Z), R̃; c) = i(P, (Z · R̃); c) = ν(Z · R̃, c).

Let P · Z = ∑_{j∈J} α_{j}Y_{j}. By [4], Prop. 5.3 the following holds

ν(Z · R̃, c) = ∑_{j∈J} α_{j} · i(Y_{j}, R̃; c) ≥ ∑_{j∈J} α_{j} · i(Y_{j}, S; c) = i((P · Z), S; c), (3)

which yields μ_{2}(Z, S)(c) ≥ i((P · Z), S; c). Apply [4], Prop. 2.6 and Thm 4.4 to get the equality ν(Z · R, c) = i((P · Z), S; c) . □

3 Index of intersection and index of contact

In this section we outline the construction of the index of intersection and the index of contact (for more details see [14], [5]). We maintain the notation and the assumptions of the previous section.

Let U be an open subset of N. B(U) := {H ⊂ U : H is a closed hypersurface in U, H ⊃ (S ∩ U)}. E(U, Z) denotes the family of systems (R, H_{1}, …, H_{d}), where R is a locally analytic set and each H_{i} ∈ B(U), which satisfy the condition:

(r_{q}(U, R)) dim[((U \ S) ∩ (Z ∩ R)) ∩ H_{1} ∩ … ∩ H_{q}] ≤ max{p − q, −1}

for q = 1, …, d.

The family H(U, Z) is defined as the set of H = (H_{1}, …, H_{n−s}), where each H_{i} ∈ B(U), such that for every a ∈ U ∩S the equality ⋂_{j=1}^{n−s} T_{a}H_{j} = T_{a}S holds and H satisfies (r_{q}(U, Z)) for q = 1, …, n − s (see [14], p. 184). For every H ∈ H(U, Z) one can define inductively an analytic cycle Z · H = Z_{0}^{S} + ⋯ + Z_{n−s}^{S} in S ∩ U ([14], Algorithm 4.1):

Z_{0} := Z ∩ U,  Z_{0}^{S} := part of Z_{0} supported by S,
Z_{j} := (Z_{j−1} − Z_{j−1}^{S}) · H_{j}, Z_{j}^{S} := part of Z_{j} supported by S,

where j = 1, …, n − s and ” · ” denotes the intersection product ([6], p. 190). The cycle Z · H has a unique decomposition Z · H = T_{(s)} + T_{(s−1)} + ⋯ + T_{(0)}, where T_{(j)} is a j-cycle (i.e. a formal combination of analytic sets of pure dimension j). One defines (see [14], [5])

p(Z, S)(c) := min{ν(Z · H, c) | H ∈ H(V, Z), V ∋ c},
g̃(Z, S)(c) := min_{lex}{ν̃(Z · H, c) | H ∈ H(V, Z), V ∋ c},
g(Z, S)(c) := the sum of the coordinates of g̃(Z, S)(c),

where ”lex” denotes the lexicographical ordering in N^{s+1} and ν̃(Z · H, c) := (ν(T_{(s)}, c), …, ν(T_{(0)}, c)). The integer p(Z, S)(c) (resp. g̃(Z, S)(c), g(Z, S)(c)) is called the index of contact (resp. the extended index of intersection, the index of intersection) of Z with the submanifold S at the point c.

Observe that if R ∈ F_{c}^{1}(Z, S) and (R, H_{1}, …, H_{d}) ∈ E(U, Z), we can apply the first d steps of [14], Algorithm 4.1 to the cycle R·Z, the submanifold S and the system (H_{1}, …, H_{d}). The result of this procedure is denoted by Z · (R, H_{1}, …, H_{d}).

For a system (Y_{1}, …, Y_{r}) of subsets of N we define (Y_{1}, …, Y_{r})|_{U} := (Y_{1} ∩ U, …, Y_{r} ∩ U).

In the section 4 we will need the following lemma:

Lemma 3.1 Let N = E^{n}, S = E^{s} × {0}^{n−s}, c = 0, dim_{0}(Z ∩ S) = p. Fix a system (H_{1}, …, H_{d}) of elements of B(U), where U ⊂ N is a neighbourhood of the point 0. Then the set of L ∈ G_{d}(C^{n−s}) which satisfy the condition

there exists a neighbourhood Ũ of 0 such that (E^{s} × L, H_{1}, …, H_{d})|Ũ (⋆)

belongs to E(Ũ, Z) and ν̃(Z · (E^{s} × L, H_{1}, …, H_{d})|Ũ, 0) = g̃(Z, S)(0) , is an open subset of G_{d}(C^{n−s}).

Proof: Since the topology of G_{d}(C^{n−s}) coincides with the topology induced by the weak topology of currents, it suffices to prove that the set defined by (⋆) is open in the latter.

Suppose that there exists a sequence {L^{j}}_{j=0}^{∞} ⊂ G_{d}(C^{m−s}) such that L^{j} → L^{0} and L^{j} satisfies (⋆) only for j = 0. We can assume that E^{s} × L^{j} and Z meet properly (see e.g. [3], proof of Cor. 3).

Let Z_{0}(j, N) denote the cycle (E^{s} × L^{j}) · Z. If Ũ is a neighbourhood of the point 0 such that (H_{1}, …, H_{d})|Ũ satisfies (r_{q}(Ũ, E^{s}×L^{j})) for q = 1, …, l, then Z_{l}(j, Ũ), Z_{l}^{S}(j, Ũ) stand for the cycles obtained in the lth step of [14], Algorithm 4.1 applied to this system, the cycle Z_{0}(j, N) and S.

According to [14], Thm 3.6 the cycles Z_{0}(j, N) converge to Z_{0}(0, N). By [14], Thm 3.4 there exists a neighbourhood Ũ_{0} of the point 0 such, that

(Z_{0}(j, N) − Z_{0}^{S}(j, N)) ∩ Ũ_{0} → (Z_{0}(0, N) − Z_{0}^{S}(0, N)) ∩ Ũ_{0}

and ν(Z_{0}^{S}(j, N), 0) = ν(Z_{0}^{S}(0, N), 0) for almost all j. Since ν((Z_{0}(0, N) − Z_{0}^{S}(0, N))^{H_{1}}, 0) = 0 we apply [14], Thm 3.4 and find a neighbourhood U_{0} ⊂ Ũ_{0} of the point 0 and a j_{0} such, that H_{1} ∩ U_{0} meets |Z_{0}(j, N) − Z_{0}^{S}(j, N)| ∩ U_{0} properly for j > j_{0}. By [14], Thm 3.6 and Thm 3.4 the cycles Z_{1}(j, U_{0}) converge to Z_{1}(0, U_{0}) and ν(Z_{1}^{S}(j, U_{0}), 0) = ν(Z_{1}^{S}(0, U_{0}), 0) for almost all j. Repeating this reasoning we find an integer j_{d} and a neighbourhood U_{d−1} such that for j > j_{d} the system (E^{s} × L^{j}, H_{1}, …, H_{d})|_{U_{d−1}} belongs to the family E(U_{d−1}, Z) and the following equality holds

ν̃(Z · (E^{s} × L^{j}, H_{1}, …, H_{d})|_{U_{d−1}}, 0) = g̃(Z, S)(0),

which contradicts our assumption about L^{j}. □

Let h(I) (resp. s(I)) denote the height (resp. the analytic spread) of an ideal I. For the proof of Cor. 4.4, we will need the following version of [2], Cor. 2.4(iii) in the case of an equidimensional catenary Noetherian local ring:

Lemma 3.2 Let (A, m) be an equidimensional and catenary Noetherian local ring and let I be a proper ideal of A. Then h(I) = s(I) if and only if precisely one of the generalized Samuel multiplicities is non-zero.

Proof: We put k := dim A, q := dim A/I, r := s(I). Let c_{j} := c_{j}(I, A) := G_{m}(G_{I}(A)) be the generalized Samuel multiplicities ([2], Def. 2.2). Since the ring G := G_{I}(A) is catenary and equidimensional (by [7], B 6.6), the equality dim G/J + dim G_{J} = dim G holds for all J ∈ Spec(G). Therefore c_{k−r} ≠ 0 by [2], Prop. 2.3(ii). According to [2], Prop. 2.3(iii) we have c_{q} ≠ 0, which yields the equality r = k − q. If h(I) = s(I), then c_{k−r} ≠ 0 by [2], Cor. 2.4(iii). □

4 Main theorem and its applications

We maintain the notation and the assumptions of previous sections. sreg(Z, S) denotes the set of points a ∈ reg(Z ∩ S) which satisfy the con-

dition: g̃(Z, S)(a) = min_{lex}{g̃(Z, S)(x) : x ∈ C}, where C is the irreducible component of Z ∩ S to which a belongs. sreg_{p}(Z ∩ S) := {a ∈ sreg(Z, S)| dim_{a}(Z ∩ S) = p}. Observe that sreg(Z, S) is open and dense in Z ∩ S (see [14], Thm 5.4 ).

In this section, we compute p(Z, S)(c), where c ∈ sreg(Z, S) and give a precise algebraic description of sreg(Z, S).

Theorem 4.1 Let c ∈ sreg_{p}(Z, S) and P ∈ T_{c}(Z, S). Then:

(a) g̃(Z, S)(c) = (0, …, 0, μ_{2}(Z, S)(c), 0, …, 0), (where (s − p) zeroes precede the integer μ_{2}(Z, S)(c)),

(b) g(Z, S)(c) = μ_{1}(Z, S)(c),

(c) g(Z, S)(c) = μ_{2}(Z, S)(c),

(d) g(Z, S)(c) = p(Z, S)(c),

(e) g(Z, S)(c) = i((P · Z), S; c),

(f) if R ∈ F_{c}^{2}(Z, S) and ν(Z ·R, c) = μ_{2}(Z, S)(c), then the germ R_{c} is smooth and satisfies the condition C_{c}(P ∩ Z, S) ∩ T_{c}R = T_{c}S, Proof: Choose a neighbourhood U of the point c such that U ∩ Z ∩ S = X, where X ⊂ sreg(Z, S) is connected, a dense subset {a_{i}}_{i=0}^{∞} of X and a sequence {P_{i}}_{i=0}^{∞} of elements of the families T_{a_{i}}(Z, S), where P_{i} ∈ T_{a_{i}}(Z, S), such that a_{0} = c, P_{0} = P. We can assume that U = E^{n}, S ∩U = E^{s}×{0}^{n−s}, c = 0 and there exists L ∈ G_{d}(C^{n−s}) and a system of smooth hypersurafces (H_{1}, …, H_{d}) satisfying the conditions:

(C^{s} × L) ∩ ⋂_{j=1}^{j=d} T_{x}H_{j} = T_{x}S for every x ∈ U ∩ S,
(E^{s} × L, H_{1}, …, H_{d}) ∈ E(U, Z),  (4)
ν̃(Z · (E^{s} × L, H_{1}, …, H_{d}), 0) = g̃(Z, S)(0).  (5)

By Lemmas 2.1, 3.1 we can choose an L which satisfies the condition

(C^{s} × L) ∩ C_{a_{i}}(P_{i} ∩ Z, S) = C^{s} × {0}^{n−s}  (6)

for every i. There exists a point a_{i_{0}} such that E^{s} × L ∈ F_{a_{i_{0}}}^{2}(Z, S). Then, by Lemma 2.2 the equality ν((E^{s} × L) · Z, a_{i_{0}}) = μ_{2}(Z, S)(a_{i_{0}}) holds.

L satisfies (4); hence, we get g̃(Z, S)(a_{i_{0}}) ≤_{lex} (0, …, μ_{2}(Z, S)(a_{i_{0}}), …, 0) (one can see that there exists a system H ∈ H(U, Z) such that the equality ν̃(Z · H, c) = (0, …, μ_{2}(Z, S)(a_{i_{0}}), …, 0) holds). By (1), p. 2 and (5) the cycle μ_{2}(Z, S)(a_{i_{0}}) · X is the p-dimensional part of Z · (E^{s} × L, H_{1}, …, H_{d}); consequently, we have

(0, …, μ_{2}(Z, S)(a_{i_{0}}), …, 0) ≤_{lex} g̃(Z, S)(0).

Since the points 0 and a_{i_{0}} belong to the same component of sreg(Z, S), we have g̃(Z, S)(0) = g̃(Z, S)(a_{i_{0}}). Consequently, (a) holds for c = a_{i_{0}} and only the (s − p + 1)th component of g̃(Z, S)(0) is non-zero, which implies that (E^{s} × L) ∈ F_{0}^{2}(Z, S) (by (5)).

(a) Since (E^{s} × L) satisfies (6) for i = 0, Lemma 2.2 yields the equality

ν((Z · (E^{s} × L))^{S}, 0) = μ_{2}(Z, S)(0) = i((P · Z), S; 0) .  (7)

(c) is an immediate consequence of (a).

(b) We can assume that there exists an analytic set R ∈ F_{0}^{1}(Z, S) such that (Z · R)^{S} = μ_{1}(Z, S)(0) · X. Fix a point a ∈ X \ |(Z · R) − (Z · R)^{S}|. Then R ∈ F_{a}^{2}(Z, S), which gives the inequality

μ_{1}(Z, S)(0) = ν(Z · R, a) ≥ μ_{2}(Z, S)(a).

By (c) we have μ_{2}(Z, S)(a) = g(Z, S)(a) = g(Z, S)(0) = μ_{2}(Z, S)(0), which concludes the proof of (b).

(d) results from (b) and the inequalities g(Z, S)(0) ≥ p(Z, S)(0) ≥ μ_{1}(Z, S)(0).

(e) is an immediate consequence of (c) and (7).

(f) If R ∈ F_{0}^{2}(Z, S) satisfies the assumptions of (f), then by (7) we have i((P · Z), R; 0) = ν(Z · R, 0) = i((P · Z), S; 0). [4], Thm 4.4 yields (f). □

We obtain the following corollaries:

Corollary 4.2 If c ∈ sreg_{p}(Z, S) and H := (H_{1}, …, H_{m−s}) ∈ H(U, Z), where U is a neighbourhod of c, then the following conditions are equivalent 1. ν̃(H · Z, c) = g̃(Z, S)(c),

2. there exists a neighbourhood Ũ of c and a submanifold P ∈ T_{c}(Z, S) such that (H_{1} ∩ … ∩ H_{k−p}) ∩ Ũ ∈ F_{c}^{2}(Z, S) and the following equality holds

C_{c}(P ∩ Z, S) ∩ T_{c}(H_{1} ∩ … ∩ H_{k−p}) = T_{c}S,

3. there exists a Ũ such that (H_{1} ∩ … ∩ H_{k−p}) ∩ Ũ ∈ F_{c}^{2}(Z, S) and for every P ∈ T_{c}(Z, S) the following equality holds

C_{c}(P ∩ Z, S) ∩ T_{c}(H_{1} ∩ … ∩ H_{k−p}) = T_{c}S.

Proof: (1 ⇒ 3). Choose a Ũ such that ⋂_{j=1}^{j=k−p} H_{j} ∩ Ũ meets Z properly. By Thm 4.1(a) we have ⋂_{j=1}^{j=k−p} H_{j} ∩ Ũ ∈ F_{c}^{2}(Z, S). Thm 4.1.(f) yields 3. (3 ⇒ 2). Obvious. (2 ⇒ 1). By Lemma 2.2 we have ν((H_{1} ∩…∩H_{k−p} ∩Ũ)·Z, c) = μ_{2}(Z, S)(c). Thm 4.1(a) gives 1. □

Corollary 4.3 Let c ∈ reg(Z ∩ S). If only one coordinate of g̃(Z, S)(c) is non-zero, then c ∈ sreg(Z, S).

Proof: Let c ∈ sreg_{p}(Z, S) and let C denote the irreducible component of Z ∩ S to which c belongs. Choose a dense subset {a_{i}}_{i=0}^{∞} ⊂ sreg(Z, S) ∩ C. By Corollary 4.2 and Lemmas 2.1, 3.1 there exists H := (H_{1}, …, H_{m−s}) ∈ H(U, Z), where U is a neighbourhod of the point c, and an a_{i} ∈ U ∩ C such that ν̃(Z · H, c) = g̃(Z, S)(c) and ν̃(Z · H, a_{i}) = g̃(Z, S)(a_{i}). Since (H_{1} ∩ … ∩ H_{k−p}) ∈ F_{c}^{2}(Z, S) ∩ F_{a_{i}}^{2}(Z, S) and the points c, a_{i} belong to C, we have g̃(Z, S)(c) = g̃(Z, S)(a_{i}). □

Let F_{0}(S, Z) denote the normal-pseudoflatness locus (see [1] for details) F_{0}(S, Z) := {a ∈ Z | h(I_{S,a}) = s(I_{S,a})}, where I_{S,a} is the ideal I_{S,a} O_{Z,a}.

Corollary 4.4 The following equality holds

reg(Z ∩ S) ∩ F_{0}(S, Z) = sreg(Z, S).

Proof: By Thm. 4.1 and Cor. 4.3 we have that c ∈ sreg(Z, S) iff only one coordinate of g̃(Z, S)(c) is non-zero. Lemma 3.2 and [3], Thm 1 (see also [12], §4) conclude the proof. □

For a cycle A = ∑_{j∈J} α_{j}C_{j} we define its geometric part A_{g} as the cycle ∑_{j∈I} α_{j}C_{j}, where {C_{j}}_{j∈I} is the family of all irreducible components of |A|. The cycle A is positive if α_{j} > 0 for all j. Let D(Z, S) denote the cycle defined by the analytically constructible function g(Z, S) (see [14], Thm 5.5 and Prop. 2.1.3).

Corollary 4.5 For every c ∈ S the following inequality holds

g(Z, S)(c) ≥ ν(D(Z, S)_{g}, c).

Proof: Let Z ∩ S = ⋃_{i∈J_{j},i=0,...,s} X_{ij}, where {X_{ij} | i ∈ J_{j}} is the family of all j-dimensional components of Z ∩ S. Fix c ∈ S. We can assume that there exists H = (H_{1}, …, H_{m−s}) ∈ H(N, Z) such that ν̃(H ·Z, c) = g̃(Z, S)(c). Let H · Z = ∑ α_{ij}X_{ij}. Since H · Z is positive, we have

ν(H · Z, c) ≥ ∑_{i∈J_{j},j=0,...,s} α_{ij} · ν(X_{ij}, c).

By Thm 4.1 we have ν(D(Z, S)_{g}, c) = ∑_{i,j} μ_{2}(Z, S)(x_{ij}) · ν(X_{ij}, c), where x_{ij} ∈ X_{ij} ∩ sreg(Z, S) satisfies (Z ∩ ⋂_{q=1}^{q=k−j} H_{q})_{x_{ij}} = (X_{ij})_{x_{ij}}. Observe that α_{ij} ≥ μ_{2}(Z, S)(x_{ij}). □

As in [1] we denote G_{1}(S, Z) := {a ∈ Z | s(I_{S,a}) − h(I_{S,a}) ≥ 1}.

Corollary 4.6 The cycle (D(Z, S) − D(Z, S)_{g})_{g} is positive. Its support coincides with the set G_{1}(S, Z).

Acknowledgements. I would like to express my gratitude to Prof. Rüdiger Achilles for suggesting improvements of Cor. 4.4 and to Prof. Piotr Tworzewski for helpful conversations.

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Sławomir Rams Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków (Poland) Email: rams@im.uj.edu.pl