Relative Tangent Cone of Analytic Sets

by

Danuta Ciesielska

IMUJ PREPRINT 2004/01 KRAKÓW

Summary. The purpose of this paper is to give a characterization of the relative tangent cone of an analytic curve and an analytic set with an isolated intersection. Moreover, we present an effective computation of the multiplicity intersection of a curve and a set with an s−parametrization.

1. Introduction. We consider an analytic curve X and an analytic set Y in a neighbourhood Ω of a in ℂ^{m} such that X ∩ Y = {a} and study the relative tangent cone of them at a (see Section 2 in [ATW]). By an analytic curve we mean an analytic set of pure dimension 1. The relative tangent cone and the intersection multiplicity of analytic curves are additive, so we restrict our attention, without lose of generality, to an analytic curves with irreducible germs at 0.

Typeset by AMS-TEX

AMS Mathematics Subject Classification: primary 32B99, secondary 14C17.

Key words: analytic curve, tangent cone, relative tangent cone, intersection multiplicity.

The main result of this paper, which is an improvement of the result from the paper [C], is the equality C_{a}(X, Y) + C_{a}(X) = C_{a}(X, Y). This theorem gives a strong geometric characterization of the relative tangent cone of an analytic curve and an analytic set.

In the last section we give an effective calculation of the intersection multiplicity of an analytic sets with s−parametrization and an analytic curve.

2. Main result. Let Ω be an open neighbourhood of a ∈ ℂ^{m} (m ⩾ 2), X analytic curve and Y analytic set, both of which in Ω, such that X ∩ Y = {a}.

By the relative tangent cone C_{a}(X, Y) of the analytic sets X, Y at the point a we mean (see [ATW]) the set: {v ∈ ℂ^{m} : ∃ {x_{ν}} ∈ X, {y_{ν}} ∈ Y, {λ_{ν}} ∈ ℂ: x_{ν} → a, y_{ν} → a, λ_{ν}(x_{ν} − y_{ν}) → v}.

The main goal of this paper is to prove the following theorem.

Theorem 2.1. If the curve X have irreducible germ at the point a and X ∩ Y = {a}, then C_{a}(X, Y) + C_{a}(X) = C_{a}(X, Y).

P r o o f. The germ of the curve X at a is irreducible, so the relative tangent cone C_{a}(X) is a complex line (see [Ł], [Ch]). If this line is transitive to the set Y, which means C_{a}(X) ∩ C_{a}(Y) = {a}, then by Property 2.9 in [ATW] we obtain the equality C_{a}(X, Y) = C_{a}(X) + C_{a}(Y).

In the second case we have C_{a}(X) ⊂ C_{a}(Y). Without loose of generality, we may assume that a = 0. Now, after a suitable biholomorphic change of coordinates, we have C_{0}(X) = ℂ_{1} := {x ∈ ℂ^{m} : x_{2} = x_{3} = ... = x_{m} = 0}.

By the Puiseux Theorem ([Ł], II. 6.2) we can parametrize X (shrinking Ω, if necessary) in the following way:

U ∋ t ↦ (t^{p}, φ(t)) ∈ X, ord_{0} φ > p

by ord_{0} φ we denote min{ord_{0} φ_{i} : i = 2, …, m}.

Fix v = (v_{1}, v_{2}, …, v_{m}) ∈ C_{a}(X, Y) and (c, 0, …, 0) ∈ ℂ_{1} = C_{a}(X). By the definition of the relative tangent cone the are sequences: {t_{n}} ∈ ℂ and {(y_{n}^{1}, y_{n}^{2}, …, y_{n}^{m})} ∈ Y such that t_{n} → 0, (y_{n}^{i}) → 0, n^{p}(t_{n}^{p} − y_{n}^{1}) → v_{1} and n^{p}(φ_{i}(t_{n}) − y_{n}^{i}) → v_{i} for i ∈ {2, 3, …, m}.

Applying Lemma 2.1 from [C] to the sequence {t_{n}} yields the sequence {h_{n}} which has the following properties:

(1) h_{n} → 0,

(2) n^{d}((t_{n} + h_{n})^{d} − t_{n}^{d}) → c,

(3) for any holomorphic function φ : Ω → ℂ defined in an open neighbourhood Ω of 0 ∈ ℂ with ord_{0} φ > d we have n^{d}(φ(t_{n} + h_{n}) − φ(t_{n})) → 0.

Substituting t_{n} + h_{n} for t_{n} in the parametrization we move a little the points on the curve X and

i) for the first coordinate we have:

n^{p}((t_{n} + h_{n})^{p} − y_{n}^{1}) = n^{p}((t_{n} + h_{n})^{p} − t_{n}) + n^{p}(t_{n}^{p} − y_{n}^{1})

and observe that the left side of the equality converges to the first coordinate of a vector in the relative tangent cone of C_{a}(X, Y), wheras components of the sum converges respectively to c and v_{1}.

ii) for the i−th coordinate (i ∈ {2, …, m}) we have

n^{p}(φ_{i}(t_{n} + h_{n}) − y_{n}^{i}) = n^{p}(φ_{i}(t_{n} + h_{n}) − φ_{i}(t_{n})) + n^{p}(φ_{i}(t_{n}) − y_{n}^{i}).

and observe that the left side of the i−th equality converges to i−th coordinate of a vector in the relative tangent cone C_{a}(X, Y), wheras the first component of the sum converges to 0 and the second converges to respective v_{i}.

Consequently, v + (c, 0, …, 0) ∈ C_{0}(X, Y). Since C_{0}(X) = ℂ_{1} we conclude that C_{0}(X, Y) + C_{0}(X) = C_{0}(X, Y) and the theorem follows.

The following corollary, which follows from Theorem 2.1, is an strong geometric descriptions of the relative tangent cone of an curve and a set:

Corollary 2.2. Let Ω be an open neighbourhood of 0 ∈ ℂ^{m} (m ⩾ 2) and let X be an analytic curve with irreducible germs at the origin and Y be an analytic set both in Ω such that X ∩ Y = {0}. If the relative tangent cone of X is the axis ℂ_{1}, then there exists an algebraic cone 𝕊 ⊂ ℂ^{m−1} such that C_{0}(X, Y) = ℂ × 𝕊.

3. Multiplicity of the intersection of analytic sets. We present an effective computation of multiplicity intersection of an analytic curve and the an analytic set with s−parametrization (for definition see [TW]). For the effective computation for two analytic curves see ([K], [KN]). Intersection multiplicity is additive, so without lose of generality we may assume that the analytic curve X has irreducible germ in a point in which we compute multiplicity. Now we state a simpler version of Theorem 4.2 from [TW], which is more suitable for our purpose.

Theorem 3.1. Let Ω ⊂ ℂ^{m} be an open set, Y be a pure dimensional analytic set of Ω, a set D ⊂ ℂ^{d} be a domain (m ≥ d) and φ : D −→ X be an s−parametrization of Y. Moreover, let mapping f : Ω −→ ℂ^{d} be holomorphic. If dimf^{−1} (0) = m − d and f^{−1} (0) ∩ Y = {a}, then

ν(Z_{f} · Y; a) = \frac{1}{s} ∑_{b∈φ^{−1} (a)} ν(Z_{f∘φ} ; b).

Let Ω be an open neigbourhood of 0 ∈ ℂ^{m}, Y an irreducible pure dimensional analytic set and X analytic curve of Ω with irreducible germ in origin. The curve X have Puiseux’s parametrization:

U ∋ t ↦ Φ(t) := (t^{p}, φ(t)) ∈ X, ord_{0} φ > p, which is 1-parametrization of the cure X. We assume that the set Y have s−parametrization:

Ψ : D ∋ τ ↦ ψ(τ) ∈ Y, where D is a domain in ℂ^{k} and X ∩ Y = {0}. Moreover, we have Ψ^{−1} (0) = {0}. To makes computations easier we will describe the relative tangent cone in a simpler way. By Corollary 2.2 there exists a k−dimensional algebraic cone 𝕊 ⊂ ℂ^{m} such that C_{0}(X, Y) = ℂ × S = ℂ_{1} + ({0} × S) ⊂ ℂ × ℂ^{m−1} = ℂ^{m} . For the linear surjection l : ℂ^{m−1} −→ ℂ^{k} we denote

f_{l} : ℂ^{m} ×ℂ^{m} ∋ (x, y) = ((x_{1}, x′), (y_{1}, y′)) ↦ (x_{1}−y_{1}, l(x′−y′)) ∈ ℂ^{k+1} .

Using the above notation and multiplicity of the holomorphic map we have

Theorem 3.2. If Ψ^{−1} (0) = {0} and Kerl ∩ S = {0}, then

i(X, Y; 0) = \frac{1}{s}μ_{0}(f_{l} ∘ (Φ × Ψ)).

P r o o f. Let T := X × Y, π : ℂ^{m} × ℂ^{m} ∋ (x, y) ↦ x − y ∈ ℂ^{m} and Δ := Kerπ. By the theory developed in [ATW], to compute the multiplicity of the isolated intersection of X and Y we should calculate the multiplicity of the isolated intersection of T with the subspace Δ at the point 0 ∈ ℂ^{m} × ℂ^{m} (see [ATW], Def. 5.1). Observe that Kerf_{l} is a linear subspace of ℂ^{m} × ℂ^{m} of codimension k + 1 and C_{0}(T, Δ) = π^{−1} (ℂ × 𝕊) = {(x, y) ∈ ℂ^{m} × ℂ^{m} : x′− y′ ∈ 𝕊} (see [ATW], Lemma 2.4). Therefore, our assumption implies Kerf_{l} ∩ C_{0}(T, Δ) = Δ and by Theorem 4.4 in [ATW] we get i(X · Y; 0) = i(T · Kerf_{l}; 0). According to Definition 4.1 in [TW], the mapping Φ × Ψ : U × D −→ T = X × Y is an s–parametrization of T and Kerf_{l} = Z_{f_{l}} is a cycle of zeros of the mapping f_{l}. By Theorem 3.1 we have ν(T · Z_{f_{l}} ; 0) = \frac{1}{s}ν(Z_{f_{l} ∘(Φ×Ψ)} ; 0) and ν(T · Z_{f_{l}} ; 0) = ν(T · Kerf_{l}; 0) = i(X, Y; 0), so ν(Z_{f_{l} ∘(Φ×Ψ)} ; 0) = μ_{0}(f_{l} ∘(Φ×Ψ)) which completes the proof.

Example 3.3. Consider the analytic curve: X = {(t^{3}, t^{5}, 0, 0) ∈ ℂ^{4} | t ∈ ℂ} and the pure dimensional analytic set: Y = {(τ^{3}, τ^{2}ϱ, τϱ^{2}, ϱ^{3}) ∈ ℂ^{4}| τ, ϱ ∈ ℂ} of dimension 2. The curve X have 1−parametrization and Y have 3–parametrization Ψ : ℂ^{3} ∋ (τ, ϱ) ↦ (τ^{3}, τ^{2}ϱ, τϱ^{2}, ϱ^{3}) ∈ Y and X ∩ Y = {0}. Moreover, the axis ℂ_{1} lays in C_{0}(X) and C_{0}(Y). The cone C_{0}(X, Y) have the form ℂ × 𝕊, for an algebraic 2−dimensional cone 𝕊. Observe that the vector (0, 1, 1) ∉ 𝕊. By Theorem 3.1, intersection multiplicity i(X, Y; 0) = \frac{1}{3}μ_{0}(f_{l}) where the mapping is following

f_{l} : ℂ^{3} ∋ (t, τ, ϱ) ↦ (t^{3} − τ^{3}, l(t^{5} − τ^{2}ϱ, τϱ^{2}, ϱ^{3})) ∈ ℂ^{3}

and a linear surjection l : ℂ^{3} −→ ℂ^{2} is such that Kerl ∩ 𝕊 = {0}. Consider l : ℂ^{3} ∋ (x_{2}, x_{3}, x_{4}) ↦ (x_{2}, x_{3} − x_{4}) ∈ ℂ^{2}, so Kerl = {(0, t, t) : t ∈ ℂ} and Kerl ∩ 𝕊 = {0}. We will compute the multiplicity μ_{0}(f_{l}) using Theorem 4.3 from [TW]. Denote

g : ℂ^{3} ∋ (t, τ, ϱ) ↦ t^{5} − τ^{2}ϱ ∈ ℂ,

h : ℂ^{3} ∋ (t, τ, ϱ) ↦ (t^{3} − τ^{3}, τϱ^{2} − ϱ^{3}) ∈ ℂ^{2}. To compute the multiplicity of f_{l} = (g, h) at 0, we observe that Z_{h} = 2A_{1} + A_{2} where A_{1} and A_{2} are the sets of three lines with respective equations t^{3} − τ^{3} = 0, ϱ = 0 and t^{3} − τ^{3} = 0, τ − ϱ = 0. Now, we have μ_{0}(f_{l}) = 3 · 2 · 5 + 3 · 3 = 39, so i(X, Y; 0) = 13.

Note that in the presented method we do not need to know the form of the relative tangent cone. It is enough to know a suitable linear subspace which allowed to chose a linear surjection.

Acknowledgements. The author wishes to thank P. Tworzewski for helpful discussions concerning these problems.

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