THE ORDER OF CONTACT OF A HOLOMORPHIC IDEAL IN ℂ^{2}

JEFFERY D. MCNEAL & ANDRÁS NÉMETHI

1. Introduction

Let I ⊂ O (ℂ^{n}, 0) be an ideal of (germs of) holomorphic functions near 0. Let Γ denote the set of holomorphic mappings γ : (ℂ, 0) ⟶ (ℂ^{n}, 0) with γ(0) = 0. For a function h : ℂ ⟶ ℂ such that h(0) = 0, let ord(h) denote the order of vanishing of h at 0, and for vector-valued mappings such as γ, let ord(γ) = min_{k} {ord(γ_{k})} (see section 2 for exact definitions). The purpose of this paper is to establish an alternate way to compute the invariant

T (I) = sup_{γ∈Γ} (inf_{f∈I} \frac{ord(f ∘ γ)}{ord(γ)}) ,  (1.1)

called the order of contact of I, when n = 2.

This invariant was introduced by D’Angelo [D’An1]. Similar notions, however, also were considered earlier, see for example [Bri-Knö]. The invariant plays a very important role in complex function theory on domains in ℂ^{n}, especially for L^{2} subelliptic estimates for the Cauchy-Riemann equations. Given a domain Ω ⊂ ℂ^{n} with smooth boundary bΩ, consider a defining function, r : ℂ^{n} ⟶ ℝ, for bΩ: bΩ = {r = 0} and dr ≠ 0 when r = 0. It was observed by Kohn [Koh] that, for certain domains Ω, if some γ ∈ Γ lies in bΩ, then the Cauchy-Riemann system ¯∂ fails to be hypoelliptic, i.e. there are ¯∂-closed (0, 1)-forms α such that every solution u, to the system of equations ¯∂u = α, has the property that singsupp u ⊈ singsupp α. Catlin [Cat1] extended this result to any smoothly bounded domain Ω. Moreover, Catlin later showed [Cat2] that holomorphic curves with high order of tangency to bΩ prevent certain analytic estimates on ¯∂, which are quantified versions of hypoellipticity, from holding. He showed that if p ∈ bΩ and there exists an analytic curve γ passing through p whose order of tangency with bΩ is M, then the subelliptic estimate

||u||_{ϵ}^{2} ≤ C (||¯∂u||^{2} + ||¯∂^{∗}u||^{2}) ,  u ∈ Dom (¯∂^{∗})  (1.2)

cannot hold in a neighborhood of p if ϵ > \frac{1}{M}. Here || · ||_{ϵ} denotes the Sobolev norm of order ϵ and || · || denotes the L^{2} norm. To clarify the connection between (1.2) and the contact order of curves γ with bΩ, D’Angelo [D’An1] defined the invariant Δ_{1}(bΩ, p), the type of p, by

Δ_{1}(bΩ, p) = sup_{γ∈Γ} (\frac{ord(r ∘ γ)}{ord(γ − γ(p))}) ,  (1.3)

and established the invariant’s basic properties. Catlin [Cat3] then proved the remarkable result that (1.2) holds in a neighborhood of p (for some ϵ > 0) if and only if Δ_{1}(bΩ, p) < ∞.

In order to establish the basic properties of Δ_{1}(bΩ, p), D’Angelo associated to the pair (r; p) – the defining function r and a point p ∈ bΩ – a family of ideals of holomorphic functions I_{β}. The exact association of I_{β} to (r; p) involves a truncation and a polarization of the defining function r; these points are not relevant to our discussion in this paper, so we refer to [D’An2] for the definition of I_{β}. However we point out that D’Angelo [D’An1] showed that Δ_{1}(bΩ, p) is bounded, from above and below, by simple functions of max_{β} T (I_{β}). Thus, in principle, the computation of Δ_{1}(bΩ, p) is reduced to computing the type of the associated ideals I_{β}.

Both authors are supported by grants from the National Science Foundation.

The difficulty in computing T (I) is the supremum, over the infinite dimensional space of curves Γ, on the right hand side of (1.1). There is also a counter-intuitive property of T (I) which stems from the fact that Γ is infinite dimensional: T (I) is not an upper semicontinuous function of parameters on I. That is, suppose that for each s ∈ (−1, 1), I_{s} is an ideal of holomorphic functions which varies continuously in s; it can happen that T (I_{0}) < T (I_{s}) for s arbitrarily near 0. (See section 5 for an explanation of this and a related discussion of the “jump phenomena” for Δ_{1}(bΩ, p).) It is, therefore, tempting to discard T (I) and replace it with another, better behaved invariant, for example one which is easier to compute (e.g. replace Γ by ℂP^{1} in (0.1)) and/or upper semicontinuous with respect to parameters (e.g. the intersection multiplicity of I). However, results such as [Cat1] show that each curve counts and may influence the function theory: any complex analytic curve γ contained in bΩ causes ¯∂ to be non-hypoelliptic, and so we cannot, apriori, ignore the order with any curve.

We want to point out that the finiteness of T (I) maybe be shown by comparison with other invariants – see [D’An2] for several ways this can be done and for sharp estimates on T (I) in terms of other, well-known invariants – and, indeed, the statement T (I) < ∞ may be verified without directly knowing the order of contact of any curves, e.g. by computing the intersection number of the ideal I. The value of T (I), however, is a different matter; the order of contact of I with all complex curves must be measured and this means grappling with the large space of curves Γ.

The following theorem shows, nevertheless, that T (I) may always be computed using only a finite number of testing curves.

1.1. Theorem. Let I ⊂ O (ℂ^{2}, 0) be an ideal and fix any set of generators f_{1}, …, f_{k} of I. For each 1 ≤ a ≤ k, let

f_{a} = ∏_{b} (f_{ab})^{p_{ab}}

be the irreducible decomposition of f_{a} in the ring O (ℂ^{2}, 0). Then

T (I) = max_{a,b} (min_{r} \frac{ord (f_{r} ∘ γ_{ab})}{ord(γ_{ab})}) ,

where γ_{ab} is a (minimal) parameterization of C_{ab} = {f_{ab} = 0}.

In other words, in order to compute T (I), we have to only consider that finite set of irreducible (parameterized) curves which appear as irreducible components in at least one of the generators f_{1}, …, f_{k}. Of course, as the choice of generators f_{1}, …, f_{k} is not unique, different choices of generators may provide a different set of testing curves C_{ab}.

Theorem 1.1 is, thus, a kind of “selection theorem” and a particularly simple one: if I is generated by k irreducible functions, then we must only test against k curves. For higher dimensional ideals, I ⊂ O_{n}, n > 2, we can also show that T (I) can be computed by testing against only a finite number of curves (rather than considering all curves in Γ), but this set is not as small as we obtain when n = 2. For this reason, and because the 2-dimensional argument is so transparent, we present only Theorem 1.1 in this paper.

The proof of Theorem 1.1 is given in section 4. In sections 2 and 3, we gather together the tools we need for this proof. We would like to emphasize the key role that the monotonicity property in Proposition 3.2 (and Proposition 3.3) plays in the proof of Theorem 1.1.

Part of the work on this paper was done while the first author was visiting Harvard University. He thanks this institution for its generous support and providing splendid working conditions. He also thanks Gordon Heier for several conversations related to Theorem 1.1 and John D’Angelo for the many stimulating discussions about type (and other matters) we have had over the years.

2. Puiseux expansions and contact orders

Let x_{1}, …, x_{n} denote the standard coordinates in ℂ^{n} and let ℂ{x_{1}, …, x_{n}} denote the ring of convergent power series centered at 0. As shorthand, let O_{n} = O (ℂ^{2}, 0) (= ℂ{x_{1}, … x_{n}}).

Let (C, 0) denote an irreducible plane curve with an isolated singularity at 0. Irreducibility here means in the sense of elements in O_{2}, not as polynomials. Thus, we may view C as given by {(x_{1}, x_{2}) : f_{C}(x_{1}, x_{2}) = 0} for an irreducible germ f_{C} ∈ O_{2}. Such a curve has a parameterization γ_{C} : (ℂ, 0) ⟶ (ℂ^{2}, 0), where γ_{C} = (γ^{1}, γ^{2}), γ^{1}, γ^{2} ∈ O_{1}, and γ_{C} is a 1-1 mapping onto C (by the Local Parameterization theorem, cf. [Gun]). Such a parameterization γ_{C} is called a normalization (or minimal parametrization).

For any h ∈ O_{n}, the order of vanishing (or multiplicity) of h at 0 is defined

ord(h) = min{l : h = ∑_{p_{1},...,p_{n}} a_{p_{1}...p_{n}}x_{1}^{p_{1}} … x_{n}^{p_{n}} such that
a_{p_{1}...p_{n}} ≠ 0 and p_{1} + ⋯ + p_{n} = l}.

2.1. Definition. Let (C, 0) and (C′, 0) be two irreducible curves plane curves. The intersection multiplicity of C and C′, denoted i(C, C′), is the integer

i(C, C′) = dim {O_{2}/ (f_{C}, f_{C′})} = ord (f_{C} ∘ γ_{C′}) ,

where f, γ are associated to C, C′ as above and (f_{C}, f_{C′}) denotes the ideal generated by f_{C} and f_{C′}.

For a proof that i(C, C′) is independent of the choices of f and γ, and that the two expressions in Definition 2.1 give the same number, see [Bri-Knö].

For a given curve C, we note that ord (f_{C}) = min_{i} {ord (γ_{C}^{i})}, where f_{C}, γ_{C} are as above. Their common value is called the multiplicity of C, which we also denote mult(C).

2.2. Proposition.(Puiseux expansion). Suppose that (C, 0) is an irreducible plane curve with an isolated singularity at 0, and the tangent cone to (C, 0) is not the x_{2}-axis.

Then there exists a parameterization γ_{C} = (x_{1}(t), x_{2}(t)) : ℂ ⟶ ℂ^{2} of the form

\{ \begin{matrix}x_{1}(t) = t^{m} \\ x_{2}(t) = a_{n}t^{n} + a_{n+1}t^{n+1} + … \end{matrix}  (2.1)

where m equals mult(C), m ≤ n, and a_{k} ∈ ℂ.

See, for example, [Walk] for a proof.

We shall also write the Puiseux parameterization (2.1) as

ζ(x_{1}) = a_{n}x_{1}^{n/m} + a_{n+1}x_{1}^{n+1/m} + … ∈ ℂ \{x1^{\frac{1}{m}} \} .

Like (2.1), this form of the expansion, called a branch, is not unique; it depends on the choice of an m-th root of unity. So, for any m-th root of unity ω, we get a different Puiseux expansion:

ζ_{ω} = a_{n}ω^{n}x_{1}^{n/m} + a_{n+1}ω^{n+1}x_{1}^{n+1/m} + …  (2.2)

For an element ζ(x_{1}) = a_{s}x_{1}^{s/m} + ⋯ ∈ ℂ\{x1^{\frac{1}{m}} \}, define the order of ζ by

ord ζ = \frac{s}{m} where s = min \{r : a_{r} ≠ 0\} .

If we have two expansions ζ_{1}(x_{1}) ∈ ℂ\{x1^{\frac{1}{m_{1}}} \} and ζ_{2}(x_{1}) ∈ ℂ\{x1^{\frac{1}{m_{2}}} \}, we may consider both as

lying in ℂ\{x1^{\frac{1}{m_{1}m_{2}}} \}. The order of contact of the two expansions ζ_{1} and ζ_{2} is then computed in this ring; by definition

ord (ζ_{1}, ζ_{2}) = ord (ζ_{1} − ζ_{2})

in ℂ\{x1^{\frac{1}{m_{1}m_{2}}} \}. If ζ_{1} = ζ_{2}, then we set ord (ζ_{1}, ζ_{2}) = ∞. The following comparison fact on orders of contact follows trivially

ord (ζ_{1}, ζ_{2}) ≥ min {ord (ζ_{1}, ζ_{3}) , ord (ζ_{2}, ζ_{3})} .  (2.3)

Using the Puiseux expansions, we define the order of contact of two curves.

2.3. Definition. Let (C, 0) and (C′, 0) be two irreducible curves plane curves. The order of contact of C and C′ (at 0) is defined

ord(C, C′) = max {ord (ζ_{ω}, ζ′_{ω′})}

where the maximum is taken over all branches ζ_{ω} of C (as ω ranges over the roots of unity) and over all branches ζ′_{ω′} of C′ (as ω ranges over the roots of unity).

We remark that, to compute ord(C, C′), we may fix one of the branches ζ′_{ω′} (i.e. fix a root of unity ω′) and take the maximum by allowing ω to vary, or vice-versa, but we cannot fix both. Also, it is clear that ord(C, C′) = ∞ if and only if C = C′.

For a curve C, we also need to consider invariants of higher order than mult(C).

2.4. Definition. (Puiseux exponents). Let (C, 0) ⊂ (ℂ^{2}, 0) be an irreducible plane curve with isolated singularity at 0. Suppose that the tangent cone to (C, 0) is not the x_{2}-axis and that C has Puiseux parameterization (2.1).

The Puiseux characteristic (m; β_{1}, …, β_{g}) is defined inductively as follows. Set e_{0} = m and

\{ \begin{matrix}β_{k} = min\{r : a_{r} ≠ 0 and k not divisible by e_{k−1}\} \\ e_{k} = gcd (e_{k−1}, β_{k}) . \end{matrix}

The inductive process stops when we arrive at e_{g} = 1 (which must occur after a finite number of steps) for some g. Conventionally, we write

β_{g+1} = ∞ and α_{k} = \frac{β_{k}}{m} , k = 1, …, g.

Remark: The set of rational numbers {α_{k}}_{k=1}^{g+1} is exactly the set of contact numbers of the different branches of the Puiseux expansion of C, i.e.

{α_{k}}_{k=1}^{g+1} = {ord (ζ_{ω}, ζ′_{ω′}) : ω, ω′are roots of unity and ζ_{ω}, ζ′_{ω′}parameterize C as in (2.1)} .

There is a relationship between the contact number of two curves C, C′ and their intersection number. This relationship seems to go back to M. Noether, but has been rediscovered many times since, e.g. [Mer, Wall].

2.5. Proposition. Let (C, 0) and (C′, 0) be two irreducible curves plane curves. Assume that C has Puisuex characteristic (m; β_{1}, …, β_{g}), and let α_{k} and e_{k} be as in Definition 2.4.

Assume that ord(C, C′) = κ and that α_{k_{0}} < κ ≤ α_{k_{0}+1}, for some k_{0}. Then

\frac{i(C, C′)}{mult(C′)} = (e_{0} − e_{1})α_{1} + ⋯ + (e_{k_{0}−1} − e_{k_{0}})α_{k_{0}} + e_{k_{0}} · κ.  (2.4)

A proof of (2.4) may be found in [Mer].

3. The Eggers tree

A convenient way to arrange the Puiseux exponents of all the irreducible components of a reduced plane curve C is the Eggers tree, T_{C}. This was introduced by Harald Eggers in his thesis [Egg], but seems to be not so well-known; we recall the construction of this tree below. See, also [Wall], where this idea is improved by labeling the edges of the tree in a useful manner. We shall then see how to (easily) measure the orders of contact that another, irreducible plane curve C′ has with the different components of C, by associating to C′ a point, P_{C′}, on T_{C}.

Let C = ∪_{q=1}^{N}C_{q} be the decomposition of C into irreducible components. Consider a fixed (irreducible) component C_{i}. Let π_{i} : [0, ∞] ⟶ [0, ∞] be a (fixed) bijective map such that π_{i}(0) = 0. Choose coordinates such that C_{i} has a Puiseux expansion of the form in Proposition 2.2 and Puiseux characteristic (m; β_{1}, …, β_{g}). The tree T_{C_{i}} is defined to be the set π_{i} ([0, ∞]) with the vertices

\begin{matrix}V_{0}^{i} = 0, V_{k}^{i} = π_{i} (α_{k}(C_{i})) , k = 1, …, g \\ V_{∞}^{i} = π_{i}(∞) \end{matrix}

(recall that α_{k} = \frac{β_{k}}{m}). We may think of this as a linear, one-dimensional tree:

[figure: 0 = V_{0}^{i}; V_{1}^{i}; V_{2}^{i}; V_{g(C_{i})}^{i}; V_{∞}^{i}; …]

T_{C_{i}} provides a simple way to visualize the operation of perturbation of the Puiseux expansions (2.2) of C_{i} at various heights. Consider a Puiseux expansion (where we drop the index i, and root of unity ω, for cleaner notation)

ζ(x_{1}) = a_{n}x_{1}^{n/m} + a_{n+1}x_{1}^{n+1/m} + …  (3.1)

and Puiseux characteristic (m; β_{1}, …, β_{g}). For any h ≥ n, the expression

ζ̃(x_{1}) = a_{n}x_{1}^{n/m} + ⋯ + a_{h}x_{1}^{h/m} + ∑_{q>h} b_{q}x_{1}^{q/m}  (3.2)

agrees with ζ up to order h/m, for any coefficients b_{q}. Thus, if α_{k−1} ≤ h < α_{k}, then all expressions of the form (3.2) agree with ζ at least up to order α_{k−1}, but not (for arbitrary coefficients b_{q}) up to order α_{k}. So, to each ζ̃ we associate the point π_{i}((h+1)/m) on T_{C_{i}} (cf. also with Definition 3.1), provided that b_{h+1} is generic, or at least b_{h+1} ≠ ω^{h+1}a_{h+1} for any m-root of unity ω. Note that (n + 1)/m represents the first order to which (3.2) disagrees with (3.1). In particular, points p on the edge of T_{C_{i}} between V_{k−1}^{i} and V_{k}^{i} correspond to curves which agree with C_{i} up to order (at least) α_{k−1}, but not to order α_{k}. The point V_{∞}^{i} corresponds, uniquely, to the curve C_{i} itself.

Now consider two irreducible components of C, C_{i} and C_{j} with ord(C_{i}, C_{j}) = κ_{ij}. Define the tree T_{C_{i}∪C_{j}} as follows:

(1) Identify the intervals π_{i} ([0, κ_{ij}]) and π_{j} ([0, κ_{ij}]).

(2) Introduce a new vertex at π_{i} (κ_{ij}) = π_{j} (κ_{ij}), and denote it by W^{ij}. (If κ_{ij} = α_{k}(C_{i}) then W^{ij} = V_{k}^{i}.)

(3) Continue the two trees T_{C_{i}} and T_{C_{j}}, separately, after the bifurcation point W^{ij}.

For example, suppose κ_{ij} is greater than α_{k−1}^{i} = α_{k−1}^{j}, but less than both α_{k}^{i} and α_{k}^{j}. Then T_{C_{i}∪C_{j}} looks like

[figure: …; …; …; V_{0}^{i} = V_{0}^{j}; V_{k−1}^{i} = V_{k−1}^{j}; V^{ij}; V_{k}^{i}; V_{k}^{j}; V_{∞}^{i}; V_{∞}^{j}]

Now consider all the irreducible components {C_{q}}_{q=1}^{N} of C. Glue the trees T_{C_{1}}, …, T_{C_{N}} together, one at a time, by the above procedure and call the resulting tree T_{C}, the Eggers tree of C. Notice that the order of gluing is irrelevant. On each “limb”, T_{C_{i}}, of T_{C} we have the natural ordering induced by π_{i}:

π_{i}(a) ≤ π_{i}(b) ⟺ a ≤ b.

This induces a partial ordering on T_{C}. In particular, each bifurcation point W^{ij} (all of which are vertices of T_{C}) is the “largest point smaller than both V_{∞}^{i} and V_{∞}^{j}”. More succinctly, we write

W^{ij} = inf (V_{∞}^{i}, V_{∞}^{j}) .

Notice also that we have a well-defined map v : T_{C} ⟶ [0, ∞], by simply defining v restricted to T_{C_{i}} to be π_{i}^{−1}.

Given a plane curve C (not necessarily irreducible) and another, irreducible plane curve C′, we associate a point P_{C′} on T_{C} which describes all contact orders of C′ with the components of C.

3.1. Definition. Let (C, 0) be a plane curve, C = ∪_{q=1}^{N}C_{q} be its decomposition into irreducible components, and (C′, 0) an irreducible plane curve. Let T_{C} be the Eggers tree of C.

The point P_{C′} ∈ T_{C} associated to C′ is defined to be that point satisfying all the equations

ord(C_{q}, C′) = v (inf (P_{C′}, V_{∞}^{q}))

for q = 1, …, N.

It is easy to see (via the partial ordering on T_{C}) that P_{C′} is well-defined. For any fixed k, by Proposition 2.5, the point P_{C′} expresses the renormalized intersection multiplicity \frac{i(C_{k},C′)}{mult(C′)}, simply in terms of the tree T_{C}, the position of P_{C′} in T_{C}, and the quantities e_{q} in Definition 2.4 associated with the component C_{k}. For our purposes, the important observation is the following.

3.2. Proposition. Let (C, 0) be a plane curve with irreducible decomposition C = ∪_{q=1}^{N}C_{q} and Eggers tree T_{C}.

For any two irreducible plane curves, C′ and C′′, if P_{C′′} ≥ P_{C′} on T_{C}, then

\frac{i(C_{k}, C′′)}{mult(C′′)} ≥ \frac{i(C_{k}, C′)}{mult(C′)}, k = 1, …, N.

Proof. P_{C′′} ≥ P_{C′} implies that, for a given component C_{k}, all the terms on the right hand side of (2.4) in the expression for \frac{i(C_{k},C′)}{mult(C′)} are also in the right hand side of the expression

for \frac{i(C_{k},C′′)}{mult(C′′)}, but that there may be more terms in the expression for \frac{i(C_{k},C′′)}{mult(C′′)}. Since every term on the right hand side of (2.4) is positive, the result follows. □

The monotonicity property in Proposition 3.2 is also true for nonreduced curves in place of the irreducible curves C_{k}. For use in the next section, we derive this simple corollary of Proposition 3.2, with hypotheses tailored to those in Theorem 1.1.

3.3. Proposition. Let f̃ ∈ O_{2} be an arbitrary (non-reduced) germ. Let C̃^{red} = {f̃ = 0} be the associated reduced curve. Suppose C′ and C′′ are irreducible curves such that P_{C′′} ≥ P_{C′} on the tree TC̃^{red}. Let γ′ resp. γ′′ be their parametrizations. Then

\frac{ord(f̃∘ γ′′)}{ord(γ′′)} ≥ \frac{ord(f̃∘ γ′)}{ord(γ′)}.

In fact, the same inequality is valid even if we replace TC̃^{red} by any Eggers tree associated with any curve C^{red} = C̃^{red} ∪ D (i.e. with any curve C^{red} with C̃^{red} ⊂ C^{red}).

Proof. Recall the simple property of the ord function: ord(gh) = ord(g) + ord(h). Suppose that f̃ = ∏_{t} f_{t}^{p_{t}} where each f_{t} is an irreducible germ in O_{2}. Write C_{t} := {f_{t} = 0}. Then we have

\frac{ord(f̃∘ γ′′)}{ord(γ′′)} = ∑_{t} p_{t} \frac{i(C_{t}, C′′)}{mult(C′′)} ≥ ∑_{t} p_{t} \frac{i(C_{t}, C′)}{mult(C′)} = \frac{ord(f̃∘ γ′)}{ord(γ′)}.

□

4. Proof of Theorem 1.1

Recall the definition of T (I):

T (I) = sup_{γ} (inf_{f} \frac{ord(f ∘ γ)}{ord(γ)}) ,  (4.1)

where γ runs over the set Γ of parameterized, irreducible curves and f runs over elements in I. Fix a generating set (f_{1}, …, f_{k}) = I, and take an arbitrary γ ∈ Γ. By simple properties of the ord function, if f is an arbitrary element of I and we write f = ∑_{r=1}^{k} g_{r}f_{r}, it follows that

ord(f ∘ γ) ≥ min_{r} ord(f_{r} ∘ γ).

Combining this with Definition 2.1, we may re-express (4.1) and eliminate the inner infimum:

T (I) = sup_{γ} (min_{r} \frac{ord(f_{r} ∘ γ)}{ord(γ)}) .  (4.2)

Next, we may apply Proposition 3.3 to f_{r} = f̃. Indeed, let T_{C} be the Eggers tree of the reduced curve ∪_{r}{f_{r} = 0}. If C_{γ′} and C_{γ′′} are two irreducible curves such that P_{C_{γ′′}} ≥ P_{C_{γ′′}} on T_{C}, then \frac{ord(f_{r}∘γ′′)}{ord(γ′′)} ≥ \frac{ord(f_{r}∘γ′)}{ord(γ′)}. That is, the function

\frac{ord(f_{r} ∘ ·)}{ord(·)} is non-decreasing

as the point associated with · moves out along T_{C}. In particular, the right hand side of (4.2) is realized at one (or more) of the relative maximum points on the tree T_{C}. However, these points are exactly the vertices V_{∞}^{rs}, which correspond to the irreducible components C_{rs}. This is the statement of Theorem 1.1.

5. The “jump phenomena” of type

Consider the family of ideals I_{s} = (f_{s}, g) for small s ≥ 0, where

\begin{matrix}f_{s} = sx_{1} + x_{1}^{3} + x_{2}^{8} \\ g = x_{1}^{2} − x_{2}^{101}. \end{matrix}  (5.1)

Notice that f_{s} and g are irreducible, for all s.

It is straightforward to compute, directly from the definition of T (I_{s}), (1.1), that

\begin{matrix}T (I_{s}) = 16 if s > 0, & while \\ T (I_{0}) = 8. \end{matrix}  (5.2)

In fact,

(1) The curves γ_{s}(t) = (\frac{t^{8}}{−s}, t), which are non-singular at 0, have the property that

min (ord(f_{s} ∘ γ_{s}), ord(g ∘ γ_{s})) = ord(g ∘ γ_{s}) = 16

and that no curve in Γ produces a number larger than \frac{16}{1} in (1.1).

(2) The curve γ(t) = (t^{101}, t^{2}), which has multiplicity 2 at 0, satisifies

min (ord(f_{0} ∘ γ), ord(g ∘ γ)) = ord(f_{0} ∘ γ) = 16

and no curve in Γ produces a number larger than \frac{16}{2} = 8 in (1.1). In this example, the type of the ideals I_{s} jumps up from the value of T (I_{0}), the “wrong”

jump direction expected by naive continuity considerations. This phenomena was first discovered by D’Angelo [D’An3], and is the source of considerable difficulties for several questions in complex analysis. The reason this jump phenomena occurs is that components of curves in Γ may “disappear” in the limit, as s → 0: the non-singular curves γ_{s} above tend, essentially, to the curve t ⟶ (t^{8}, 0) as s → 0, a curve with high multiplicity which has no bearing on the computation of T (I_{0}).

Theorem 1.1 provides a better explanation of the jump phenomena. It is particularly cogent for the case of two, irreducible generators.

5.1. Proposition. Suppose that I ⊂ O_{2} is an ideal generated by irreducible germs f_{1}, f_{2}, I = (f_{1}, f_{2}). Then

T (I) = \frac{i(f_{1}, f_{2})}{min (mult(f_{1}), mult(f_{2}))}.  (5.3)

Proof. By Theorem 1.1, there are only two parametrizations which need to be considered: the normalizations γ′ and γ′′ of {f_{1} = 0} respectively {f_{2} = 0}. Then

ord (f_{1} ∘ γ_{1}) = ∞ and ord (f_{1} ∘ γ_{1}) = i(f_{1}, f_{2})

and, similarly,

ord (f_{1} ∘ γ_{2}) = i(f_{1}, f_{2}) and ord (f_{1} ∘ γ_{1}) = ∞.

The order of contact of a holomorphic ideal 9

(5.3) now follows immediately. □

The point of Proposition 5.1 is that it expresses T (I) as the ratio of 2 upper semicontinuous functions, the intersection multiplicity i(f_{1}, f_{2}) and (the minimum of) the multiplicities mult(f_{1}), mult(f_{2}). Both the numerator and denominator of this ratio act as typical geometric invariants and only jump “down” as parameters vary. However, and this is the key point, they may jump down by different amounts. This potential for unequal decrease is exactly what allows the type to jump up (in certain examples), and accounts for the lack of upper semicontinuity of the type function.

We should also point out, however, that the type does not jump up unrestrictedly. The main result in [D’An1] shows that T (I_{s}) ≤ T (I_{0})^{power}, for a continuous family of holomorphic ideals I_{s}, s near 0, or, in other words, that type is a locally bounded function. In particular, the different rates of decrease, that the numerator and denominator in (5.3) may experience, are correlated by D’Angelo’s theorem.

Finally, we mention that examples like (5.1) may be used to show that Δ_{1}(bΩ, p) is not upper semicontinuous in p, for domains Ω in ℂ^{n}, n ≥ 3. Consider the domain Ω given by {(x_{1}, x_{2}, x_{3}) ∈ ℂ^{3} : r(x_{1}, x_{2}, x_{3}) = 0} where

r = 2Rex_{3} + |x_{3}x_{1} + x_{1}^{3} + x_{2}^{8}|^{2} + |x_{1}^{2} − x_{2}^{101}|^{2}

which corresponds to the example (5.1). The same computations that gave (5.2) show that

Δ_{1}(bΩ, 0) = 8 while Δ_{1}(bΩ, p_{ϵ}) = 16

where p_{ϵ} = (0, 0, iϵ) ∈ bΩ.

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[D’An1] J.P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. 115 (1982),

615-637. [D’An2] Several complex variables and the geometry of real hypersurfaces, CRC press, Boca Raton,

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Department of Mathematics, Ohio State University, Columbus, OH 43210,USA

   E-mail address: mcneal@math.ohio-state.edu
   E-mail address: nemethi@math.ohio-state.edu