ON SPECIAL VALUES FOR PENCILS OF PLANE CURVE SINGULARITIES

by Arkadiusz Płoski

IMUJ PREPRINT 2002/08 KRAKÓW

Abstract

Let (F_{t} : t ∈ P^{1}) be a pencil of plane curve singularities and let μ_{0}^{t} be the Milnor number of the fiber F_{t}. We prove a formula for the jumps μ_{0}^{t} − inf{μ_{0}^{t} : t ∈ P^{1}}. As application we give a description of the special values of the pencil (F_{t} : t ∈ P^{1}).

Introduction

Let (F_{t} : t ∈ P^{1}), P^{1} = C ∪ {∞} be a pencil of plane curve singularities defined by two coprime power series f, g ∈ C{X, Y } without constant term. That is F_{t} = f − tg for t ∈ C and F_{∞} = g. Let μ_{0}^{t} be the Milnor number of the fiber F_{t} and let

μ_{0}^{min} = inf{μ_{0}^{t} : t ∈ P^{1}} .

2000 Mathematics Subject Classification: 14H20, 32S10. Key words and phrases: pencil of plane curve singularities, Milnor number, special value. Supported in part by the KBN grant No 2P03A01522

Our aim is to give a formula for the jumps μ_{0}^{t} − μ_{0}^{min} by means of the meromorphic fraction f/g considered on the branches of the jacobian curve

j(F) = \frac{∂f}{∂X} \frac{∂g}{∂Y} − \frac{∂f}{∂Y} \frac{∂g}{∂X} = 0 .

Roughly speaking we will show that μ_{0}^{t} − μ_{0}^{min} = the number of zeros of f/g − t if t ∈ C and μ_{0}^{∞} − μ_{0}^{min} = the number of poles of f/g on the branches of the jacobian curve j(F) = 0 provided that μ_{0}^{t} ≠ +∞ (resp. μ_{0}^{∞} ≠ +∞). Then we prove a known result on the special values of the pencil (F_{t} : t ∈ P^{1}).

1 Preliminaries

Let f ∈ C{X, Y } be a non-zero power series without constant term. We say that the curve f = 0 is singular if ∂f/∂X(0, 0) = ∂f/∂Y (0, 0) = 0. A branch P is a prime ideal of C{X, Y } generated by an irreducible power series p. Let B be the set of all branches. For any curve f = 0 we put

B(f) = {P ∈ B : f ≡ 0(mod P)} .

We put by definition (f, g)_{0} = dim_{C}C{X, Y }/(f, g) the intersection multiplicity of f and g. Note that (f, g)_{0} = +∞ if and only if f and g have a common factor. The Milnor number μ_{0}(f) is defined to be μ_{0}(f) = (∂f/∂X, ∂f/∂Y)_{0}. Then μ_{0}(f) = +∞ if and only if the curve f = 0 is not reduced (i. e. the power series f has a multiple factor).

The following lemma is well-known (see [GLM], [Ca] and [P]).

Lemma 1.1 Let f = 0 and g = 0 be two curves without common branch. Let j(f, g) = (∂f/∂X)(∂g/∂Y) − (∂f/∂Y)(∂g/∂X). Then

(f, j(f, g))_{0} = μ_{0}(f) + (f, g)_{0} − 1 .

In particular the curve f = 0 is not reduced if and only if the curves f = 0 and j(f, g) = 0 share a common branch.

Proof. We may assume that f(0, Y) ≠ 0. Using Delgado’s formula ([Ca], Prop. 7.4.1) we get

(f, j(f, g))_{0} = (f, ∂f/∂Y)_{0} + (f, g)_{0} − (f, X)_{0} .  (1)

On the other hand by Teissier’s formula ([T], Chap. II, Prop. 1.2) one can write

(f, ∂f/∂Y)_{0} = μ_{0}(f) + (f, X)_{0} − 1  (2)

and the lemma follows.

For every branch P we denote by M_{P} the field of fractions of the ring C{X, Y }/P. Let f, g ∈ C{X, Y } be coprime power series. Put

D(f/g) = {P ∈ B : g ≢ 0(mod P)} .

Then for every P ∈ D(f/g) the fraction f/g defines an element of M_{P} which we denote also f/g. We put for P ∈ D(f/g):

ord_{P} (f/g) = (f, p)_{0} − (g, p)_{0} ,

where p is a generator of P. Clearly ord_{P} is a valuation of the field M_{P}.

Let P^{1} = C ∪ {∞} and let us denote by f/g ↦ (f/g)(P) ∈ P^{1} the place associated with ord_{P}.

Recall that (f/g)(P) = ∞ if ord_{P} f < ord_{P} g < +∞ and (f/g)(P) = 0 if ord_{P} g < ord_{P} f.

Lemma 1.2 Suppose that ord_{P} f/g ≥ 0 for a P ∈ D(f/g) and let t_{0} = (f/g)(P). Then ord_{P} (f − tg) = ord_{P} g if t ≠ t_{0} and ord_{P} (f − t_{0}g) > ord_{P} g.

Proof. Obvious.

2 Result

Let f = 0 and g = 0 be two curves without common branch. We put F = (f, g) and j(F) = j(f, g). Let us consider the pencil (F_{t} : t ∈ P^{1}) where F_{t} = f − tg for t ∈ C and F_{∞} = g. Let μ_{0}^{t} = μ_{0}(F_{t}). If j(F)(0, 0) ≠ 0 then μ_{0}^{t} = 0 for all t ∈ P^{1}. In the sequel we assume that j(F)(0, 0) = 0.

Proposition 2.1 Let t ∈ P^{1}. Then the following two conditions are equivalent:

(i) μ_{0}^{t} = +∞,

(ii) the curves j(F) = 0 and F_{t} = 0 share a common branch.

Proof. We use Lemma 1.1 to power series F_{t}, g if t ∈ C and to power series F_{∞} = g and f if t = ∞.

Proposition 2.1 implies Bertini’s theorem “the set {t ∈ P^{1} : F_{t} is not reduced} is finite”. Indeed, it is easy to check that

#{t ∈ P^{1} : μ_{0}^{t} = +∞} ≤ B(j(F)) .

Let μ_{0}^{min} = inf{μ_{o}^{t} : t ∈ P^{1}}. By Bertini’s theorem μ_{0}^{min} is an integer.

Let us put

U(F) = {P ∈ B(j(F)) : ord_{P} f ≥ ord_{P} g}

and

U(F)^{c} = {P ∈ B(j(F)) : ord_{P} f < ord_{P} g} .

Thus U(F) ⊂ D(f/g) and U(F)^{c} ⊂ D(f/g) provided that μ_{0}^{∞} < +∞.

For every branch P of the jacobian curve j(F) = 0 we denote by m(P) the multiplicity of P i.e. the greatest integer m > 0 such that j(F) ≡ 0(mod P^{m}). By convention a sum extended over an empty set equals zero.

Our main result is the following

Theorem 2.2 With the notation introduced above

(i) if μ_{0}^{t} ≠ +∞ for a t ∈ C, then

μ_{0}^{t} − μ_{0}^{min} = ∑_{P ∈U(F)} m(P)ord_{P} (f/g − t) .

(ii) If μ_{0}^{∞} ≠ +∞, then

μ_{0}^{∞} − μ_{0}^{min} = − ∑_{P ∈U(F)^{c}} m(P)ord_{P} (f/g) .

Proof. Let us fix a t ∈ C such that μ_{0}^{t} ≠ +∞. We have j(F_{t}, g) = j(f, g) and (F_{t}, g)_{0} = (f, g)_{0}. Applying Lemma 1.1 to F_{t} and g we get

μ_{0}^{t} = (F_{t}, j(f, g))_{0} − (f, g)_{0} + 1 .  (3)

Let us write

j(f, g) = ∏_{i=1}^{k} p_{i}

with irreducible p_{i} ∈ C{X, Y } and let P_{i} = (p_{i})C{X, Y }. Therefore (P_{i})_{i=1....,k} is a sequence of branches of j(f, g) = 0 counted with multiplicities. Let

I = {i ∈ [1, k] : ord_{P_{i}}f ≥ ord_{P_{i}}g}

and observe that ord_{P_{i}}F_{t} = ord_{P_{i}}f for i ∉ I. Then

(F_{t}, j(f, g))_{0} = ∑_{i=1}^{k} ord_{P_{i}}F_{t} = ∑_{i∈I} ord_{P_{i}}F_{t} + ∑_{i∉I} ord_{P_{i}}f

and by (3) we get

μ_{0}^{t} = ∑_{i∈I} ord_{P_{i}}F_{t} + ∑_{i∉I} ord_{P_{i}}f − (f, g)_{0} + 1 .  (4)

If i ∈ I then by Lemma 1.2 we have ord_{P_{i}}F_{t} ≥ ord_{P_{i}}g with equality for t ≠ (f/g)(P_{i}). Using (4) we get

μ_{0}^{min} = ∑_{i∈I} ord_{P_{i}}g + ∑_{i∉I} ord_{P_{i}}f − (f, g)_{0} + 1 .  (5)

and consequently

μ_{0}^{t} − μ_{0}^{min} =∑_{i∈I} ord_{P_{i}}F_{t} − ord_{P_{i}}g = ∑_{i∈I} ord_{P_{i}}(F_{t}/g)
= ∑_{P ∈U(F)} m(P)ord_{P} (f/g − t)

for F_{t}/g = f/g − t. We have thus proved (i).

Let us suppose that μ_{0}^{∞} ≠ +∞. By Lemma 1.1 applied to g and f we get

μ_{0}^{∞} = (g, j(f, g))_{0} − (f, g)_{0} + 1 = ∑_{i∈I} ord_{P_{i}}g + ∑_{i∉I} ord_{P_{i}}g − (f, g)_{0} + 1 . (6)

Now, by (5) and (6) we get

μ_{0}^{∞} − μmin_{0} =∑_{i∉I} ord_{P_{i}}g − ord_{P_{i}}f = − ∑_{i∉I} ord_{P_{i}}(f/g)
=− ∑_{P ∈U(F)^{c}} m(P)ord_{P} (f/g)

which proves (ii).

3 Description of special values

Let

Λ(F) = {t ∈ P^{1} : μ_{0}^{t} > μ_{0}^{min} }

be the set of special values of the pencil (F_{t} : t ∈ P^{1}) (see [LW]). We put by convention (f/g)(P) = ∞ if g ≡ 0(mod P).

The following description of the special values is due to different authors: Theorem 3.1 (see [MM], Théorème 1, [LO], pp 410-411, [Ca], 7.4). One has

Λ(F) = {(f/g)(P) : P ∈ B(j(F))} .

Proof. First we prove the following

{t ∈ C : μ_{0}^{t} > μ_{0}^{min} } = {(f/g)(P) : P ∈ U(F)} .  (7)

Fix t ∈ C. We will check that μ_{o}^{t} > μ_{0}^{min} if and only if there exists a P ∈ U(F) such that (f/g)(P) = t. If μ_{0}^{t} = +∞ then F_{t} has multiple factors. Thus there exists a branch P such that F_{t} ≡ 0(mod P^{2}). It is easy to check that P ∈ U(F) and (f/g)(P) = t.

Now suppose that μ_{0}^{t} < +∞. According to Theorem 2.2(i) the inequality μ_{0}^{t} > μ_{0}^{min} holds if and only if there exists P ∈ U(F) such that ord_{P} (f −tg) > ord_{P} g. The last inequality is equivalent to the condition (f/g)(P) = t. This proves (7).

Let us check the following property

μ_{0}^{∞} > μ_{0}^{min} if and only if U(F)^{c} ≠ ∅ .  (8)

Indeed, if μ_{0}^{∞} = +∞ then there is a branch P such that g ≡ 0(mod P) and P ∈ B(j(F)) by Proposition 2.1. Obviously f ≢ 0(mod P) and we get ord_{P} g = +∞ > ord_{P} f. Thus P ∈ U(F)^{c}. If μ_{0}^{∞} < +∞ then (8) follows from Theorem 2.2(ii). Theorem 3.1 follows from (7) and (8) for (f/g)(P) = ∞ if P ∈ U(F)^{c}.

When studying the singularities at infinity of a polynomial in two complex variables of degree N > 1 one considers the pencil defined by F = (f, l^{N}) where l = 0 is a smooth curve which is not a component of the curve f = 0. Clearly μ_{0}^{∞} = +∞. Using Theorem 3.1 we get Corollary 3.2 ((see [GBP], Prop. 2.2))

Λ(F) ∩ C = {(f/l^{N})(P) : P ∈ B(j(f, l)) and ord_{P} f/ord_{P} l ≥ N} .

References

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[GLM] A. Gabrielov, J.M. Lion, R. Moussu, Ordre de contact de courbes intégral du plan, C. R. Acac. Sci. Paris, t.319, Série I, p. 219–221, 1994.

[LW] Lê Dung Trang, C. Weber, Équisingularité dans les pinceaux de germes de courbes planes et C^{0}-suffisance, L’Enseignement Mathématique, t.43, (1997), 355–380.

[LO] Lê Vanh Than, M. Oka, Note on estimation of the number of the critical values at infinity, Kodai Math. J. 17 (1994), 409–414.

[MM] H. Maugendre, F. Michel, Fibration associée à un pinceau de courbes planes, Prépublication de l’Université de Provence, Mars 2001.

[P] A. Płoski, The Milnor Number of a plane algebroid curve, Materiały XVI Konferencji Szkoleniowej z Analizy i Geometrii Zespolonej, Łódź (1995), 73–82.

[T] B. Teissier, Cycles évanescentes, section planes et conditions de Whitney, Astérisque, No. 7-8 (1973).

Department of Mathematics, Technical University, AL. 1000 L PP 7, 25-314 Kielce, Poland

e-mail: matap@tu.kielce.pl