Remark on the equisingularity of families of affine plane curves

POLONICI MATHEMATICI LXVIII.3 (1998)

Remark on the equisingularity of families of affine plane curves

by H` a Huy Vui (Hanoi) and Pha . m Tien Son (Dalat)

Abstract. We give some criteria for the equisingularity of families of affine plane curves.

1. Introduction. Let f

: C

→ C be a family of polynomials whose coefficients are polynomial functions of α ∈ C

. Consider the family of affine curves V

:= {(x, y) ∈ C

| f

(x, y) = 0}, α ∈ C

. The aim of this paper is to give certain necessary and sufficient conditions for the family {V

}

to be equisingular . These conditions read as follows: apart from the requirement that the curves V

satisfy Whitney’s conditions at each common critical point (or equivalently, µ

(V

) = const at such a point a, where µ

(V

) denotes the Milnor number of the curve V

at a) they need to have good behavior at infinity (i.e., in a sense, V

satisfy the so-called Whitney’s affine conditions at infinity).

We now suppose that the affine curves V

, α ∈ C

, all have the same critical points, say a

= (x

, y

) ∈ C

, i = 1, . . . , s.

1.1. Definition. The family of affine curves V

is said to be equisingular if for all α

∈ C

there exist a neighborhood U

of α

and a diffeomorphism h such that h(a

, α) = (a

, α), i = 1, . . . , s, and the diagram

{(x, y) | f

(x, y) = 0} × U

{(x, y, α) | f

(x, y) = 0} ∩ π

(U

)

U

U

π



h

//

π



//

1991 Mathematics Subject Classification: 32S55, 57M25, 57Q45.

Key words and phrases: affine curves, singularity, equisingularity, Milnor number, Euler characteristic, Whitney’s conditions.

Supported in part by the National Basic Research Program in Natural Sciences, Viet- nam. The second author would like to thank “l’Association d’Aubonne Culture et Edu- cation France-Vietnam” for its financial support.

[275]

is commutative, where π is the second projection. Let Γ := {(x, y, α) ∈ C

× C

| f

(x, y) = 0}.

1.2. Definition. The family of affine curves V

is said to be good at infinity if for each α

∈ C

, there exist c > 0 and a neighborhood U

of α

such that the tangent hyperplane T

(Γ ∩ {α = β}) is transverse within the plane {α = β} ⊂ C

×C

to the line {x = u, α = β} for all (u, v, β) ∈ Γ , β ∈ U

, |u| ≥ c.

1.3. Remark. (a) Although the above definition is based on a specific (and explicit) choice of the line {x = u, α = β}, it is easily seen that we can choose {l

x + l

y = l

u + l

v, α = β}, (l

: l

) ∈ CP

, instead.

(b) By definition, the family {V

}

is good at infinity if and only if there exist c > 0 and a neighborhood U

of α

such that

∂f

∂y 6= 0 for all α ∈ U

, x, y ∈ C with |x| ≥ c, f

(x, y) = 0.

(c) The following assumptions will be made throughout this paper:

• the curves {f

(x, y) = 0} are all reduced;

• d := deg(f

) = deg

(f

).

The second assumption implies that the restriction map l|

: V

→ C, (x, y) 7→ x,

is proper. Let δ(x, α) := disc

(f

(x, y)) be the discriminant of f

with re- spect to y. Then we can write

δ(x, α) = q

(α)x

+ q

(α)x

+ . . . + q

(α), q

6≡ 0,

where q

(α), i = 0, . . . , k, are polynomials of α. Therefore, by the properties of resultants, the family {V

}

is good at infinity if and only if q

(α) = const 6= 0.

2. The main result of this paper is the following theorem.

2.1. Theorem. Suppose that the affine curves V

, α ∈ C

, have the same critical points, say a

= (x

, y

) ∈ C

, i = 1, . . . , s. Then the following two conditions are equivalent :

(a) the family {V

}

is equisingular ;

(b) µ

(V

) = const, i = 1, . . . , s; and the family {V

}

is good at infinity.

For the proof we need the below lemma.

2.2. Lemma ([4]). Let there be given a polynomial F of two complex variables and the map

l : C

→ C, (x, y) 7→ x,

such that the restriction map l|

, V := F

(0), is proper. Moreover , suppose that the curve V is reduced. Then

χ(F

(0)) = d − deg disc

F (x, y),

where d := deg

(F ) and χ(F

(0)) is the Euler characteristic of F

(0).

Proof of Theorem 2.1. (a)⇒(b) is easy. Indeed, the Milnor number is a topological invariant for isolated curve singularities [5]; hence,

µ

(V

) = const, i = 1, . . . , s.

Moreover, by the definition of equisingularity, χ(V

) = const. Therefore, according to Lemma 2.2, q

(α) = const. So the family {V

}

is good at infinity by Remark 1.3(c).

(b)⇒(a). We denote by grad f the vector grad f := (∂f /∂x, ∂f /∂y), so the chain rule may be expressed by the inner product ∂f /∂v = hv, grad f i.

Assume that α

∈ C

. Since µ

(V

) = const, i = 1, . . . , s, there exists a neighborhood U

of α

such that the family {V

}

satisfies Whitney’s conditions along {a

} × U

at a

, i = 1, . . . , s (see [6], [2]). Thus there exist closed balls D

small enough centered at a

such that D

∩ D

= ∅ (i 6= j) and there exist integrable vector fields ξ

(x, y, α), η

(x, y, α), i = 1, . . . , s, j = 1, . . . , n, nowhere zero on the set

Γ

:= {(x, y, α) ∈ D

× U

| f

(x, y) = 0}

such that the following equations are satisfied:

hξ

(x, y, α), grad f

(x, y)i + ∂f

∂α

(x, y) = 0, hη

(x, y, α), grad f

(x, y)i + √

−1 ∂f

∂α

(x, y) = 0,

for all (x, y, α) ∈ {(x, y, α) ∈ ∂D

× U

| f

(x, y) = 0}. Moreover, by the same method as in [4], integrating the above vector fields, we can get the diffeomorphisms h

, i = 1, . . . , s, such that the diagrams

{(x, y) ∈ D

| f

(x, y) = 0} × U

{(x, y, α) ∈ D

× U

| f

(x, y) = 0}

U

U

π



hi

//

π



//

are commutative and h

(a

, α) = (a

, α).

Further, by Remark 1.3(b), there exist c > 0 and a neighborhood of α

, which we may assume to be precisely U

, such that

(1) ∂f

∂y 6= 0 for all (x, y, α) ∈ Γ

∪ Ω, where

Γ

:= {(x, y, α) ∈ C

× C

| f

(x, y) = 0, α ∈ U

, |x| ≥ c}

and Ω is an open neighborhood of the set

{(x, y, α) ∈ C

× C

| f

(x, y) = 0, α ∈ U

, |x| = c}.

On the other hand, one has

(2) grad f

6= 0 for all (x, y, α) ∈

 C

\

s

[

i=1

D



× U

with f

(x, y) = 0, |x| ≤ c.

From (1) and (2) we conclude that there exist smooth vector fields ξ

(x, y, α) = (ξ

(x, y, α), ξ

(x, y, α)),

η

(x, y, α) = (η

(x, y, α), η

(x, y, α)), j = 1, . . . , n, such that

hξ

(x, y, α), grad f

(x, y)i + ∂f

∂α

(x, y) = 0, hη

(x, y, α), grad f

(x, y)i + √

−1 ∂f

∂α

(x, y) = 0, on the set X := {(x, y, α) ∈ C

× C

| f

(x, y) = 0, α ∈ U

}, and

ξ

(x, y, α) = 0, η

(x, y, α) = 0, for all (x, y, α) ∈ Γ

∪ Ω.

Furthermore, the restrictions of ξ

(resp., η

), j = 1, . . . , n, on Γ

, i = 1, . . . , s, are precisely ξ

(resp., η

). (We can construct such vector fields locally and then extend them over X by a smooth partition of unity.) The resulting vector fields on X are the ones we are looking for. Using them, we may follow again the method of [4] to obtain the global diffeomorphism h such that the diagram

{(x, y) | f

(x, y) = 0} × U

{(x, y, α) | f

(x, y) = 0} ∩ π

(U

)

U

U

π



h

//

π



//

is commutative and h|

= h

, i = 1, . . . , s. The theorem is proved.

2.3. Remark. By definition and the above theorem, it is reasonable to say that the family {V

}

satisfies Whitney’s affine conditions at infinity provided it is good at infinity.

3. In CP

we consider the family of curves

V

:= {(x : y : z) ∈ CP

| z

f

(x/z, y/z) = 0}, α ∈ C

.

Clearly, V

is the compactification of V

. From now on we make the as- sumption that B := V

∩ {z = 0} ⊂ CP

is a finite set, say {b

, . . . , b

}, i.e., the degree d homogeneous parts of f

are independent of α.

3.1. Lemma. Suppose that the Milnor numbers of V

are independent of α, i.e., P

a∈Vα

µ

(V

) = const. Then χ(V

) − χ(V

) =

m

X

j=1

[µ

(V

) − µ

(V

)].

P r o o f. According to [3], χ(V

) = χ(V

) − m

= 2 − (d − 1)(d − 2) + X

a∈Vα

µ

(V

) +

m

X

j=1

µ

(V

) − m, which completes the proof.

From Theorem 2.1, we obtain the following corollary.

3.2. Corollary. Under the hypotheses of Theorem 2.1, (a), hence (b), is equivalent to

(c) µ

(V

) = const, i = 1, . . . , s, and µ

(V

) = const, j = 1, . . . , m.

P r o o f. We need only prove (b)⇔(c). From Lemma 2.2 and Remark 1.3(c), the family {V

}

is good at infinity if and only if χ(V

) = const.

But, by Lemma 3.1, this is equivalent to

m

X

j=1

µ

(V

) = const,

or equivalently (using the semicontinuity of the Milnor number), µ

(V

) = const, j = 1, . . . , m.

3.3. Remark. According to [6], [1] and [2], µ

(V

) = const, j =

1, . . . , m, if and only if the family {V

}

satisfies Whitney’s conditions

along {b

} × C

at b

. In the case of a family of affine plane curves, the

equisingularity is therefore equivalent to Whitney’s conditions at each com-

mon singular point of the curves (including such singular points at infin-

ity).

References

[1] J. B r i a n ¸ c o n et J.-P. S p e d e r, Les conditions de Whitney impliquent “µ

constant”, Ann. Inst. Fourier (Grenoble) 26 (2) (1976), 153–163.

[2] A. D i m c a, Singularities and Topology of Hypersurfaces, Universitex, Springer, New York, 1992.

[3] H ` a H u y V u i et L ˆ e D ˜ u n g T r ´ a n g, Sur la topologie des polynˆ omes complexes, Acta Math. Vietnam. 9 (1984), 21–32.

[4] H ` a H u y V u i and P h a. m T i e n S o n, Topology of families of affine plane curves, Institute of Mathematics, Hanoi, Vietnam, preprint 1, 1997.

[5] L ˆ e D ˜ u n g T r ´ a n g, Topologie des singularit´ es des hypersurfaces complexes, Singu- larit´ es ` a Carg` ese, Ast´ erisque 7–8 (1973), 171–182.

[6] B. T e i s s i e r, Cycles ´ evanescents, sections planes et conditions de Whitney , ibid., 285–362.

Institute of Mathematics Department of Mathematics

P.O. Box 631 Dalat University

Bo-Ho, Hanoi, Vietnam Dalat, Vietnam

Re¸ cu par la R´ edaction le 28.6.1997