1. Introduction. Let f

POLONICI MATHEMATICI LXXI.2 (1999)

Topology of families of affine plane curves

by H` a Huy Vui (Hanoi) and Pham Tien Son (Dalat)

Abstract. We determine bifurcation sets of families of affine curves and study the topology of such families.

1. Introduction. Let f

(x, y) be a family of polynomials of two complex variables (x, y) ∈ C

whose coefficients are polynomial functions of α ∈ C

. We consider the family of affine curves {(x, y) ∈ C

| f

(x, y) = 0}.

In this paper, we first determine the bifurcation set B

, i.e., the smallest set of parameters α such that the family is equisingular outside this set.

Then applying this result, we introduce the notions of semi-cycles vanishing at infinity and study the topology of the family. Finally, we show that our results imply some well-known facts on the topology of polynomial functions ([2], [3], [6], [7]).

2. Bifurcation set of families of affine plane curves. Let f

(x, y) :=

P (x, y, α), α ∈ C

, be a family of polynomials of two variables whose coef- ficients are polynomials of α.

2.1. Definition. The family of affine curves {(x, y) ∈ C

| f

(x, y) = 0}, α ∈ C

, is said to be equisingular outside a set B ⊂ C

if for all α

6∈ B there exist a neighborhood U

of α

and a diffeomorphism h such that the diagram

{(x, y) | f

(x, y) = 0} × U

{(x, y, α) | f

(x, y) = 0} ∩ π

(U

)

U

U

h

//

π





//

is commutative, where π is the second projection.

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

Key words and phrases: affine curves, singularity, equisingularity, Milnor number.

Supported in part by the National Basic Research Program in Natural Sciences, Viet- nam. The second author would like to thank the AUBONNE for financial support.

[129]

Let B

be the smallest set of parameters α such that the family {f

(x, y) = 0} is equisingular ouside B

. We call B

the bifurcation set of the family.

2.2. The following assumptions will be needed throughout the paper:

• deg(f

) = deg

(f

) = d = const;

• the curves {f

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

Let l be the linear function defined by l(x, y) = x. By assumption, the restriction map

l

: f

(0) → C, (x, y) 7→ x, is proper for each α ∈ C

.

Let δ(x, α) := disc

(f

(x, y)) be the discriminant of f

with respect to y.

Then we may write

δ(x, α) = q

(α)x

+ q

(α)x

+ . . . where q

(α), i = 0, . . . , k, are polynomials of α. Put

B

:= {α | q

(α) = 0}.

Denote by C

(f

) the set of critical values of f

.

2.3. Theorem. Assume that 0 6∈ C

(f

) for generic α. Then the bi- furcation set of the family of affine curves {f

(x, y) = 0} is precisely the set

B = {α | 0 ∈ C

(f

)} ∪ B

.

P r o o f. We first prove that the family of affine curves {f

(x, y) = 0} is equisingular outside B. For each polynomial f

, define

grad f

:= (∂f

/∂x, ∂f

/∂y).

Assume that α

6∈ B. Let U

:= {α ∈ C

| kα − α

k < δ} so that C

\ B contains the closure of U

.

By the definition of B, q

(α) 6= 0 for all α ∈ U

; hence there exists c

> 0 such that if α ∈ U

, then δ(x, α) 6= 0 on the set {x ∈ C | |x| ≥ c

}.

By the properties of resultants, the system of equations

 f

(x, y) = 0,

∂f

/∂y = 0,

has no solution on the set {(x, y) ∈ C

| |x| ≥ c

} for any fixed α ∈ U

. Thus

(1) ∂f

/∂y 6= 0 for all (x, y, α) ∈ V

∪ Ω,

where

V

:= {(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, by the definition of U

, one has

(2) grad f

6= 0 for all (x, y, α) ∈ C

× 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

(3)



 



 



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 (4)

( ξ

(x, y, α) = 0, η

(x, y, α) = 0,

for all (x, y, α) ∈ V

∪ Ω. (We can construct such vector fields locally and then extend them over X by a smooth partition of unity.)

To shorten notation, we write e

= (0, . . . , 0, 1, 0, . . . , 0) with 1 in the jth place.

Let ϕ

(x

, y

, α

, τ ) := (ϕ

(x

, y

, α

, τ ), ϕ

(x

, y

, α

, τ ), α(x

, y

, α

, τ )), j = 1, . . . , n, be solutions of the system

(5)



 

 

 

 



 

 

 

 

 dx(τ )

dτ = ξ

(x(τ ), y(τ ), α(τ )), dy(τ )

dτ = ξ

(x(τ ), y(τ ), α(τ )), dα(τ )

dτ = e

,

x(0) = x

, y(0) = y

, α(0) = α

,

and ψ

(u

, v

, β

, τ ) := (ψ

(u

, v

, β

, τ ), ψ

(u

, v

, β

, τ ), β(u

, v

, β

, τ )),

j = 1, . . . , n, be solutions of the system

(6)



 

 

 

 



 

 

 

 

 du(τ )

dτ = η

(u(τ ), v(τ ), β(τ )), dv(τ )

dτ = η

(u(τ ), v(τ ), β(τ )), dβ(τ )

dτ = √

−1e

,

u(0) = u

, v(0) = v

, β(0) = β

,

where (x

, y

, α

), (u

, v

, β

) ∈ X. We conclude from (3) and (5) that d

dτ P (ϕ

(x

, y

, α

, τ )) = ∂P

∂x

∂ϕ

∂τ + ∂P

∂y

∂ϕ

∂τ +

n

X

k=1

∂P

∂α

∂α

∂τ

= ∂P

∂x

∂ϕ

∂τ + ∂P

∂y

∂ϕ

∂τ + ∂P

∂α

∂α

∂τ = 0, hence that

P (ϕ

(x

, y

, α

, τ ), ϕ

(x

, y

, α

, τ ), e

τ + α

) = 0.

Further, it follows from (4) and (5) that d

dτ |ϕ

(x

, y

, α

, τ )|

= d

dτ hϕ

(x

, y

, α

, τ ), ϕ

(x

, y

, α

, τ )i

=  dϕ

(x

, y

, α

, τ )

dτ , ϕ

(x

, y

, α

, τ )

+

ϕ

(x

, y

, α

, τ ), dϕ

(x

, y

, α

, τ ) dτ

= 0, hence

(7) |ϕ

(x

, y

, α

, τ )| = const for each (x

, y

, α

) ∈ V

. Analogously,

P (ψ

(u

, v

, β

, τ ), ψ

(u

, v

, β

, τ ), √

−1 e

τ + β

) = 0;

moreover,

(8) |ψ

(u

, v

, β

, τ )| = const for each (u

, v

, β

) ∈ V

. Since deg(f

) = deg

(f

) = d = const, the restriction map

l

: {(x, y) | f

(x, y) = 0, α ∈ U

} → C, (x, y) 7→ x,

is proper. In addition, it is clear from (7) and (8) that the solutions

ϕ

(x

, y

, α

, τ ) and ψ

(u

, v

, β

, τ ), j = 1, . . . , n, can be extended over

their maximal intervals.

Now define

h : {(x, y) | f

(x, y) = 0} × U

→ {(x, y, α) | f

(x, y) = 0} ∩ π

(U

) by

h(x, y, α) = ψ

(ϕ

(. . . (ψ

(ϕ

(x, y, α

, Re α

− Re α

), Im α

− Im α

), . . . , Re α

− Re α

), Im α

− Im α

) for (x, y) ∈ f

(0) and α ∈ U

. We can easily check that the map h is a diffeomorphism, and that π ◦ h(x, y, α) = π(x, y, α) = α. This gives a trivialization of the fibration over the set U

.

We next prove that the set B is smallest. By contradiction, assume that for some α

∈ B there exist a neighborhood U

of α

and a diffeomorphism h : {(x, y) | f

(x, y) = 0} × U

→ {(x, y, α) | f

(x, y) = 0} ∩ π

(U

).

Consider the following two cases:

Case 1: 0 ∈ C

(f

). Since h is a diffeomorphism of {(x, y) | f

(x, y)=0}

× U

onto {(x, y, α) | f

(x, y) = 0} ∩ π

(U

), it follows that 0 ∈ C

(f

) for all α ∈ U

, a contradiction.

Case 2: α

∈ B

. In this case, the next lemma is needed.

2.4. Lemma. Let F be a polynomial of two complex variables such that the restriction map l|

, V := F

(0), is proper where

l : C

→ C, (x, y) 7→ x.

Suppose that the curve V is reduced. Then

χ(F

(0)) = d − deg disc

F (x, y).

P r o o f. Let x

, β = 1, . . . , p, be the critical values of l|

and (x

, y

), j = 1, . . . , i

, be the corresponding critical points of l|

with multiplicity l

. We may write

disc

F (x, y) = a(x − x

)

· · · (x − x

)

, where a 6= 0 and γ

= P

j=1

l

, β = 1, . . . , p.

Take x

∈ C\{x

, . . . , x

}. Then l|

(x

) consists of d := deg(F ) distinct points. In the x-plane, we consider a system of paths T

, . . . , T

connecting x

, . . . , x

to x

such that

(i) no path T

has self-intersection points;

(ii) T

∩ T

= {x

} (i 6= j).

Put

S(V ) := l| b

 [

i=1

T



.

Then b S(V ) is a union of 1-dimensional curves. Let ˇ S(V ) be the set of all curves in b S(V ) \ l|

(x

) which contain a point of Σ := {(x

, y

) | j = 1, . . . , i

, β = 1, . . . , p}.

Since S

i=1

T

is a deformation retract of C and the restriction map l|

: F

(0) \ Σ → C \ {x

, . . . , x

}, (x, y) 7→ x,

is a locally trivial fibration, it follows that b S(V ) is a deformation retract of V .

It is not hard to see that

S(V ) := ˇ S(V ) ∪ l|

(x

)

is a deformation retract of b S(V ) and also of V = F

(0). (The set S(V ) is called the skeleton of the curve V ([2]).) Hence,

χ(F

(0)) = χ(S(V )).

The set S(V ) can be identified with a 1-dimensional graph of d + P

i

vertices and P

P

j=1

(l

+ 1) edges. Thus, χ(S(V )) =

 d +

p

X

β=1

i



−

p

X

β=1 iβ

X

j=1

(l

+ 1)

= d −

p

X

β=1 iβ

X

j=1

l

= d −

p

X

β=1

γ

= d − deg disc

F (x, y).

We now return to Case 2.

For each α ∈ C

, let k(α) := max{j ∈ {0, . . . , k} | q

(α) 6= 0}. By Lemma 2.4, we conclude that χ(f

(0)) = d − k(α). Since α

∈ B

= q

(0) and α 6∈ B

for generic α, one gets

χ(f

(0)) = d − k < χ(f

(0)), a contradiction, which ends the proof of Theorem 2.3.

2.5. Remark. From the construction of B

, it is reasonable to call each α

∈ B

a bifurcation value corresponding to the singularity at infinity of the family {f

(x, y) = 0}.

3. Topology of families of affine plane curves. From now on, we assume that the curves {f

= 0} are smooth for generic α.

Consider the family of affine curves {f

= 0}. By Lemma 2.4, χ(f

(0))

= d − k(α). On the other hand, by the definition of the set B

, we see that

α

∈ B

iff k(α

) < k = k(α) for generic α. Hence χ(f

(0)) < χ(f

(0))

for generic α.

Thus we can reformulate Theorem 2.3 as follows.

3.1. Theorem. α

∈ C

is a bifurcation value of the family of affine curves {f

(x, y) = 0} if and only if either

(i) the curve {(x, y) | f

(x, y) = 0} is singular , or (ii) χ(f

(0)) < χ(f

(0)) for generic α.

3.2. Remark. In the general case, it is not sufficient to use the group H

(f

(0)) to distinguish the generic curve from the special curve. For example, consider the family of polynomials f

(x, y) = y

+ xy

+ y − α, α ∈ C. Then B

= {0} and rank H

(f

(0)) = rank H

(f

(0)) = 1.

There is no loss of generality in assuming that the map l

= x is simple for each α near a given α

(l

is said to be simple iff l

(x) consists of d − 1 distinguished points for every critical value x of l

). Then the number of singular points of l

is exactly k(α). Let

(x

(α), y

(α)), . . . , (x

(α), y

(α))

be the critical points of the map l

, α 6∈ B

. Now we use the notations as in the proof of Lemma 2.4. Suppose that x

∈ C is a common regular value of l

and l

for all α near α

and let e

(α) := l

(T

) ∩ S(f

(0)), j = 1, . . . , k, be the cycles of the group H

(f

(0), l

(x

)) corresponding to the singular points (x

(α), y

(α)). These elements define a basis of the group H

(f

(0), l

(x

)). By definition, α

∈ B

iff there exist critical points (x

(α), y

(α)) of l

such that k(x

(α), y

(α))k → ∞ as α → α

. Moreover, since the map l

is simple, the number of singular points of l

tending to infinity as α → α

is r := k − k(α

). Therefore, we may assume without loss of generality that such critical points are

(x

(α), y

(α)), . . . , (x

(α), y

(α)).

3.3. Definition. We call e

(α), j = 1, . . . , r, the semi-cycles vanishing at infinity as α → α

.

From Theorem 3.1 and the above definition we easily obtain the follow- ing.

3.4. Theorem. α

is a bifurcation value of the family of affine curves {f

(x, y) = 0} if and only if either

(i) the curve {(x, y) | f

(x, y) = 0} is singular , or

(ii) there exist semi-cycles e

(α) vanishing at infinity as α → α

. The number of such semi-cycles is exactly k − k(α

).

3.5. Remark. The number of semi-cycles e

(α) vanishing at infinity can

be given in other ways as follows.

(i) Let (F, G) be the intersection number of two curves {F = 0} and {G = 0}. Then

χ(f

(0)) − χ(f

(0)) = k − k(α

)

= (f

, ∂f

/∂y) − (f

, ∂f

/∂y)

= dim

C[x, y]/(f

, ∂f

/∂y)

− dim

C[x, y]/(f

, ∂f

/∂y).

(ii) In CP

we consider the family of curves

Γ

:= {(x : y : z) | z

f

x/z, y/z
= 0}.

Clearly, Γ

is the compactification of Γ

:= f

(0). In addition, we assume that the homogeneous part of degree d of f

does not depend on α. Then the curves Γ

intersect the line z = 0 at the same points A

, . . . , A

for any α. Let µ

(Γ

) be the Milnor number of the germ of the analytic curve Γ

at A

. By arguments of [3], we can show that

k − k(α

) = χ(f

(0)) − χ(f

(0))

=

s

X

i=1

[µ

(Γ

) − µ

(Γ

)].

Next, we describe the change in the homotopy type of the curve {f

(x, y) = 0} as α → α

, α

∈ B

.

3.6. Definition. An operation of attaching a 1-dimensional cell to an affine curve V is a map

ϕ : I := [0, 1] → C

such that

(i) ϕ(I) is diffeomorphic to I;

(ii) ϕ(I) ∩ V = {ϕ(0), ϕ(1)}.

The set V

= V ∪ ϕ(I) is called V with a 1-cell attached .

3.7. Theorem. Let α

be a bifurcation value corresponding to the singularity at infinity of the family {f

(x, y) = 0} such that the curve {f

(x, y) = 0} is smooth. Then a generic curve f

(0) may be obtained from f

(0), up to homotopy type, by attaching exactly k − k(α

) 1-dimen- sional cells.

P r o o f. Let S(f

(0)) (resp. S(f

(0))) be the skeleton of the affine

plane curve {f

(x, y) = 0} (resp. {f

(x, y) = 0}) as in the proof of

Lemma 2.4. According to the construction of skeletons, the set S(f

(0))

(resp. S(f

(0))) is a graph with d + k (resp. d + k(α

)) vertices and 2k

(resp. 2k(α

)) edges.

Furthermore, S(f

(0)) is obtained from S(f

(0)) by deleting k−k(α

) vertices (x

(α), y

(α)), β = 1, . . . , r, and k−k(α

) pairs of edges. These pairs of edges connect a point (x

(α), y

(α)) tending to infinity to two distinct points in l

(x

). In other words, the set S(f

(0)) is S(f

(0)) with k − k(α

) 1-cells attached.

On the other hand, the graph S(f

(0)) (resp. S(f

(0))) is a deforma- tion retract of the curve f

(0) (resp. f

(0)). This proves the theorem.

3.8. Corollary. If α

is a bifurcation value at infinity of the family of curves {f

= 0}, then the number of connected components of the curve f

(0) is greater than or equal to the one for f

(0).

Moreover, we can describe a change mechanism of the number of con- nected components when passing from the general curves f

(0) to the special curve f

(0). For that, we need:

3.9. Definition. A subgraph B(α) of the graph S(f

(0)) is said to be a block vanishing at infinity as α → α

if the following four conditions are satisfied.

(i) B(α) is connected;

(ii) each vertex of B(α) either belongs to l

(x

) or tends to infinity as α → α

;

(iii) the number of connected components of S(f

(0)) is different from that of S(f

(0)) \ B(α);

(iv) B(α) is minimal in the sense that there exists no subgraph B

(α) ! B(α) of S(f

(0)) satisfying (i)–(iii).

Let v(α

) be the number of blocks vanishing at infinity as α → α

, and let b

(α) and b

(α

) be the numbers of connected components of f

(0) and f

(0), respectively. By Theorem 3.7, we obtain the following.

3.10. Theorem. b

(α

) − b

(α) = v(α

).

4. Corollaries

4.1. We begin by recalling some facts on the topology of polynomials of two variables.

Let F : C

→ C be a polynomial function. It is well known that there exists a finite set C(F ) ⊂ C, called the bifurcation set of F , such that the restriction

F : C

\ F

(C(F )) → C \ C(F )

is a locally trivial C

-fibration (see, for example, [5], [6], [7], [3]).

We say that a value t

∈ C is regular at infinity if there exist a small

δ > 0 and a compact K ⊂ C

such that the restriction

F : F

(D

) \ K → D

, D

:= {t | |t − t

| < δ}, is a trivial C

-fibration ([4]).

If t

is not regular at infinity, it is called a critical value at infinity of F . Denote by C

(F ) the set of critical values at infinity of F . It is known ([3]) that C(F ) = C

(F ) ∪ C

(F ).

Let F be a polynomial with isolated critical points only. Denote by µ

(F ) the fibre Milnor number of F at c.

Let f

(x, y) := F (x, y) − α, α ∈ C. Using the notation of Remark 3.5, we put

λ

(F ) :=

s

X

i=1

[µ

(Γ

) − µ

(Γ

)], for α

∈ C and generic α.

Now, let us mention an important consequence of the above results.

4.2. Corollary. The following statements are equivalent.

(i) α

∈ B

; (ii) 0 ∈ C(f

);

(iii) µ

(f

) + λ

(f

) > 0.

P r o o f. We first show that

(9) B

= {α | 0 ∈ C

(f

)}.

In fact, let ∆(x, α, t) := disc

(f

(x, y) − t). Then we may write

∆(x, α, t) = Q

(α, t)x

+ Q

(α, t)x

+ . . . According to [1],

C

(f

) = {t ∈ C | Q

(α, t) = 0}.

On the other hand, since 0 6∈ C

(f

) for generic α, Q

(α, 0) 6≡ 0.

Furthermore, because δ(x, α) = ∆(x, α, 0), we have Q

(α, 0) ≡ q

(α) and m(α) ≡ k.

Therefore,

B

= {α | q

(α) = 0} = {α | Q

(α, 0) = 0} = {α | 0 ∈ C

(f

)}.

We now prove the theorem. By [1], t

∈ C

(f

) iff

(10) λ

(f

) > 0.

By (9), (10) and the definition of B

, one has B

= {α | 0 ∈ C(f

)}, from which the assertion easily follows.

In a special case, the following corollary is well known.

4.3. Corollary ([3], [6]). Suppose that F ∈ C[x, y] is a polynomial of two complex variables. Then α

∈ C(F ) if and only if either

(i) α

is a singular value of F , or

(ii) χ(F

(α)) < χ(F

(α

)) for generic α.

P r o o f. Let f

(x, y) := F (x, y) − α, α ∈ C. Then the conclusion follows from Theorem 3.1 and Corollary 4.2.

4.4. Remark. Let f

(x, y) = F (x, y)−α. Then the results of §3 also give us the corresponding results of [2] on the semi-cycles vanishing at infinity and on the construction of the homotopy type of the generic fiber for a global Milnor fibration.

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[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.

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Institute of Mathematics P.O. Box 631

Bo-Ho, Hanoi, Vietnam E-mail: hhvui@ioit.ncst.ac.vn

Department of Mathematics Dalat University Dalat, Vietnam

Re¸ cu par la R´ edaction le 17.11.1997

R´ evis´ e le 11.5.1998