On stable linear deformations of Brieskorn …fj_singularities/FJV2014/...Introduction Main Results Sketch of the proofs On stable linear deformations of Brieskorn singularities of

IntroductionMain Results

Sketch of the proofs

On stable linear deformations of Brieskornsingularities of two variables

(Joint work with K. Inaba, M. Ishikawa and M. Kawashima)Hokkaido, August 26th, 2014

(Joint work with K. Inaba, M. Ishikawa and M. Kawashima) Hokkaido, August 26th , 2014On stable linear deformations of Brieskorn singularities



Introduction

Let f , g : C2 → C be polynomial mappings, f has isolatedsingularity at the origin.

f + tg is a deformation of singularity (f , 0) at 0, t ∈ [0, ε).

For generic g (e.g. linear functions) the deformation f + tghas only Morse singularities in a neighborhood of the origin, tsmall.

Aim of this talk: give an observation about deformation ofsingularities with the conjugates of linear functions:

f (u, v) + au + bv : R4 → R2, a, b ∈ C.




Introduction





f (u, v) + au + bv : R4 → R2, a, b ∈ C.




Introduction





f (u, v) + au + bv : R4 → R2, a, b ∈ C.




Introduction




Aim of this talk: give an observation about deformation ofsingularities with the conjugates of linear functions

:

f (u, v) + au + bv : R4 → R2, a, b ∈ C.




Introduction





f (u, v) + au + bv : R4 → R2, a, b ∈ C.




Introduction

Let F : X n → Y 2 be a smooth map, n ≥ 2.

p ∈ X is a fold singularity of F if one can choose coordinates(u, z1, . . . , zn−1) at p and (U,V ) at F (p) such that in aneighborhood of p, F has the form:

U = u,V =n−1∑i=1

±z2i .

p ∈ X is a cusp singularity of F if one can choose coordinates(u, x , z1, . . . , zn−2) at p and (U,V ) at F (p) such that in aneighborhood of p, F has the form:

U = u,V = x3 + ux +n−2∑i=1

±z2i .




Introduction



U = u,V =n−1∑i=1

±z2i .


U = u,V = x3 + ux +n−2∑i=1

±z2i .




Introduction



U = u,V =n−1∑i=1

±z2i .


U = u,V = x3 + ux +n−2∑i=1

±z2i .




Introduction

The map F is called generic if it has only fold or cuspsingularities.

Jk(X ,Y ): bundle of k-jets of maps from X to Y .

Si (X ,Y ) = {j1(h)(p) : rank(dh)(p) = 2− i} ⊂ J1(X ,Y ).

Si (h) := {p ∈ X : rank(dh)(p) = 2− i}, h ∈ C∞(X ,Y ).

S21 (X ,Y ) = {j2(h)(p) : j1(h)(p) ∈ S1(X ,Y ), j1(h)−tp

S1(X ,Y ), rank d(h|S1(h))(p) = min(2, dimS1(h))− 1}.




Introduction

The map F is called generic if it has only fold or cuspsingularities.

Jk(X ,Y ): bundle of k-jets of maps from X to Y .

Si (X ,Y ) = {j1(h)(p) : rank(dh)(p) = 2− i} ⊂ J1(X ,Y ).

Si (h) := {p ∈ X : rank(dh)(p) = 2− i}, h ∈ C∞(X ,Y ).

S21 (X ,Y ) = {j2(h)(p) : j1(h)(p) ∈ S1(X ,Y ), j1(h)−tp

S1(X ,Y ), rank d(h|S1(h))(p) = min(2, dimS1(h))− 1}.




Introduction

It is known that F is generic if and only if:

(a) J1(F ) : X → J1(X ,Y ) is transversal to S1(X ,Y ) andS2(X ,Y );

(b) J2(F ) : X → J2(X ,Y ) is transversal to S21 (X ,Y ).

{ Generic maps} ⊂ C∞(X ,Y ) is an open, dense subset.




Introduction

It is known that F is generic if and only if:

(a) J1(F ) : X → J1(X ,Y ) is transversal to S1(X ,Y ) andS2(X ,Y );

(b) J2(F ) : X → J2(X ,Y ) is transversal to S21 (X ,Y ).

{ Generic maps} ⊂ C∞(X ,Y ) is an open, dense subset.




Introduction

The smooth map h : X → Y is called stable if for any map h′

in a neighborhood of h in C∞(X ,Y ) there existdiffeomorphisms Φ : X → X and Ψ : Y → Y such thath = Ψ ◦ h′ ◦ Φ.

Stability =⇒ Genericity.

F is locally stable at p if and only if either p is a regularpoint, or a fold or a cusp singularity of F .




Introduction








Introduction








Results

Let f be the Brieskorn polynomial f (u, v) = up + vq.

Theorem 1

Let f (u, v) = up + vq, p, q ≥ 2, be a Brieskorn polynomial of twovariables. If a, b ∈ C are generic then f (u, v) + au + bv is a genericmap.

Theorem (K. Inaba 2014)

Let p(z) and q(z) be 2-variable convenient weighted homogeneouscomplex polynomials such that p(z)q(z) has an isolated singularityat 0 and U be a small neighborhood of 0. Then there exists adeformation Ft(z) of p(z)q(z) such that any singularity of Ft(z) isan indefinite fold singularity in U \ {0} for any t small.




Results


Theorem 1







Results


Theorem 1







Results

Theorem 2

Suppose that F (u, v) := f (u, v) + au + bv is a generic map. Thenthe number of cusps of F is between (p + 1)(q − 1) and(p − 1)(q + 1).

Corollary

Assume that p = q and F (u, v) is a generic map. Then thenumber of cusps is (p − 1)(p + 1).




Results

Theorem 2


Corollary

Assume that p = q and F (u, v) is a generic map. Then thenumber of cusps is (p − 1)(p + 1).




Example

Let F (u, v) = u2 + v2 + au + bv .

(1) For any a, b ∈ R>0 the map F is generic.

(2) If |a| = |b| and arg a− arg b = 32(2k + 1)π + 2l

3 π(k , l ∈ Z)then F (sing F ) = {pt}. Otherwise, F (sing F ) is a closedone-dimensional curve whose singularities are three cusps.




Example

Let F (u, v) = u2 + v2 + au + bv .







Example

Let F (u, v) = u2 + v2 + au + bv .







Higher differentialsProof of Theorem 1Proof of Theorem 2

Second differentials

Let f : X n → Y 2 smooth, n ≥ 2.

df : T (X )→ f −1(TY ) and dfx := df |TxX .

Si (f ) := {x ∈ X : k(x) := rank dfx = min(n, 2)− i}.

For x ∈ X , y ∈ Y , let U,V be small neighborhoods of x , ysuch that f (U) ⊂ V and T (X )|U,T (Y )|V are trivial.

Choose bases {ui}, {vj} of those bundles and let {u∗i }, {v∗j }be the associate dual bases.

Let wj := vj ◦ f ,w∗j := v∗j ◦ f . Then

df =∑ij

aiju∗i ⊗ wj ,

where(df (ui ))x =

∑i ,j

aij(x)wj(x).








Si (f ) := {x ∈ X : k(x) := rank dfx = min(n, 2)− i}.For x ∈ X , y ∈ Y , let U,V be small neighborhoods of x , ysuch that f (U) ⊂ V and T (X )|U,T (Y )|V are trivial.



df =∑ij

aiju∗i ⊗ wj ,

where(df (ui ))x =

∑i ,j

aij(x)wj(x).











df =∑ij

aiju∗i ⊗ wj ,

where(df (ui ))x =

∑i ,j

aij(x)wj(x).











df =∑ij

aiju∗i ⊗ wj ,

where(df (ui ))x =

∑i ,j

aij(x)wj(x).






Let E := T (X )|Si (f ),F := f −1(TY )|Si (f ).

Define L and G by the exactness of the following:

0→ L→ Edf→F

π1→G → 0.

One sees that L is a (n− k)-bundle and G is a (2− k)-bundle(k = min(n, 2)− i).

Define map ϕ1 : E → L∗ ⊗ F by, x ∈ Si (f ), k ∈ TxX andt ∈ Lx :

ϕ1x(k)(t) :=

∑i ,j

〈k, daij(x)〉〈t, u∗i (x)〉wj(x).

The second differential d2f : E → L∗ ⊗ G of f is defined asd2fx(k)(t) := π1(ϕ1

x(k)(t)).








0→ L→ Edf→F

π1→G → 0.



ϕ1x(k)(t) :=

∑i ,j



x(k)(t)).








0→ L→ Edf→F

π1→G → 0.



ϕ1x(k)(t) :=

∑i ,j



x(k)(t)).








0→ L→ Edf→F

π1→G → 0.



ϕ1x(k)(t) :=

∑i ,j



x(k)(t)).








0→ L→ Edf→F

π1→G → 0.



ϕ1x(k)(t) :=

∑i ,j



x(k)(t)).






Proposition 1 (H.I. Levine (topology 1965))

The map J1(f ) is transversal to Si (X ,Y ) at x ∈ Si (f ) if and onlyif the map

d2fx : TxX → L∗x ⊗ Gx

has rank (n − k)(2− k) = i(|n − 2|+ i).





Third differentials

Assume that J1(f )−t Si (X ,Y ) at x ∈ Si (f ).

Let ψ := d2fx | ker dfx , then ψ is defined by a(n − k)× (n − k)(p − k) matrix (k = k(x)).

We say that x ∈ Si ,h(f ) if and only if rankψ = n − k − h.

Define T by the following exact sequence:

0→ T → Ed2f→L∗ ⊗ G → 0.

Similarly, by restricting on Si ,h(f ) we receive the exactsequence:

0→ R → Ld2f→L∗ ⊗ G

π2→K → 0,

where dimRx = h anddimKx = h(p − k) + (n − k − h)(p − k − 1).





Third differentials





0→ T → Ed2f→L∗ ⊗ G → 0.


0→ R → Ld2f→L∗ ⊗ G

π2→K → 0,






Third differentials





0→ T → Ed2f→L∗ ⊗ G → 0.


0→ R → Ld2f→L∗ ⊗ G

π2→K → 0,






Third differentials





0→ T → Ed2f→L∗ ⊗ G → 0.


0→ R → Ld2f→L∗ ⊗ G

π2→K → 0,






Third differentials





0→ T → Ed2f→L∗ ⊗ G → 0.


0→ R → Ld2f→L∗ ⊗ G

π2→K → 0,






Third differentials

we define the bundle R∗© K by the exact sequence:

0→ R∗© K → R∗ ⊗ Kξ→R∗ ∧ R∗ ⊗ G → 0,

where ξ is the composition of

R∗ ⊗ K → R∗ ⊗ R∗ ⊗ G → R∗ ∧ R∗ ⊗ G .

We have the projection maps:

π1 : f −1(TY )→ coker df , π2 : L∗ ⊗ G → coker d2f

which induce:

p1 : R∗⊗ L∗⊗ F → R∗⊗ L∗⊗G , p2 : R∗⊗ L∗⊗G → R∗⊗K .





Third differentials

we define the bundle R∗© K by the exact sequence:

0→ R∗© K → R∗ ⊗ Kξ→R∗ ∧ R∗ ⊗ G → 0,

where ξ is the composition of

R∗ ⊗ K → R∗ ⊗ R∗ ⊗ G → R∗ ∧ R∗ ⊗ G .

We have the projection maps:

π1 : f −1(TY )→ coker df , π2 : L∗ ⊗ G → coker d2f

which induce:

p1 : R∗⊗ L∗⊗ F → R∗⊗ L∗⊗G , p2 : R∗⊗ L∗⊗G → R∗⊗K .





Third differentials

Let {z1, . . . , zn−k} be a part of a basis of sections in E oversome neighborhood U of x such that {zj |Si (f ) ∩ U} is a basisfor L|U ∩ Si (f ).

We define ϕ2 : T → R∗ ⊗ L∗ ⊗ F by

ϕ2x(t)(r⊗l) :=

∑i ,j ,m

〈t, d(〈zi , damj〉)(x)〉〈l , z∗m(x)〉〈r , z∗i (x)〉wj(x),

r ∈ R, l ∈ L.

The third differential d3f : T → R∗ ⊗ K of f is defined as

d3fx := p2p1ϕ2x .





Third differentials



ϕ2x(t)(r⊗l) :=

∑i ,j ,m


r ∈ R, l ∈ L.


d3fx := p2p1ϕ2x .





Third differentials



ϕ2x(t)(r⊗l) :=

∑i ,j ,m


r ∈ R, l ∈ L.


d3fx := p2p1ϕ2x .





Third differentials

Lemma (H. I. Levine (Topology 1965))

The map d3f is well defined over Si ,h(f ) and d3f (T ) ⊂ R∗© K .

Proposition 2 (H. I. Levine (Topology 1965))

The map J2(f ) is transversal to S21 (X ,Y ) at x if and only if

d3fx : Tx → R∗© K is surjective.





Third differentials

Lemma (H. I. Levine (Topology 1965))

The map d3f is well defined over Si ,h(f ) and d3f (T ) ⊂ R∗© K .

Proposition 2 (H. I. Levine (Topology 1965))

The map J2(f ) is transversal to S21 (X ,Y ) at x if and only if

d3fx : Tx → R∗© K is surjective.





Fold and cusp singularities

The map f is generic at p if and only if:

(1) p is a regular point of f , or

(2) p ∈ S1(f ), d2fp : TpX → L∗p ⊗ Gp is surjective; and

(a) p /∈ S1,1(f ) (a fold singularity), or(b) p ∈ S1,1(f ) and d3fp : Tp → R∗p © Kp is surjective (a cusp

singularity).






Let P = (P1,P2) : R4 → R2 be a smooth map and p ∈ S1(P).

Assume that (x1, x2, x3, x4) is the local coordinate at p suchthat p = 0, gradP1(0) = (1, 0, 0, 0) andgradP2(0) = (0, 0, 0, 0).

d2P0 is given by the matrix:

M = (∂2P2

∂xi∂xj: i = 1, 2, 3, 4; j = 2, 3, 4).

The restriction d2P0| ker df0 is given by:

H = (∂2P2

∂xi∂xj: i = 2, 3, 4; j = 2, 3, 4)

Then 0 ∈ S1,1(P) means rankH = 2.









M = (∂2P2

∂xi∂xj: i = 1, 2, 3, 4; j = 2, 3, 4).


H = (∂2P2

∂xi∂xj: i = 2, 3, 4; j = 2, 3, 4)










M = (∂2P2

∂xi∂xj: i = 1, 2, 3, 4; j = 2, 3, 4).


H = (∂2P2

∂xi∂xj: i = 2, 3, 4; j = 2, 3, 4)







If 0 ∈ S1,1(P), assume that ∂2P2∂x4∂xj

(0) = 0 for all j = 2, 3, 4.

The map d3P0 : T0 → R∗ ⊗ K is given by

t 7→ t∂3P2

∂x34

(0)〈., dx4〉

which is surjective if and only if ∂3P2

∂x34

(0) 6= 0.






If 0 ∈ S1,1(P), assume that ∂2P2∂x4∂xj

(0) = 0 for all j = 2, 3, 4.

The map d3P0 : T0 → R∗ ⊗ K is given by

t 7→ t∂3P2

∂x34

(0)〈., dx4〉

which is surjective if and only if ∂3P2

∂x34

(0) 6= 0.






Conclusion:

If detH 6= 0 then 0 is a fold singularity of P.

If rankH = 2 and ∂2P2∂x4∂xj

(0) = 0 for j = 2, 3, 4 then 0 is a cusp

singularity of P if and only if ∂3P2

∂x34

(0) 6= 0.





Proof of Theorem 1

We are interested in the mixed polynomialF (u, v) := up + vq + au + bv : C2 → C.

For ab 6= 0, let c1, c2 ∈ C \ {0} satisfying cp1 = ac1 andcq2 = bc2.

Changing the coordinates as z := c1u and w := c2v andsetting µ = c1/c2, then

F (z ,w) = bc2(µ(up + u) + vq + v).

P(u, v ;µ) := µ(up + u) + vq + v , p, q ≥ 2, µ ∈ C \ {0}.Study when P is a generic map instead of F .





Proof of Theorem 1




F (z ,w) = bc2(µ(up + u) + vq + v).






Proof of Theorem 1




F (z ,w) = bc2(µ(up + u) + vq + v).






Proof of Theorem 1




F (z ,w) = bc2(µ(up + u) + vq + v).






Proof of Theorem 1

Let Q(u, v ;µ) and R(u, v ;µ) be the real and imaginary partof P(u, v ;µ), i.e., P(u, v ;µ) = Q(u, v ;µ) + iR(u, v ;µ).

Set r1 = |u|, θ1 = arg u, r2 = |v | and θ2 = arg v , so that(r1, θ1, r2, θ2) are regarded as the polar coordinates of C2.

Lemma (M. Oka (Kodai J. Math. 2008))

A singular point z0 = (u0, v0) of the mixed polynomial P is givenby

∂P

∂u(z0) = α

∂P

∂u(z0) and

∂P

∂v(z0) = α

∂P

∂v(z0),

where α is a complex number with |α| = 1.





Proof of Theorem 1





∂P

∂u(z0) = α

∂P

∂u(z0) and

∂P

∂v(z0) = α

∂P

∂v(z0),






Proof of Theorem 1





∂P

∂u(z0) = α

∂P

∂u(z0) and

∂P

∂v(z0) = α

∂P

∂v(z0),






Proof of Theorem 1

A singular point z0 = (u0, v0) of P satisfies p|u0|p−1 = q|v0|q−1 = 1,

p − 1

2arg u0 + argµ =

q − 1

2arg v0 + nπ,

where n is some integer.





Proof of Theorem 1

(k1, k2, k3, k4) := (Qr1(z0),Qθ1(z0),Qr2(z0),Qθ2(z0)).

Denote

Θ1 =p + 1

2arg u0, Θ2 =

p − 1

2arg u0 + arg µ,

Θ3 =q + 1

2arg v0, Θ4 =

q − 1

2arg v0.

Then

k1 = 2|µ| cos Θ1 cos Θ2, k2 = −2|µ||u0| sin Θ1 cos Θ2,k3 = 2 cos Θ3 cos Θ4, k4 = −2|v0| sin Θ3 cos Θ4.





Proof of Theorem 1


Denote

Θ1 =p + 1

2arg u0, Θ2 =

p − 1

2arg u0 + arg µ,

Θ3 =q + 1

2arg v0, Θ4 =

q − 1

2arg v0.

Then






Proof of Theorem 1


Denote

Θ1 =p + 1

2arg u0, Θ2 =

p − 1

2arg u0 + arg µ,

Θ3 =q + 1

2arg v0, Θ4 =

q − 1

2arg v0.

Then






Case Qr1(z0) 6= 0

Let (r ′1, θ′1, r′2, θ′2) be new coordinates in a small neighborhood

of z0 defined by

(r ′1, θ′1, r′2, θ′2) = (k1r1 + k2θ1 + k3r2 + k4θ2, θ1, r2, θ2).

(Qr ′1(z0),Qθ′1(z0),Qr ′2

(z0),Qθ′2(z0)) = (1, 0, 0, 0).

Let R = R − sQ, where s =Rr′

1(z0)

Qr′1(z0)

then P ∼ (Q, R) and

grad R(z0) = 0.





Case Qr1(z0) 6= 0


of z0 defined by


(Qr ′1(z0),Qθ′1(z0),Qr ′2

(z0),Qθ′2(z0)) = (1, 0, 0, 0).


1(z0)

Qr′1(z0)


grad R(z0) = 0.





Case Qr1(z0) 6= 0


of z0 defined by


(Qr ′1(z0),Qθ′1(z0),Qr ′2

(z0),Qθ′2(z0)) = (1, 0, 0, 0).


1(z0)

Qr′1(z0)


grad R(z0) = 0.





Case Qr1(z0) 6= 0

Let H be the Hessian of R with variables (θ′1, r′2, θ′2).

At the singular point z0 = (u0, v0) of P:

detH(z0) = − 1

k21

4(p − 1)(q − 1)|µ|2 cos Θ2φ(z0),

where

φ(z0) = (−1)n(p − 1)|v0| sin Θ3 + (q − 1)|µ||u0| sin Θ1.

Lemma

If a singular point z0 = (u0, v0) of P satisfies k1 6= 0 and φ(z0) 6= 0then it is a fold.





Case Qr1(z0) 6= 0



detH(z0) = − 1

k21

4(p − 1)(q − 1)|µ|2 cos Θ2φ(z0),

where


Lemma






Case Qr1(z0) 6= 0



detH(z0) = − 1

k21

4(p − 1)(q − 1)|µ|2 cos Θ2φ(z0),

where


Lemma






Case Qr1(z0) 6= 0 and Rθ′1θ′1(z0) 6= 0

If detH = 0 then rankH = 2.

z0 is a cusp singularity if and only if ψ(z0) 6= 0 where

ψ(z0) = (p+1)(q−1) sin Θ3 cos Θ1−(p−1)(q+1) sin Θ1 cos Θ3.

For generic choice of µ then there is no z0 such thatφ(z0) = 0 and ψ(z0) = 0. It means z0 is a fold singularity or acusp singularity.





Case Qr1(z0) 6= 0 and Rθ′1θ′1(z0) 6= 0









Case Qr1(z0) 6= 0 and Rθ′1θ′1(z0) 6= 0









Case Qr1(z0) 6= 0 and Rθ′1θ′1(z0) = 0 or Qr1(z0) = 0

By the same argument, one can show that for generic µ then themap has only fold singularities.





Theorem 2


Remark

If P : R4 → R2 is a generic map. Then S1(P) is a submanifold ofR4. The cusp singularities are singular points of the restriction:P|S1(P) : S1(P)→ R2.





Theorem 2


Remark

If P : R4 → R2 is a generic map. Then S1(P) is a submanifold ofR4. The cusp singularities are singular points of the restriction:P|S1(P) : S1(P)→ R2.





Case ab 6= 0

F ∼ P(u, v) = µ(up + u) + vq + v , µ ∈ C∗.

The set of singular points of P consisting r curvesCk , k = 0, . . . , r − 1 on the torus{(u, v) ∈ C2 : |u| = A, |v | = B}.Each of which is parameterized as

(u, v) = (Ae(q−1rθ+ck )i ,Be

p−1rθi), where

r = gcd(p − 1, q − 1),A = 1/p1/(p−1),B = 1/q1/(q−1), ck =1

p−1(−2 argµ+ 2kπ).

Let Pk := P|Ck: Ck → C as

Pk(θ) = P((Ae(q−1rθ+ck )i ,Be

p−1rθi)).

Since P is generic, the number of cusps of P corresponding tothe number of zeros of dPk

dθ (θ) = 0 on [0, 2π].





Case ab 6= 0

F ∼ P(u, v) = µ(up + u) + vq + v , µ ∈ C∗.The set of singular points of P consisting r curvesCk , k = 0, . . . , r − 1 on the torus{(u, v) ∈ C2 : |u| = A, |v | = B}.Each of which is parameterized as


p−1rθi), where

r = gcd(p − 1, q − 1),A = 1/p1/(p−1),B = 1/q1/(q−1), ck =1

p−1(−2 argµ+ 2kπ).



p−1rθi)).


dθ (θ) = 0 on [0, 2π].





Case ab 6= 0



p−1rθi), where

r = gcd(p − 1, q − 1),A = 1/p1/(p−1),B = 1/q1/(q−1), ck =1

p−1(−2 argµ+ 2kπ).



p−1rθi)).


dθ (θ) = 0 on [0, 2π].





Case ab 6= 0



p−1rθi), where

r = gcd(p − 1, q − 1),A = 1/p1/(p−1),B = 1/q1/(q−1), ck =1

p−1(−2 argµ+ 2kπ).



p−1rθi)).


dθ (θ) = 0 on [0, 2π].





Case ab 6= 0

dPkdθ = −2e

(p−1)(q−1)2r

θiΦ(θ, |µ|), with

Φ(θ, |µ|) = (−1)k |µ|q−1r A sin

((p+1)(q−1)

2r θ + p+12 ck

)+

p−1r B sin

((q+1)(p−1)

2r θ).

Lemma

If p > q (resp. p < q) then the number of solutions ofΦ(θ, |µ|) = 0 is monotone decreasing (resp. increasing) accordingto the parameter |µ| ∈ (0,∞).





Case ab 6= 0

dPkdθ = −2e

(p−1)(q−1)2r

θiΦ(θ, |µ|), with

Φ(θ, |µ|) = (−1)k |µ|q−1r A sin

((p+1)(q−1)

2r θ + p+12 ck

)+

p−1r B sin

((q+1)(p−1)

2r θ).

Lemma

If p > q (resp. p < q) then the number of solutions ofΦ(θ, |µ|) = 0 is monotone decreasing (resp. increasing) accordingto the parameter |µ| ∈ (0,∞).





Case ab 6= 0

Lemma

Suppose p = q. Then, the number of the solutions of Φ(θ, |µ|) = 0

is p2−1r except the finite number of points on |µ| = 1 satisfying

sin(p+12 ck) = 0.





Proof of Theorem 2

If p 6= q.

(a) When |µ| is small, the number of cusps of P on Ck is the

number of solutions of sin( (p−1)(q+1)2r θ) = 0 which is (p−1)(q+1)

r .(b) If |µ| is large, it is equal to the number of solutions of

sin( (p+1)(q−1)2r θ + (p + 1)/2ck) = 0 which is (p+1)(q−1)

r .

If p = q. The above Lemma implies the assertion except for|µ| = 1 and sin((p + 1)/2ck) = 0. But for these maps, usingthe argument in previous Section, one can prove that eitherthe map is not generic or the number of cusps is(p − 1)(p + 1).





Proof of Theorem 2

If p 6= q.



r .

(b) If |µ| is large, it is equal to the number of solutions of


r .






Proof of Theorem 2

If p 6= q.





r .






Proof of Theorem 2

If p 6= q.





r .






Case ab = 0

Assume a 6= 0, b = 0. Then F ∼ Q = up + u + vq.

If q > 2 then d2Q| ker dfx has rank less than 3 for any singularpoint x . It implies that any singular point is not a fold. Thenthe map is not generic.

If q = 2: the singular set is parameterized by (Aeθi , 0). Wehave

dQ(Aeθi , 0)

dθ= −2µAe

p−12θi sin

p + 1

2θ.

Therefore either Q has p + 1 cusps or it is not a generic map.





Case ab = 0




dQ(Aeθi , 0)

dθ= −2µAe

p−12θi sin

p + 1

2θ.






Case ab = 0




dQ(Aeθi , 0)

dθ= −2µAe

p−12θi sin

p + 1

2θ.






Conjecture

Let f (u, v) = up + vq, p, q ≥ 2, be a Brieskorn polynomial of twovariables. Let ft be a deformation of f into a generic map. Thenthe number of cusps of ft , t > 0, appearing in a fixed smallneighborhood of the origin is between (p − 1)(q + 1) and(p + 1)(q − 1).





Thank you for your attention!


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