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DEFORMATIONS OF LEGENDRIAN CURVES

MARCO SILVA MENDES AND ORLANDO NETO

Abstract. We construct versal and equimultiple versal deformations

of the parametrization of a Legendrian curve.

1. Contact Geometry

Let (X, OX) be a complex manifold of dimension 3. A diﬀerential form ω

of degree 1 is said to be a contact form if ω∧dω never vanishes. Let ωbe

a contact form. By Darboux’s theorem for contact forms there is locally a

system of coordinates (x, y, p) such that ω=dy −pdx. If ωis a contact form

and fis a holomorphic function that never vanishes, f ω is also a contact

form. We say that a locally free subsheaf Lof Ω1

Xis a contact structure on

Xif Lis locally generated by a contact form. If Lis a contact structure on

Xthe pair (X, L) is called a contact manifold. Let (X1,L1) and (X2,L2) be

contact manifolds. Let χ:X1→X2be a holomorphic map. We say that χ

is a contact transformation if χ∗ωis a local generator of L1whenever ωis

a local generator of L2.

Let θ=ξdx +ηdy denote the canonical 1-form of T∗C2=C2×C2. Let

π:P∗C2=C2×P1→C2be the projective cotangent bundle of C2, where

π(x, y;ξ:η)=(x, y). Let U[V] be the open subset of P∗C2deﬁned by

η6= 0 [ξ6= 0]. Then θ/η [θ/ξ] deﬁnes a contact form dy −pdx [dx −qdy] on

U[V], where p=−ξ/η [q=−η/ξ]. Moreover, dy −pdx and dx −qdy deﬁne

a structure of contact manifold on P∗C2.

If Φ(x, y) = (a(x, y), b(x, y)) with a, b ∈C{x, y}is an automorphism of

(C2,(0,0)), we associate to Φ the germ of contact transformation

χ: (P∗C2,(0,0; 0 : 1)) →P∗C2,(0,0; −∂xb(0,0) : ∂xa(0,0)

deﬁned by

(1.1) χ(x, y;ξ:η)=(a(x, y), b(x, y); ∂ybξ −∂xbη :−∂yaξ +∂xaη).

If DΦ(0,0) leaves invariant {y= 0}, then ∂xb(0,0) = 0, ∂xa(0,0) 6= 0 and

χ(0,0; 0 : 1) = (0,0; 0 : 1). Moreover,

χ(x, y, p) = (a(x, y), b(x, y),(∂ybp +∂xb)/(∂yap +∂xa)) .

Let (X, L) be a contact manifold. A curve Lin Xis called Legendrian if

ω|L= 0 for each section ωof L.

Date: June 24, 2018.

1

arXiv:1607.02873v1 [math.AG] 11 Jul 2016

Let Zbe the germ at (0,0) of an irreducible plane curve parametrized by

(1.2) ϕ(t) = (x(t), y(t)).

We deﬁne the conormal of Zas the curve parametrized by

(1.3) ψ(t)=(x(t), y(t); −y0(t) : x0(t)).

The conormal of Zis the germ of a Legendrian curve of P∗C2.

We will denote the conormal of Zby P∗

ZC2and the parametrization (1.3)

by Con ϕ.

Assume that the tangent cone C(Z) is deﬁned by the equation ax+by = 0,

with (a, b)6= (0,0). Then P∗

ZC2is a germ of a Legendrian curve at (0,0; a:

b).

Let f∈C{t}. We say the fhas order kand write ord f =kor ordtf=k

if f/tkis a unit of C{t}.

Remark 1.1.Let Zbe the plane curve parametrized by (1.2). Let L=P∗

ZC2.

Then:

(i) C(Z) = {y= 0}if and only if ord y > ord x. If C(Z) = {y= 0},L

admits the parametrization

ψ(t)=(x(t), y(t), y0(t)/x0(t))

on the chart (x, y, p).

(ii) C(Z) = {y= 0}and C(L) = {x=y= 0}if and only if ord x <

ord y < 2ord x.

(iii) C(Z) = {y= 0}and {x=y= 0}*C(L)⊂ {y= 0}if and only if

ord y ≥2ord x.

(iv) C(L) = {y=p= 0}if and only if ord y > 2ord x.

(v) mult L ≤mult Z. Moreover, mult L =mult Z if and only if ord y ≥

2ord x.

If Lis the germ of a Legendrian curve at (0,0; a:b), π(L) is a germ of a

plane curve of (C2,(0,0)). Notice that all branches of π(L) have the same

tangent cone.

If Zis the germ of a plane curve with irreducible tangent cone, the union

Lof the conormal of the branches of Zis a germ of a Legendrian curve. We

call Lthe conormal of Z.

If C(Z) has several components, the union of the conormals of the branches

of Zis a union of several germs of Legendrian curves.

If Lis a germ of Legendrian curve, Lis the conormal of π(L).

Consider in the vector space C2, with coordinates x, p, the symplectic

form dp ∧dx. We associate to each symplectic linear automorphism

(p, x)7→ (αp +βx, γ p +δx)

of C2the contact transformation

(1.4) (x, y, p) = (γp +δx, y +1

2αγp2+βγxp +1

2βδx2, αp +βx).

We call (1.4) a paraboloidal contact transformation.

2

In the case α=δ= 0 and γ=−β= 1 we get the so called Legendre

transformation

Ψ(x, y, p)=(p, y −px, −x).

We say that a germ of a Legendrian curve Lof (P∗C2,(0,0; a:b)) is in

generic position if C(L)6⊃ π−1(0,0).

Remark 1.2.Let Lbe the germ of a Legendrian curve on a contact manifold

(X, L) at a point o. By the Darboux’s theorem for contact forms there is

a germ of a contact transformation χ: (X, o)→(U, (0,0,0)), where U=

{η6= 0}is the open subset of P∗C2considered above. Hence C(π(χ(L))) =

{y= 0}. Applying a paraboloidal transformation to χ(L) we can assume

that χ(L) is in generic position. If C(L) is irreducible, we can assume

C(χ(L)) = {y=p= 0}.

Following the above remark, from now on we will always assume that every

Legendrian curve germ is embedded in (C3

(x,y,p), ω), where ω=dy −pdx.

Example 1.3.(1) The plane curve Z={y2−x3= 0}admits a parametriza-

tion ϕ(t)=(t2, t3). The conormal Lof Zadmits the parametrization

ψ(t)=(t2, t3,3

2t). Hence C(L) = π−1(0,0) and Lis not in generic

position. If χis the Legendre transformation, C(χ(L)) = {y=p=

0}and Lis in generic position. Moreover, π(χ(L)) is a smooth curve.

(2) The plane curve Z={(y2−x3)(y2−x5) = 0}admits a parametriza-

tion given by

ϕ1(t1) = (t12, t13), ϕ2(t2)=(t22, t25).

The conormal Lof Zadmits the parametrization given by

ψ1(t1) = (t12, t13,3

2t1), ψ2(t2)=(t22, t25,5

2t23).

Hence C(L1) = π−1(0,0) and Lis not in generic position. If χis the

paraboloidal contact transformation

χ: (x, y, p)7→ (x+p, y +1

2p2, p),

then χ(L) has branches with parametrization given by

χ(ψ1)(t1) = (t12+3

2t1, t13+9

8t12,3

2t1),

χ(ψ2)(t2) = (t22+5

2t23, t25+25

8t26,5

2t23).

Then

C(χ(L1)) = {y=p−x= 0}, C(χ(L2)) = {y=p= 0}

and Lis in generic position.

3

2. Relative Contact Geometry

Set x= (x1, . . . , xn) and z= (z1, . . . , zm). Let Ibe an ideal of the ring

C{z}. Let e

Ibe the ideal of C{x,z}generated by I.

Lemma 2.1. (a) Let f∈C{x,z},f=Pαaαxαwith aα∈C{z}. Then

f∈e

Iif and only if aα∈Ifor each α.

(b) If f∈e

I, then ∂xif∈e

Ifor 1≤i≤n.

(c) Let a1, . . . , an−1∈C{x,z}. Let b, β0∈e

I. Assume that ∂xnβ0= 0. If β

is the solution of the Cauchy problem

(2.1) ∂xnβ−

n−1

X

i=1

ai∂xiβ=b, β −β0∈C{x,z}xn,

then β∈e

I.

Proof. There are g1, . . . , g`∈C{z}such that I= (g1, . . . , g`). If aα∈I

for each α, there are hi,α ∈C{z}such that aα=P`

i=1 hi,αgi. Hence f=

P`

i=1(Pαhi,α xα)gi∈e

I.

If f∈e

I, there are Hi∈C{x,z}such that f=P`

i=1 Higi. There are

bi,α ∈C{z}such that Hi=Pαbi,αxα. Therefore aα=P`

i=1 bi,αgi∈I.

We can perform a change of variables that rectiﬁes the vector ﬁeld ∂xn−

Pn−1

i=1 ai∂xi, reducing the Cauchy problem (2.1) to the Cauchy problem

∂xnβ=b, β −β0∈C{x,z}xn.

Hence statements (b) and (c) follow from (a).

Let Jbe an ideal of C{z}contained in I. Let X, S and Tbe analytic

spaces with local rings C{x},C{z}/I and C{z}/J. Hence X×Sand X×T

have local rings O:= C{x,z}/e

Iand e

O:= C{x,z}/e

J. Let a1,...,an−1,b∈

Oand g∈ O/xnO. Let ai, b ∈e

Oand g∈e

O/xne

Obe representatives of ai,b

and g. Consider the Cauchy problems

(2.2) ∂xnf+

n−1

X

i=1

ai∂xif=b, f +xne

O=g

and

(2.3) ∂xnf+

n−1

X

i=1

ai∂xif=b,f+xnO=g.

Theorem 2.2. (a) There is one and only one solution of the Cauchy prob-

lem (2.2).

(b) If fis a solution of (2.2), f=f+e

Iis a solution of (2.3).

(c) If fis a solution of (2.3) there is a representative fof fthat is a

solution of (2.2).

4

Proof. By Lemma 2.1, ∂xie

I=e

I. Hence (b) holds.

Assume J= (0). The existence and uniqueness of the solution of (2.2) is

a special case of the classical Cauchy-Kowalevski Theorem. There is one and

only one formal solution of (2.2). Its convergence follows from the majorant

method.

The existence of a solution of (2.3) follows from (b).

Let f1,f2be two solutions of (2.3). Let fjbe a representative of fjfor

j= 1,2. Then ∂xn(f2−f1) + Pn−1

i=1 ai∂xi(f2−f1)∈e

Iand f2−f1+xne

O ∈

e

I+xne

O. By Lemma 2.1, f2−f1∈e

I. Therefore f1=f2. This ends the

proof of statement (a). Statement (c) follows from statements (a) and (b).

Set Ω1

X|S=Ln

i=1 Odxi. We call the elements of Ω1

X|Sgerms of rel-

ative diﬀerential forms on X×S. The map d:O → Ω1

X|Sgiven by

df =Pn

i=1 ∂xif dxiis called the relative diﬀerential of f.

Assume that dim X = 3 and let Lbe a contact structure on X. Let

ρ:X×S→Xbe the ﬁrst projection. Let ωbe a generator of L. We will

denote by LSthe sub O-module of Ω1

X|Sgenerated by ρ∗ω. We call LSa

relative contact structure of X×S. We call (X×S, LS) a relative contact

manifold. We say that an isomorphism of analytic spaces

(2.4) χ:X×S→X×S

is a relative contact transformation if χ(0, s)=(0, s), χ∗ω∈ LSfor each

ω∈ LSand the diagram

(2.5) X

_

idX//X

_

X×S

χ//X×S

SidS//S

commutes.

The demand of the commutativeness of diagram (2.5) is a very restrictive

condition but these are the only relative contact transformations we will

need. We can and will assume that the local ring of Xequals C{x, y, p}and

that Lis generated by dy −pdx.

Set O=C{x, y, p, z}/e

Iand e

O=C{x, y, p, z}/e

J. Let mXbe the maximal

ideal of C{x, y, p}. Let m[e

m] be the maximal ideal of C{z}/I [C{z}/J]. Let

n[e

n] be the ideal of O[e

O] generated by mXm[mXe

m].

5

Remark 2.3.If (2.4) is a relative contact transformation, there are α, β , γ ∈n

such that ∂xβ∈nand

(2.6) χ(x, y, p, z)=(x+α, y +β, p +γ, z).

Theorem 2.4. (a) Let χ:X×S→X×Sbe a relative contact transfor-

mation. There is β0∈nsuch that ∂pβ0= 0,∂xβ0∈n,βis the solution

of the Cauchy problem

(2.7) 1 + ∂α

∂x +p∂α

∂y ∂β

∂p −p∂α

∂p

∂β

∂y −∂α

∂p

∂β

∂x =p∂α

∂p , β −β0∈pO

and

(2.8) γ=1 + ∂α

∂x +p∂α

∂y −1∂β

∂x +p∂β

∂y −∂α

∂x −p∂α

∂y .

(b) Given α, β0∈nsuch that ∂pβ0= 0 and ∂xβ0∈n, there is a unique

contact transformation χverifying the conditions of statement (a). We

will denote χby χα,β0.

(c) Given a relative contact transformation eχ:X×T→X×Tthere is one

and only one contact transformation χ:X×S→X×Ssuch that the

diagram

(2.9) X×S

_

χ//X×S

_

X×T

eχ//X×T

commutes.

(d) Given α, β0∈nand eα, e

β0∈e

nsuch that ∂pβ0= 0, ∂pe

β0= 0,∂xβ0∈n,

∂xe

β0∈e

nand eα, e

β0are representatives of α, β0, set χ=χα,β0,eχ=χeα,

e

β0.

Then diagram (2.9) commutes.

Proof. Statements (a) and (b) are a relative version of Theorem 3.2 of [1]. In

[1] we assume S={0}. The proof works as long Sis smooth. The proof in

the singular case depends on the singular variant of the Cauchy-Kowalevski

Theorem introduced in 2.2. Statement (c) follows from statement (b) of

Theorem 2.2. Statement (d) follows from statement (c) of Theorem 2.2.

Remark 2.5.(1) The inclusion S →Tis said to be a small extension if

the surjective map OTOShas one dimensional kernel. If the kernel is

generated by ε, we have that, as complex vector spaces, OT=OS⊕εC.

Every extension of Artinian local rings factors through small extensions.

Theorem 2.6. Let S →Tbe a small extension such that

OS∼

=C{z},

OT∼

=C{z, ε}/(ε2, εz1, . . . εzm) = C{z} ⊕ Cε.

6

Assume χ:X×S→X×Sis a relative contact transformation given at

the ring level by

(x, y, p)7→ (H1, H2, H3),

α, β0∈mX, such that ∂pβ0= 0 and β0∈(x2, y). Then, there are uniquely

determined β, γ ∈mXsuch that β−β0∈pOXand eχ:X×T→X×T,

given by

eχ(x, y, p, z, ε)=(H1+εα, H2+εβ, H3+εγ, z, ε),

is a relative contact transformation extending χ(diagram (2.9) commutes).

Moreover, the Cauchy problem (2.7) for eχtakes the simpliﬁed form

(2.10) ∂β

∂p =p∂α

∂p , β −β0∈C{x, y, p}p

and

(2.11) γ=∂β

∂x +p(∂β

∂y −∂α

∂x )−p2∂α

∂y .

Proof. We have that eχis a relative contact transformation if and only if

there is f:= f0+εf00 ∈ OT{x, y, p}with f /∈(x, y , p)OT{x, y, p}, f0∈

OS{x, y, p}, f 00 ∈C{x, y, p}=OXsuch that

(2.12) d(H2+εβ)−(H3+εγ)d(H1+εα) = f(dy −pdx).

Since χis a relative contact transformation we can suppose that

dH2−H3dH1=f0(dy −pdx).

Using the fact that εmOTwe see that (2.12) is equivalent to

∂β

∂p =p∂ α

∂p ,

γ=∂β

∂x +p(∂ β

∂y −∂ α

∂x )−p2∂ α

∂y ,

f00 =∂β

∂y −p∂ α

∂y .

As β−β0∈(p)C{x, y, p}we have that β, and consequently γ, are completely

determined by αand β0.

Remark 2.7.Set α=Pkαkpk,β=Pkβkpk,γ=Pkγkpk, where αk, βk, γk∈

C{x, y}for each k≥0 and β0∈(x2, y). Under the assumptions of Theo-

rem 2.6,

(i) βk=k−1

kαk−1, k ≥1 .

(ii) Moreover,

γ0=∂β0

∂x , γ1=∂β0

∂y −∂α0

∂x , γk=−1

k

∂αk−1

∂x −1

k−1

∂αk−2

∂y , k ≥2.

Since, ∂

∂y γ0=∂

∂x (∂α0

∂x +γ1),

β0is the solution of the Cauchy problem

∂β0

∂x =γ0,∂β0

∂y =∂α0

∂x +γ1, β0∈(x2, y).

7

3. Categories of Deformations

A category Cis called a groupoid if all morphisms of Care isomorphisms.

Let p:F→Cbe a functor.

Let Sbe an object of C. We will denote by F(S) the subcategory of F

given by the following conditions:

•Ψ is an object of F(S) if p(Ψ) = S.

•χis a morphism of F(S) if p(χ) = idS.

Let χ[Ψ] be a morphism [an object] of F. Let f[S] be a morphism [an

object] of C. We say that χ[Ψ] is a morphism [an object] of Fover f[S] if

p(χ) = f[p(Ψ) = S].

A morphism χ0: Ψ0→Ψ of Fover f:S0→Sis called cartesian if for

each morphism χ00 : Ψ00 →Ψ of Fover fthere is exactly one morphism

χ: Ψ00 →Ψ0over idS0such that χ0◦χ=χ00.

If the morphism χ0: Ψ0→Ψ over fis cartesian, Ψ0is well deﬁned up to

a unique isomorphism. We will denote Ψ0by f∗Ψ or Ψ ×SS0.

We say that Fis a ﬁbered category over Cif

(1) For each morphism f:S0→Sin Cand each object Ψ of Fover S

there is a morphism χ0: Ψ0→Ψ over fthat is cartesian.

(2) The composition of cartesian morphisms is cartesian.

A ﬁbered groupoid is a ﬁbered category such that F(S) is a groupoid for

each S∈C.

Lemma 3.1. If p:F→Csatisﬁes (1) and F(S)is a groupoid for each

object Sof C, then Fis a ﬁbered groupoid over C.

Proof. Let χ: Φ →Ψ be an arbitrary morphism of F. It is enough to show

that χis cartesian. Set f=p(χ). Let χ0: Φ0→Ψ be another morphism

over f. Let f∗Ψ→Ψ be a cartesian morphism over f. There are morphisms

α: Φ0→f∗Ψ, β: Φ →f∗Ψ such that the solid diagram

(3.1) f∗Ψ

!!

Φ

β

oo

χ

Φ0

oo

α

tt

χ0

Ψ

commutes. Hence β−1◦αis the only morphism over fsuch that diagram

(3.1) commutes.

Let Anbe the category of analytic complex space germs. Let 0 denote the

complex vector space of dimension 0. Let p:F→Anbe a ﬁbered category.

Deﬁnition 3.2. Let Tbe an analytic complex space germ. Let ψ[Ψ] be an

object of F(0) [F(T)]. We say that Ψ is a versal deformation of ψif given

•a closed embedding f:T00 →T0,

8

•a morphism of complex analytic space germs g:T00 →T,

•an object Ψ0of F(T0) such that f∗Ψ0∼

=g∗Ψ,

there is a morphism of complex analytic space germs h:T0→Tsuch that

h◦f=gand h∗Ψ∼

=Ψ0.

If Ψ is versal and for each Ψ0the tangent map T(h) : TT0→TTis determined

by Ψ0, Ψ is called a semiuniversal deformation of ψ.

Let Tbe a germ of a complex analytic space. Let Abe the local ring of T

and let mbe the maximal ideal of A. Let Tnbe the complex analytic space

with local ring A/mnfor each positive integer n. The canonical morphisms

A→A/mnand A/mn→A/mn+1

induce morphisms αn:Tn→Tand βn:Tn+1 →Tn.

A morphism f:T00 →T0induces morphisms fn:T00

n→T0

nsuch that the

diagram

T00 f//T0

T00

n

?

α00

n

OO

fn//T0

n

?

α0

n

OO

T00

n+1

?

β00

n

OO

fn+1 //T0

n+1

?

β0

n

OO

commutes.

Deﬁnition 3.3. We will follow the terminology of Deﬁnition 3.2. Let gn=

g◦α00

n. We say that Ψ is a formally versal deformation of ψif there are

morphisms hn:T0

n→Tsuch that

hn◦fn=gn, hn◦β0

n=hn+1 and h∗

nΨ∼

=α0

n

∗Ψ0.

If Ψ is formally versal and for each Ψ0the tangent maps T(hn) : TT0

n→TT

are determined by α0

n

∗Ψ0, Ψ is called a formally semiuniversal deformation

of ψ.

Theorem 3.4 ([4], Theorem 5.2).Let F→Cbe a ﬁbered groupoid. Let

ψ∈F(0). If there is a versal deformation of ψ, every formally versal [semi-

universal ]deformation of ψis versal [semiuniversal ].

Let Zbe a curve of Cnwith irreducible components Z1, . . . , Zr. Set

¯

C=Fr

i=1 ¯

Ciwhere each ¯

Ciis a copy of C. Let ϕibe a parametrization of

Zi, 1 ≤i≤r. Let ϕ:¯

C→Cnbe the map such that ϕ|¯

Ci=ϕi, 1 ≤i≤r.

We call ϕthe parametrization of Z.

9

Let Tbe an analytic space. A morphism of analytic spaces Φ : ¯

C×T→

Cn×Tis called a deformation of ϕover Tif the diagram

¯

C

_

ϕ//Cn

_

¯

C×T

Φ//Cn×T

TidT//T

commutes. The analytic space Tis called de base space of the deformation.

We will denote by Φithe composition

¯

Ci×T →¯

C×TΦ

−→ Cn×T→Cn,1≤i≤r.

The maps Φi, 1 ≤i≤r, determine Φ.

Let Φ be a deformation of ϕover T. Let f:T0→Tbe a morphism of

analytic spaces. We will denote by f∗Φ the deformation of ϕover T0given

by

(f∗Φ)i= Φi◦(id ¯

Ci×f).

We call f∗Φ the pullback of Φ by f.

Let Φ0:¯

C×T→Cn×Tbe another deformation of ϕover T. A

morphism from Φ0into Φ is a pair (χ, ξ) where χ:Cn×T→Cn×Tand

ξ:¯

C×T→¯

C×Tare isomorphisms of analytic spaces such that the diagram

T¯

C×T

ooΦ//Cn×T//T

¯

C

?

OO

_

ϕ//Cn× {0}

?

OO

_

T

idT

OO

¯

C×T

oo

ξ

CC

Φ0

//Cn×T

χ

[[

//T

idT

OO

commutes.

Let Φ0be a deformation of ϕover Sand f:S→Ta morphism of analytic

spaces. A morphism of Φ0into Φover fis a morphism from Φ0into f∗Φ.

There is a functor pthat associates Tto a deformation Ψ over Tand fto

a morphism of deformations over f.

Given t∈Tlet Ztbe the curve parametrized by the composition

¯

C× {t}→¯

C×TΦ

−→ Cn×T→Cn.

We call Ztthe ﬁber of the deformation Φat the point t.

10

Let ϕ:¯

C→C2be the parametrization of a plane curve Z. We will denote

by Defϕ[Def em

ϕ] the category of deformations [equimultiple deformations]

Φof (the parametrization ϕof ) the plane curve Z.

Consider in C3the contact structure given by the diﬀerential form dy −

pdx. Let ψ:¯

C→C3be the parametrization of a Legendrian curve L. We

say that a deformation Ψ of ψis a Legendrian deformation of ψif all of its

ﬁbers are Legendrian. We say that (χ, ξ) is an isomorphism of Legendrian

deformations if χ:X×T→X×Tis a relative contact transformation.

We will denote by d

Defψ[d

Def em

ψ] the category of Legendrian [equimultiple

Legendrian] deformations of ψ. All deformations are assumed to have trivial

sections (see [3]).

Assume that ψ=Con ϕ parametrizes a germ of a Legendrian curve L, in

generic position, in (C3

(x,y,p), ω). If Φ ∈ Defϕis given by

(3.2) Φi(ti,s) = (Xi(ti,s), Yi(ti,s)) ,1≤i≤r,

such that Pi(ti,s) := ∂tYi(ti,s)/∂tXi(ti,s)∈C{ti,s}for 1 ≤i≤r, then

(3.3) Ψi(ti,s)=(Xi(ti,s), Yi(ti,s), Pi(ti,s)) .

deﬁnes a deformation Ψ of ψwhich we call conormal of Φ. Notice that in

this case all ﬁbers of Φ have the same tangent space {y= 0}. We will denote

Ψ by Con Φ. If Ψ ∈d

Defψis given by (3.3), we call plane projection of Ψ to

the deformation Φ of ϕgiven by (3.2). We will denote Φ by Ψπ.

Let us consider the full subcategory →

Defϕof the deformations Φ ∈ Def em

ϕ

such that all ﬁbers of Φ have the same tangent space {y= 0}.

Remark 3.5.We see immediately that if Φ ∈→

Defϕthen Con Φ exists. How-

ever, it should be noted that there are more deformations for which the

conormal is deﬁned:

Let Φ be the deformation of ϕ= (t3, t10) given by

X(t, s) = st +t3;Y(t, s) = 5

12st8+t10.

Then Con Φ exists, but Φ is not equimultiple.

We deﬁne in this way the functors

Con :→

Defϕ→d

Defψ, π :d

Defψ→ Defϕ.

Notice that the conormal of the plane projection of a Legendrian deformation

always exists and we have that Con (Ψπ) = Ψ for each Ψ ∈d

Defψand

(Con Φ)π= Φ where Φ ∈→

Defϕ.

Let us denote by

Defϕthe subcategory of equimultiple deformations Φ of

ϕsuch that all ﬁbres of Φ have ﬁxed tangent space {y= 0}with conormal

in generic position. Then

Defϕ⊂→

Defϕand if Φ ∈→

Defϕis given by 3.2,

11

then Φ ∈

Defϕiﬀ

(3.4) ordtiYi≥2ordtiXi,1≤i≤r.

Because we demand that Φ is equimultiple and all branches have tangent

space {y= 0}, 3.4 is equivalent to

(3.5) ordtiYi≥2mi,1≤i≤r,

where miis the multiplicity of the component Ziof Z.

Lemma 3.6. Under the assumptions above,

Con (

Defϕ)⊂d

Def em

ψand (d

Def em

ψ)π⊂

Defϕ.

Proof. Let mibe the multiplicity of the component Ziof Z. Let Zi,s[Li,s ] be

the ﬁber of Φ[Ψ] (given by 3.2 [3.3]) at s. If Φ ∈

Defϕ,C(Li,s)6⊃ π−1(0,0)

for each s, so ordtiYi≥2ordtiXi= 2mi. Hence ordtiPi≥miand Ψ is

equimultiple.

If Ψ ∈d

Def em

ψ,ordtiPi≥ordtiXiand we get that C(Li,s)6⊃ π−1(0,0)

for each s. Each component Li,s has multiplicity mifor each s. Hence

mult Zi,s ≥mifor each s. Since multiplicity is semicontinuous, mult Zi,s =

mifor each sand Φ is equimultiple.

Lemma 3.7. If Cis one of the categories d

Defψ,d

Def em

ψ,p:C→Anis a

ﬁbered groupoid.

Proof. Let f:S→Tbe a morphism of An. Let Ψ be a deformation over

T. Then, ( eχ, e

ξ) : f∗Ψ→Ψ is cartesian, with

e

ξ(ti,s)=(ti,s),eχ(x, y, p, s)=(x, y, p, s).

This is because if (χ, ξ):Ψ0→Ψ is a morphism over f, then by deﬁnition

of morphism of deformations over diﬀerent base spaces, (χ, ξ ) is a morphism

from Ψ0into f∗Ψ over idS.

4. Equimultiple Versal Deformations

For Sophus Lie a contact transformation was a transformation that takes

curves into curves, instead of points into points. We can recover the initial

point of view. Given a plane curve Zat the origin, with tangent cone

{y= 0}, and a contact transformation χfrom a neighbourhood of (0; dy)

into itself, χacts on Zin the following way: χ·Zis the plane projection

of the image by χof the conormal of Z. We can deﬁne in a similar way

the action of a relative contact transformation on a deformation of a plane

curve Z, obtainning another deformation of Z.

We say that Φ ∈

Defϕ(T) is trivial (relative to the action of the group

of relative contact transformations over T) if there is χsuch that χ·Φ :=

π◦χ◦ Con Φ is the constant deformation of φover T, given by

(ti,s)7→ ϕi(ti), i = 1, . . . , r.

12

Let Zbe the germ of a plane curve parametrized by ϕ:¯

C→C2. In the

following we will identify each ideal of OZwith its image by ϕ∗:OZ→ O¯

C.

Hence

OZ=C

x1

.

.

.

xr

,

y1

.

.

.

yr

⊂

r

M

i=1

C{ti}=O¯

C.

Set ˙

x= [ ˙x1,..., ˙xr]t, where ˙xiis the derivative of xiin order to ti, 1 ≤i≤r.

Let

˙ϕ:= ˙

x∂

∂x +˙

y∂

∂y

be an element of the free O¯

C-module

(4.1) O¯

C

∂

∂x ⊕ O ¯

C

∂

∂y .

Notice that (4.1) has a structure of OZ-module induced by ϕ∗.

Let mibe the multiplicity of Zi, 1 ≤i≤r. Consider the O¯

C-module

(4.2) r

M

i=1

tmi

iC{ti}∂

∂x !⊕ r

M

i=1

t2mi

iC{ti}∂

∂y !.

Let m¯

C˙ϕbe the sub O¯

C-module of (4.2) generated by

(a1, . . . , ar)˙

x∂

∂x +˙

y∂

∂y ,

where ai∈tiC{ti},1≤i≤r. For i= 1, . . . , r set pi= ˙yi/˙xi. For each

k≥0 set

pk=hpk

1, . . . , pk

rit.

Let b

Ibe the sub OZ-module of (4.2) generated by

pk∂

∂x +k

k+ 1pk+1 ∂

∂y , k ≥1.

Set

c

Mϕ=Lr

i=1 tmi

iC{ti}∂

∂x ⊕Lr

i=1 t2mi

iC{ti}∂

∂y

m¯

C˙ϕ+ (x, y)∂

∂x ⊕(x2, y)∂

∂y +b

I.

Given a category Cwe will denote by Cthe set of isomorphism classes of

elements of C.

Theorem 4.1. Let ψbe the parametrization of a germ of a Legendrian curve

Lof a contact manifold X. Let χ:X→C3be a contact transformation

such that χ(L)is in generic position. Let ϕbe the plane projection of χ◦ψ.

Then there is a canonical isomorphism

d

Def em

ψ(Tε)∼

−→ c

Mϕ.

13

Proof. Let Ψ ∈d

Def em

ψ(Tε). By Lemma 3.6, Ψ is the conormal of its projec-

tion Φ ∈

Defϕ(Tε). Moreover, Ψ is given by

Ψi(ti, ε)=(xi+εai, yi+εbi, pi+εci),

where ai, bi, ci∈C{ti}, ord ai≥mi, ord bi≥2mi, i = 1, . . . , r. The defor-

mation Ψ is trivial if and only if Φ is trivial for the action of the relative

contact transformations. Φ is trivial if and only if there are

ξi(ti) = e

ti=ti+εhi,

χ(x, y, p, ε)=(x+εα, y +εβ, p +εγ, ε),

such that χis a relative contact transformation, ξiis an isomorphism,

α, β, γ ∈(x, y, p)C{x, y, p}, hi∈tiC{ti},1≤i≤r, and

xi(ti) + εai(ti) = xi(e

ti) + εα(xi(e

ti), yi(e

ti), pi(e

ti)),

yi(ti) + εbi(ti) = yi(e

ti) + εβ(xi(e

ti), yi(e

ti), pi(e

ti)),

for i= 1, . . . , r. By Taylor’s formula xi(e

ti) = xi(ti) + ε˙xi(ti)hi(ti), yi(e

ti) =

yi(ti) + ε˙yi(ti)hi(ti) and

εα(xi(e

ti), yi(e

ti), pi(e

ti)) = εα(xi(ti), yi(ti), pi(ti)),

εβ(xi(e

ti), yi(e

ti), pi(e

ti)) = εβ(xi(ti), yi(ti), pi(ti)),

for i= 1, . . . , r. Hence Φ is trivialized by χif and only if

ai(ti) = ˙xi(ti)hi(ti) + α(xi(ti), yi(ti), pi(ti)),(4.3)

bi(ti) = ˙yi(ti)hi(ti) + β(xi(ti), yi(ti), pi(ti)),(4.4)

for i= 1, . . . , r. By Remark 2.7 (i), (4.3) and (4.4) are equivalent to the

condition

a∂

∂x +b∂

∂y ∈m¯

C˙ϕ+ (x, y)∂

∂x ⊕(x2, y)∂

∂y +b

I.

Set

Mϕ=Lr

i=1 tmi

iC{ti}∂

∂x ⊕Lr

i=1 tmi

iC{ti}∂

∂y

m¯

C˙ϕ+ (x, y)∂

∂x ⊕(x, y)∂

∂y

,

Mϕ=Lr

i=1 tmi

iC{ti}∂

∂x ⊕Lr

i=1 t2mi

iC{ti}∂

∂y

m¯

C˙ϕ+ (x, y)∂

∂x ⊕(x2, y)∂

∂y

.

By Proposition 2.27 of [3],

Def em

ϕ(Tε)∼

=Mϕ.

A similar argument shows that

Defϕ(Tε)∼

=

Mϕ.

14

We have linear maps

(4.5) Mϕ

ı

←

Mϕc

Mϕ.

Theorem 4.2 ([3], II Theorem 2.38 (3)).Set k=dim Mϕ. Let aj,bj∈

Lr

i=1 tmi

iC{ti},1≤j≤k. If

(4.6) aj∂

∂x +bj∂

∂y =

aj

1

.

.

.

aj

r

∂

∂x +

bj

1

.

.

.

bj

r

∂

∂y ,

1≤j≤k, represents a basis of Mϕ, the deformation Φ : ¯

C×Ck→C2×Ck

given by

(4.7) Xi(ti,s) = xi(ti) +

k

X

j=1

aj

i(ti)sj, Yi(ti,s) = yi(ti) +

k

X

j=1

bj

i(ti)sj,

i= 1, . . . , r, is a semiuniversal deformation of ϕin Def em

ϕ.

Lemma 4.3. Set

k=dim

Mϕ. Let aj∈Lr

i=1 tmi

iC{ti},bj∈Lr

i=1 t2mi

iC{ti},

1≤j≤→

k. If (4.6) represents a basis of

Mϕ, the deformation

Φgiven by

(4.7), 1≤i≤r, is a semiuniversal deformation of ϕin

Defϕ. Moreover,

Con

Φis a versal deformation of ψin d

Def em

ψ.

Proof. We will only show the completeness of

Φ and Con

Φ. Since the lin-

ear inclusion map ıreferred in (4.5) is injective, the deformation

Φ is the

restriction to

Mϕof the deformation Φ introduced in Theorem 4.2. Let

Φ0∈

Defϕ(T). Since Φ0∈ Def em

ϕ(T), there is a morphism of analytic

spaces f:T→Mϕsuch that Φ0∼

=f∗Φ. Since Φ0∈

Defϕ(T), f(T)⊂

Mϕ.

Hence f∗

Φ = f∗Φ.

If Ψ ∈d

Def em

ψ(T), Ψπ∈

Defϕ(T). Hence there is f:T→

Mϕsuch that

Ψπ∼

=f∗

Φ. Therefore Ψ = Con Ψπ∼

=Con f ∗

Φ = f∗Con

Φ.

Theorem 4.4. Let aj∈Lr

i=1 tmi

iC{ti},bj∈Lr

i=1 t2mi

iC{ti},1≤j≤.

Assume that (4.6) represents a basis [a system of generators ]of c

Mϕ. Let Φ

be the deformation given by (4.7), 1≤i≤r. Then Con Φis a semiuniversal

[versal ]deformation of ψin d

Def em

ψ.

Proof. By Theorem 3.4 and Lemma 4.3 it is enough to show that Con Φ is

formally semiuniversal [versal].

Let ı:T0→Tbe a small extension. Let Ψ ∈d

Def em

ψ(T). Set Ψ0=ı∗Ψ.

Let η0:T0→C`be a morphism of complex analytic spaces. Assume that

15

(χ0, ξ0) deﬁne an isomorphism

η0∗Con Φ∼

=Ψ0.

We need to ﬁnd η:T→C`and χ, ξ such that η0=η◦ıand χ, ξ deﬁne an

isomorphism

η∗Con Φ∼

=Ψ

that extends (χ0, ξ0). Let A[A0] be the local ring of T[T0]. Let δbe the

generator of Ker(AA0). We can assume A0∼

=C{z}/I, where z=

(z1, . . . , zm). Set

e

A0=C{z}and e

A=C{z, ε}/(ε2, εz1, . . . , εzm).

Let mAbe the maximal ideal of A. Since mAδ= 0 and δ∈mA, there is a

morphism of local analytic algebras from e

Aonto Athat takes εinto δsuch

that the diagram

(4.8) e

A

//e

A0

A//A0

commutes. Assume e

T[e

T0] has local ring e

A[e

A0]. We also denote by ıthe

morphism e

T0→e

T. We denote by κthe morphisms T →e

Tand T0→e

T0.

Let e

Ψ∈d

Def em

ψ(e

T) be a lifting of Ψ.

We ﬁx a linear map σ:A0→e

A0such that κ∗σ=idA0. Set eχ0=

χσ(α),σ(β0), where χ0=χα,β0. Deﬁne eη0by eη0∗si=σ(η0∗ si), i= 1, . . . , l. Let

e

ξ0be the lifting of ξ0determined by σ. Then

e

Ψ0:= eχ0−1◦eη0∗ Con Φ◦e

ξ0−1

is a lifting of Ψ0and

(4.9) eχ0◦e

Ψ0◦e

ξ0=eη0∗Con Φ.

By Theorem 2.4 it is enough to ﬁnd liftings eχ, e

ξ, eηof eχ0,e

ξ0,eη0such that

eχ·e

Ψπ◦e

ξ=eη∗Φ

in order to prove the theorem.

16

Consider the following commutative diagram

¯

C×e

T0

e

Ψ0

//¯

C×e

T

e

Ψ

//¯

C×C`

Con Φ

C3×e

T0

pr

//C3×e

T

pr

//C3×C`

e

T0//

eη077

e

T

eη//C`.

If Con Φ is given by

Xi(ti,s), Yi(ti,s), Pi(ti,s)∈C{s, ti},

then eη0∗ Con Φ is given by

Xi(ti,eη0(z)), Yi(ti,eη0(z)), Pi(ti,eη0(z)) ∈e

A0{ti}=C{z, ti}

for i= 1, . . . , r. Suppose that e

Ψ0is given by

U0

i(ti,z), V 0

i(ti,z), W 0

i(ti,z)∈C{z, ti}.

Then, e

Ψ must be given by

Ui=U0

i+εui, Vi=V0

i+εvi, Wi=W0

i+εwi∈e

A{ti}=C{z, ti} ⊕ εC{ti}

with ui, vi, wi∈C{ti}and i= 1, . . . , r. By deﬁnition of deformation we

have that, for each i,

(Ui, Vi, Wi) = (xi(ti), yi(ti), pi(ti)) mod me

A.

Suppose eη0:e

T0→C`is given by (eη0

1,...,eη0

`), with eη0

i∈C{z}. Then eηmust

be given by eη=eη0+εeη0for some eη0= (eη0

1,...,eη0

`)∈C`. Suppose that

˜χ0:C3×e

T0→C3×e

T0is given at the ring level by

(x, y, p)7→ (H0

1, H0

2, H0

3),

such that H0=id mod me

A0with H0

i∈(x, y, p)A0{x, y, p}. Let the automor-

phism e

ξ0:¯

C×e

T0→¯

C×e

T0be given at the ring level by

ti7→ h0

i

such that h0=id mod me

A0with h0

i∈(ti)C{z, ti}.

Then, from 4.9 follows that

Xi(ti,eη0) = H0

1(U0

i(h0

i), V 0

i(h0

i), W 0

i(h0

i)),

Yi(ti,eη0) = H0

2(U0

i(h0

i), V 0

i(h0

i), W 0

i(h0

i)),(4.10)

Pi(ti,eη0) = H0

3(U0

i(h0

i), V 0

i(h0

i), W 0

i(h0

i)).

17

Now, eη0must be extended to eηsuch that the ﬁrst two previous equations

extend as well. That is, we must have

Xi(ti,eη)=(H0

1+εα)(Ui(h0

i+εh0

i), Vi(h0

i+εh0

i), Wi(h0

i+εh0

i),(4.11)

Yi(ti,eη)=(H0

2+εβ)(Ui(h0

i+εh0

i), Vi(h0

i+εh0

i), Wi(h0

i+εh0

i).

with α, β ∈(x, y, p)C{x, y, p},h0

i∈(ti)C{ti}such that

(x, y, p)7→ (H0

1+εα, H0

2+εβ, H 0

3+εγ)

gives a relative contact transformation over e

Tfor some γ∈(x, y, p)C{x, y, p}.

The existence of this extended relative contact tranformation is guaranteed

by Theorem 2.6. Moreover, again by Theorem 2.6 this extension depends

only on the choices of αand β0. So, we need only to ﬁnd α,β0,eη0and h0

i

such that (4.11) holds. Using Taylor’s formula and ε2= 0 we see that

Xi(ti,eη0+εeη0) = Xi(ti,eη0) + ε

`

X

j=1

∂Xi

∂sj

(ti,eη0)eη0

j

(εme

A= 0) = Xi(ti,eη0) + ε

`

X

j=1

∂Xi

∂sj

(ti,0)eη0

j,(4.12)

Yi(ti,eη0+εeη0) = Yi(ti,eη0) + ε

`

X

j=1

∂Yi

∂sj

(ti,0)eη0

j.

Again by Taylor’s formula and noticing that εme

A= 0, εme

A0= 0 in e

A,

h0=id mod me

A0and (Ui, Vi)=(xi(ti), yi(ti)) mod me

Awe see that

Ui(h0

i+εh0

i) = Ui(h0

i) + ε˙

Ui(h0

i)h0

i

=U0

i(h0

i) + ε( ˙xih0

i+ui),(4.13)

Vi(h0

i+εh0

i) = V0

i(h0

i) + ε( ˙yih0

i+vi).

Now, H0=id mod me

A0, so

∂H 0

1

∂x = 1 mod me

A0,∂H 0

1

∂y ,∂H0

1

∂p ∈me

A0e

A0{x, y, p}.

In particular,

ε∂H 0

1

∂y =ε∂H0

1

∂p = 0.

By this and arguing as in (4.12) and (4.13) we see that

(H0

1+εα)(U0

i(h0

i) + ε( ˙xih0

i+ui), V 0

i(h0

i) + ε( ˙yih0

i+vi), W 0

i(h0

i) + ε( ˙pih0

i+wi))

=H0

1(U0

i(h0

i), V 0

i(h0

i), W 0

i(h0

i)) + ε(α(U0

i(h0

i), V 0

i(h0

i), W 0

i(h0

i)) + 1( ˙xih0

i+ui))

=H0

1(U0

i(h0

i), V 0

i(h0

i), W 0

i(h0

i)) + ε(α(xi, yi, pi) + ˙xih0

i+ui),

(H0

2+εβ)(U0

i(h0

i) + ε( ˙xih0

i+ui), V 0

i(h0

i) + ε( ˙yih0

i+vi), W 0

i(h0

i) + ε( ˙pih0

i+wi))

=H0

2(U0

i(h0

i), V 0

i(h0

i), W 0

i(h0

i)) + ε(β(xi, yi, pi) + ˙yih0

i+vi)

18

Substituting this in (4.11) and using (4.10) and (4.12) we see that we have

to ﬁnd eη0= (eη0

1,...,eη0

`)∈C`,h0

isuch that

(ui(ti), vi(ti)) =

`

X

j=1 eη0

j∂Xi

∂sj

(ti,0),∂Yi

∂sj

(ti,0)−(4.14)

−h0

i(ti)(( ˙xi(ti),˙yi(ti)) −(α(xi(ti), yi(ti), pi(ti)), β(xi(ti), yi(ti), pi(ti))).

Note that, because of Remark 2.7 (i), (α(xi(ti), yi(ti), pi(ti)), β(xi(ti), yi(ti), pi(ti))) ∈

b

Ifor each i. Also note that e

Ψ∈d

Def em

ψ(e

T) means that ui∈tmi

iC{ti}, vi∈

t2mi

iC{ti}. Then, if the vectors

∂X1

∂sj

(t1,0),...,∂Xr

∂sj

(tr,0)∂

∂x +∂Y1

∂sj

(t1,0),...,∂Yr

∂sj

(tr,0)∂

∂y

= (aj

1(t1), . . . , aj

r(tr)) ∂

∂x + (bj

1(t1), . . . , bj

r(tr)) ∂

∂y , j = 1, . . . ,

form a basis of [generate] c

Mϕ, we can solve (4.14) with unique eη0

1,...,eη0

`

[respectively, solve] for all i= 1, . . . , r. This implies that the conormal of Φ

is a formally semiuniversal [respectively, versal] equimultiple deformation of

ψover C`.

5. Versal Deformations

Let f∈C{x1, . . . , xn}. We will denote by Rf dxithe solution of the

Cauchy problem

∂g

∂xi

=f, g ∈(xi).

Let ψbe a Legendrian curve with parametrization given by

(5.1) ti7→ (xi(ti), yi(ti), pi(ti)) i= 1, . . . , r.

We will call fake plane projection of (5.1) to the plane curve σwith parametriza-

tion given by

(5.2) ti7→ (xi(ti), pi(ti)) i= 1, . . . , r.

We will denote σby ψπf.

Given a plane curve σwith parametrization (5.2), we will cal fake conor-

mal of σto the Legendrian curve ψwith parametrization (5.1), where

yi(ti) = Zpi(ti) ˙xi(ti)dti.

We will denote ψby Confσ. Applying the construction above to each

ﬁbre of a deformation we obtain functors

πf:d

Defψ→ Defσ,Conf:Defσ→d

Defψ.

Notice that

(5.3) Conf(Ψπf)=Ψ,(Conf(Σ))πf= Σ

19

for each Ψ ∈d

Defψand each Σ ∈ Defσ.

Let ψbe the parametrization of a Legendrian curve given by (5.1). Let σ

be the fake plane projection of ψ. Set ˙σ:= ˙

x∂

∂x +˙

p∂

∂p . Let Ifbe the linear

subspace of

m¯

C

∂

∂x ⊕m¯

C

∂

∂p = r

M

i=1

tiC{ti}∂

∂x !⊕ r

M

i=1

tiC{ti}∂

∂p !

generated by

α0

∂

∂x −∂α0

∂x +∂α0

∂y pp∂

∂p ,∂β0

∂x +∂β0

∂y p∂

∂p ,

and

αkpk∂

∂x −1

k+ 1 ∂αk

∂x pk+1 +∂αk

∂y pk+2 ∂

∂p , k ≥1,

where αk∈(x, y), β0∈(x2, y) for each k≥0. Set

Mf

σ=m¯

C∂

∂x ⊕m¯

C∂

∂p

m¯

C˙σ+If.

Theorem 5.1. Assuming the notations above, d

Defψ(Tε)∼

=Mf

σ.

Proof. Let Ψ ∈d

Defψ(Tε) be given by

Ψi(ti, ε)=(Xi, Yi, Pi)=(xi+εai, yi+εbi, pi+εci),

where ai, bi, ci∈C{ti}tiand Yi=RPi∂tiXidti,i= 1, . . . , r. Hence

bi=Z( ˙xici+ ˙aipi)dti, i = 1, . . . , r.

By (5.3) Ψ is trivial if and only if there an isomorphism ξ:¯

C×Tε→¯

C×Tε

given by

ti→e

ti=ti+εhi, hi∈C{ti}ti, i = 1, . . . , r,

and a relative contact transformation χ:C3×Tε→C3×Tεgiven by

(x, y, p, ε)7→ (x+εα, y +εβ, p +εγ, ε)

such that

Xi=xi(e

ti) + εα(xi(e

ti), yi(e

ti), pi(e

ti)),

Pi=pi(e

ti) + εγ(xi(e

ti), yi(e

ti), pi(e

ti)),

i= 1, . . . , r. Following the argument of the proof of Theorem 4.1, Ψπfis

trivial if and only if

ai(ti) = ˙xi(ti)hi(ti) + α(xi(ti), yi(ti), pi(ti)),

ci(ti) = ˙pi(ti)hi(ti) + γ(xi(ti), yi(ti), pi(ti)),

i= 1, . . . , r. The result follows from Remark 2.7 (ii).

20

Lemma 5.2. Let ψbe the parametrization of a Legendrian curve. Let Φbe

the semiuniversal deformation in Defσof the fake plane projection σof ψ.

Then ConfΦis a versal deformation of ψin d

Defψ.

Proof. It follows the argument of Lemma 4.3.

Theorem 5.3. Let aj,cj∈m¯

Csuch that

(5.4) aj∂

∂x +cj∂

∂p =

aj

1

.

.

.

aj

r

∂

∂x +

cj

1

.

.

.

cj

r

∂

∂p ,

1≤j≤, represents a basis [a system of generators ]of Mf

σ. Let Φ∈ Defσ

be given by

(5.5) Xi(ti,s) = xi(ti) +

`

X

j=1

aj

i(ti)sj, Pi(ti,s) = pi(ti) +

`

X

j=1

cj

i(ti)sj,

i= 1, . . . , r. Then ConfΦis a semiuniversal [versal ]deformation of ψin

d

Defψ.

Proof. It follows the argument of Theorem 4.4, using Remark 2.7 (ii).

6. Examples

Example 6.1.Let ϕ(t) = (t3, t10), ψ(t) = (t3, t10 ,10

3t7), σ(t) = (t3,10

3t7).

The deformations given by

•X(t, s) = t3, Y (t, s) = s1t4+s2t5+s3t7+s4t8+t10 +s5t11 +s6t14;

•X(t, s) = s1t+s2t2+t3, Y (t, s) = s3t+s4t2+s5t4+s6t5+s7t7+s8t8+

+t10 +s9t11 +s10t14;

are respectively

•an equimultiple semiuniversal deformation;

•a semiuniversal deformation

of ϕ. The conormal of the deformation given by

X(t, s) = t3, Y (t, s) = s1t7+s2t8+t10 +s3t11;

is an equimultiple semiuniversal deformation of ψ. The fake conormal of the

deformation given by

X(t, s) = s1t+s2t2+t3, P (t, s) = s3t+s4t2+s5t4+s6t5+10

3t7+s7t8;

is a semiuniversal deformation of the fake conormal of σ. The conormal of

the deformation given by

X(t, s) = s1t+s2t2+t3, Y (t, s) = α2t2+α3t3+α4t4+α5t5+α6t6+

+α7t7+α8t8+α9t9+α10t10 +α11t11;

21

with

α2=s1s3

2, α3=s1s4+ 2s2s3

3, α4=3s3+ 2s2s4

4,

α5=3s4+s1s5

5, α6=2s2s5+s1s6

6, α7=3s5+ 2s2s6

7,

α8=10s1+ 9s6

24 , α9=3s1s7+ 20s2

27 , α10 = 1 + s2s7

5,

α11 =3s7

11 ,

is a semiuniversal deformation of ψ.

Example 6.2.Let Z={(x, y)∈C2: (y2−x5)(y2−x7)=0}. Consider the

parametrization ϕof Zgiven by

x1(t1) = t2

1, y1(t1) = t5

1x2(t2) = t2

2, y2(t2) = t7

2.

Let σbe the fake projection of the conormal of ϕgiven by

x1(t1) = t2

1, p1(t1) = 5

2t3

1x2(t2) = t2

2, p2(t2) = 7

2t5

2.

The deformations given by

•X1(t1,s) = t2

1, Y1(t1,s) = s1t3

1+t5

1,

X2(t2,s) = t2

2, Y2(t2,s) = s2t2

2+s3t3

2+s4t4

2+s5t5

2+s6t6

2+t7

2+

+s7t8

2+s8t10

2+s9t12

2;

•X1(t1,s) = s1t1+t2

1, Y1(t1,s) = s3t1+s4t3

1+t5

1,

X2(t2,s) = s2t2+t2

2, Y2(t2,s) = s5t2+s6t2

2+s7t3

2+s8t4

2+s9t5

2+s10t6

2+

+t7

2+s11t8

2+s12t10

2+s13t12

2;

are respectively

•an equimultiple semiuniversal deformation;

•a semiuniversal deformation

of ϕ. The conormal of the deformation given by

X1(t1,s) = t2

1, Y1(t1,s) = t5

1,

X2(t2,s) = t2

2, Y2(t2,s) = s1t4

2+s2t5

2+s3t6

2+t7

2+s4t8

2;

is an equimultiple semiuniversal deformation of the conormal of ϕ. The fake

conormal of the deformation given by

X1(t1,s) = s1t1+t2

1, P1(t1,s) = s3t1+5

2t3

1,

X2(t2,s) = s2t2+t2

2, P2(t2,s) = s4t2+s5t2

2+s6t3

2+s7t4

2+7

2t5

2+s8t6

2;

22

is a semiuniversal deformation of the fake conormal of σ. The conormal of

the deformation given by

X1(t1,s) = s1t1+t2

1, Y1(t1,s) = α2t2

1+α3t3

1+α4t4

1+t5

1,

X2(t2,s) = s2t2+t2

2, Y2(t2,s) = β2t2

2+β3t3

2+β4t4

2+β5t5

2+β6t6

2+

+β7t7

2+β8t8

2;

with

α2=s1s3

2, α3=2s3

3, α4=5s1

8,

β2=s2s4

2, β3=2s4+s2s5

3, β4=2s5+s2s6

4,

β5=2s6+s2s7

5, β6=4s7+ 7s2

12 , β7= 1 + s2s8

7,

β8=2s8

8,

is a semiuniversal deformation of the conormal of ϕ.

References

[1] A. Ara´ujo and O. Neto, Moduli of Germs of Legendrian Curves, Ann. Fac. Sci. Toulouse

Math.,Vol. XVIII, 4, 2009, pp. 645–657.

[2] J. Cabral and O. Neto, Microlocal versal deformations of the plane curves yk=xn,

C. R. Acad. Sci. Paris, Ser. I 347 (2009), pp. 1409–1414.

[3] G. -M. Greuel, C. Lossen and E. Shustin, Introduction to Singularities and Deforma-

tions, Springer (2007).

[4] H. Flenner, Ein Kriterium f¨ur die Oﬀenheit der Versalit¨at, Math. Z. 178 (1981),

pp. 449–473.

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