S S - univ-toulouse.fr · 2 DIMENSION OF THE MODULI ASPCE OF A CURVE IN THE COMPLEX PLANE. The rst computations of the generic dimension of M(S) go back to Zariski [19, 20]

DIMENSION OF THE MODULI SPACE OF A CURVE IN THE

COMPLEX PLANE.

Yohann Genzmer1

Abstract. In this article, we prove a formula that computes the generic di-mension of the moduli space of a germ of irreducible curve in the complexplane. It is obtained from the study of the Saito module associated to thecurve, which is the module of germs of holomorphic 1-forms letting the curveinvariant.

1. Introduction

Let S be a germ of irreducible curve in the complex plane. Its moduli space M (S)is dened as the set of germs of curves topologically equivalent to S up to analyticequivalence. It is an non Hausdor analytic manifold. In this article, we intend togive a simple formula for the generic dimension of M (S).

Theorem 1. Let E = E1 · · · EN be the minimal resolution of S. Let ci be the

center of Ei. Then

dimgenM (S) =

N∑i=1

σ(νci

((E1 · · · Ei−1)

−1(S)))

where ν? is the algebraic multiplicity and σ (k) =

(k−3)2

4 if k is odd(k−2)(k−4)

4 else.

Example. In [20], Zariski showed that the dimension of the generic component ofthe moduli space of S =

yn+1 − xn = 0

is σ (n) . After one blowing-up E1, the

strict transform of S by E1 is a smooth curve, thus for any i ≥ 2, the multiplicitysatisfy

νci

((E1 · · · Ei−1)

−1(S))≤ 3.

Hence, Theorem 1 recovers the result of Zariski.

Corollary. A germ of irreducible curve S is rigid if and only if it belongs to the

following list

(1) ν (S) = 1, i.e., S is smooth.

(2) ν (S) = 2(3) ν (S) = 3 and its sole Puiseux pair is equal to (3, 4) , (3, 5) , (3, 7) or (3, 8) .(4) ν (S) = 4 and its Puiseux pairs are equal to (4, 5) , (2, 3) , (2, 2k + 1) , k ≥ 3

or (4, 7).

1The author is partially supported by ANR-13-JS01-0002-0

1

2 DIMENSION OF THE MODULI SPACE OF A CURVE IN THE COMPLEX PLANE.

The rst computations of the generic dimension of M (S) go back to Zariski [19, 20]who produced number of formulas in particular cases. Briançon, Granger andMaisonobe in [1, 8] proposed an eective algorithm to get this dimension for apossibly reducible quasi-homogeneous curve xa + yb = 0. In [11, 12, 10], Hefezand Hernandes completely solved the problem of the analytic classication of anirreducible germ of curve but still through an algorithm from which it seems dicultto extract a general formula. In [5, 6], Paul and the author proposed a dierentapproach based upon foliated techniques and obtained an expression in the samecase as the one of Briançon, Granger and Maisonobe. However, at the very nalstep, the computation is also based upon an algorithmic procedure that does notlead to what we can properly called an explicit formula.

The proof of Theorem 1 is based upon the study of the Saito module of S. In [17],Saito shows that the module Ω1 (S) of holomorphic 1−forms tangent to a givengerm of curve S in C2 is free O−module of rank 2. If f is a reduced equation of S,the family ω1, ω2 is a basis of Ω1 (S) if and only if there exists a germ of unityu ∈ O, u (0) 6= 0 such that the exterior product of ω1 and ω2 is written

ω1 ∧ ω2 = ufdx ∧ dy.In other words, the tangency locus between ω1 and ω2 is reduced to the sole curveS. Beyond this characterization, very few is known about these two generators. Atrst glance, we can say the following: among all the possible basis ω1, ω2, thereis one for which the sum of the algebraic multiplicities

ν (ω1) + ν (ω2) ≤ ν (S)

is maximal. It is easily seen that

Proposition. The couple of multiplicities (ν (ω1) , ν (ω2)) up to order that maxi-

mizes its sum is an analytic invariant of S.

However, these two integers are not topologically invariant and along the equisin-gularity class of a curve, they may vary widely.

Example. Let S be the curve y7 − x6 = 0. Then the family7xdy − 6ydx,d

(y7 − x6

)is a basis of the Saito module since

(7xdy − 6ydx) ∧ d(y7 − x6

)= −42

(y7 − x6

)dx ∧ dy.

In that case, the couple of valuation is (1, 5) whose sum is exactly 6. However,perturbing a bit S leads to dierent values of the multiplicities. For instance, if S ifthe curve y6 − x7 + x4y4 = 0 which is topologically but not analytically equivalentto y6 = x7, one can show that the couple

ω1 =5

4x4dx− 20

21x2y3dy +

(8

21xy3 + y

)(6xdy − 7ydx)

ω2 =20

21x3y3dx+

(10

7y4 − 80

147xy6

)dy +

(x2 +

32

147y6

)(6xdy − 7ydx)

is a basis for Ω1 (S). The multiplicities are respectively 2 and 3 whose sum is strictlysmaller than the multiplicity of S. Finally, if S is given by y6 − x7 + y2x5 = 0 thenS admits a basis ω1, ω2 with ν (ω1) = ν (ω2) = 3.

DIMENSION OF THE MODULI SPACE OF A CURVE IN THE COMPLEX PLANE. 3

This example leads us to introduce the following class of curves.

Denition. A curve S, reducible or not, is said to admit a balanced basis if thereexists a basis ω1, ω2 of Ω1 (S) with

• ν (ω1) = ν (ω2) = ν(S)2 if ν (S) is even,

• ν (ω1) = ν (ω2)− 1 = ν(S)−12 else.

A direction d for S is either an empty set, a smooth germ of curve or the union oftwo transverse smooth curves. The interest of d will be highlighted in the courseof the article. We will denote by Sd the union S ∪ d.

Theorem 2. For a generic irreducible curve S and any direction d, one has

minω∈Ω1(Sd)

ν (ω) =

[ν (Sd)

2

]where [·] stands for the integer part function. Moreover, if S is generic then the

curve Sd admits a balanced basis.

The rst section of this article is devoted to the proof of Theorem 2. The secondfocuses on the proof of Theorem 1 as a consequence of Theorem 2.

2. Balanced basis for a generic irreducible curve.

For any basis ω1, ω2 of Ω1 (Sd), the criterion of Saito ensures that

ν (ω1) + ν (ω2) ≤ ν (Sd) .

Thus at least one of these multiplicities is smaller than[ν(Sd)

2

], which proves one

part of the equality in Theorem 2. However, to obtain the whole equality we willneed some more informations about these generators. In this section, we are going

to construct quite explicitly an element of Ω1 (Sd) with multiplicity[ν(Sd)

2

].

2.1. The auxiliary foliation F [Sd]. In this section, we are going to construct afoliation associated to Sd denoted by F [Sd] thanks to a result of Alcides Lins-Neto[14] that is a kind of recipe to construct germs of singular foliations in the complexplane.

Let E be the minimal resolution of S. We denote it by

E : (M, D)→(C2, 0

).

The map E is a nite sequence of elementary blowing-ups of points

E = E1 E2 E3 · · · EN .

In general, if Σ is a germ of curve at(C2, 0

)or a divisor, ΣE will stand for the

strict transform of Σ by E, i.e., the closure inM of E−1 (Σ \ 0) . The exceptional


divisor of E, D = E−1 (0), is an union of a nite number of exceptional smoothrational curves intersecting transversely

D =

N⋃i=1

Di.

The components are numbered such that Di appears exactly after i blowing-ups.We can encode the map E in a square matrix E of size N the following way. The

rst two rows of E are

1 −10 1 . . .0 0...

...

︸︷︷︸

N

. The ith row Ci is dened by (Ci)i = 1

and (Ci)i−1 = −1 ; if Ei is the blowing-up of the point Di−1 ∩Dj then (Ci)j = −1

; for any other index j, (Ci)j = 0. Notice that the prole of E is decreasing.

We will denote by Ek the truncated process Ek · · · EN and Dk =⋃Ni=kDi the

exceptional divisor of Ek. Let Si be the strict transform of S by E1 · · · Ei−1 fori ≥ 2 and S1 = S. The map Ek is the minimal resolution of the total transform ofS1 by E1 · · · Ei−1. The following lemma is classical.

Lemma 3. The matrix E−1 has the following form1

0. . . ekl. . . 1

0 0 1

where for k < l, ekl is the multiplicity of Dl in the blowing-up process Ek. Further-more, the matrix EtE is the intersection matrix of D.

Example 4. Let us consider S =y5 = x13

. Then the matrix E is written

E =

1 −1 0 0 0 00 1 −1 −1 0 00 0 1 −1 −1 00 0 0 1 −1 −10 0 0 0 1 −10 0 0 0 0 1

.

The inverse matrix is written

E−1 =

1 1 1 2 3 50 1 1 2 3 50 0 1 1 2 30 0 0 1 1 20 0 0 0 1 10 0 0 0 0 1

.

The associated sequence of process of blowing-upsEkk=1..5

is presented in Figure

(2.1).


Figure 2.1. Sequence of process of blowing-ups associated to theresolution of y5 − x13 = 0.

The next lemma is the one upon which the construction of the auxiliary foliationF [Sd] is based.

Lemma 5. Let δ1 ∈ 0, 1, 2 be the number of components of the direction d. In the

same way, consider the number δi of branches of (E1 · · · Ei−1)−1

(d) meeting Sifor 2 ≤ i ≤ N. For i ≥ 2, δi ∈ 1, 2 . Let us consider the vector of integers dened

by

(2.1)

p1

p2

...

pN

= E

[ν(S1)−δ1

2

]+ 1[

ν(S2)−δ22

]+ 1

...[ν(SN )−δN

2

]+ 1

.

Then

(1) any integer pi is bigger or equal to −1. The case pi = −1 may occur only

when δi = 2, δi+1 = 1 and ν (Si) is odd.

(2) If Di ∩Dj 6= ∅ then one cannot have both pi = −1 and pj = −1.

(3) Let us consider D the exceptional divisor D deprived of DN and of the

components Di for which pi = −1. Then in each connected component of

D, there exists either at least one component Dj for which pj > 0 or a

component of dE .

Proof. The proof is an induction on the length of the resolution of Sd. Let usconsider that E is written

E =

1 −1 −1 · · · −1 0−1 −1

1 · · ·. . . −1

1 −11

.... . .

Expanding the expression of p1, we nd

p1 =

[ν (S1)− δ1

2

]+ 1−

n∑j=2

([ν (Sj)− δj

2

]+ 1

)


where n ≥ 2 is the greatest integer for which e1n = −1. Consider a Puiseuxparametrization of S1 = S,

S1 :

x = tp

y = tq + · · ·

with p < q. Following to the resolution of S1, the multiplicities satisfy : ν (S1) = p; for 2 ≤ j ≤ n− 1, ν (Sj) = q− p and ν (Sn) = (n− 1) p− (n− 2) q. If n = 2, thenthe following possibilities may occur

p1 =

[p− δ1

2

]−[p− δ2

2

]=

δ1 = 0

δ2 = 1

0 if p is odd

1 else

δ2 = 2 1

δ1 = 1

δ2 = 1 0

δ2 = 2

1 if p is odd

0 else

δ2 = 2

δ2 = 1

−1 if pis odd

0 else

δ2 = 2 0

.

When n ≥ 3, p1 is written

p1 =

[p− δ1

2

]−n−1∑j=2

[q − p− δj

2

]−[

(n− 1) p− (n− 2) q − δn2

]− n+ 2.

Notice that the integer δ1 belongs to 0, 1, 2 , δ2 belongs to 1, 2 and for 2 ≤ j ≤ n,δj = 2. Let us start with some remarks about the values of p1 when δ1 = 2.

• Suppose that δ1 = 2 and δ2 = 1. If p and q are both even or odd, then

p1 = 0. If p is even and q is odd, then p1 =

n−3

2 if n is oddn−4

2 elseis positive.

If p is odd and q is even, then p1 =

n−5

2 if n is oddn−4

2 else. In this situation,

p1 = −1 if and only if n = 3.• Suppose that δ1 = 2 and δ2 = 2. If p and q are both even or odd, then

p1 = 0. If p is even and q is odd, then p1 =

n−1

2 if n is oddn−2

2 else. If p is odd

and q is even then p1 =

n−3

2 if n is oddn−2

2 else.

Now, we are able to study the general behavior of p1 and to prove the proposition.

First case: δ1 = 0. Suppose rst that n > 2. The direction d is empty thus δ2 = 1and δj = 2 for 3 ≤ j ≤ n.

• Suppose that p and q are both even or both odd then p1 = 1. The connectedcomponent of D1 in D contains at least D1 with p1 > 0. Since, the other


components of D appear only in the remaining resolution, which is the oneof S2, Lemma 5 applied inductively to S2 establishes the result for S1.

• Suppose that p is odd and q is even then p1 =

n−3

2 if n is odd.n−2

2 else. If

n ≥ 4, the same argument as above proves the proposition . If n = 3, thenp1 = 0. In D, the component D1 meets D3. We know that δ3 = 2. Let usshow that δ4 = 2. If δ4 = 1 then S3 is neither tangent to D1 nor to D2.Looking at the Puiseux parametrization of S3 yields

q − p = 2p− q

which is impossible since p is odd. Thus δ4 = 2. Applying the initialremarks about p1 to p3 shows that p3 cannot be equal to −1. Therefore,the connected component of D1 in D meets the connected component ofD3 in D2. Thus, according to the induction hypothesis, this componentcontains a component Dj such that pj > 0.

• Suppose that p is even and q is odd then p1 =

n2 if n is oddn−2

2 else. Thus p1

is strictly positive and the arguments of the rst case yield the proposition.

Suppose nally that n = 2. Then the formula reduces to

p1 =[p

2

]−[p− 1

2

]=

1 if p is even

0 else.

If p is odd, the component D1 is attached to D2 in D. Since, δ2 = 1, it is im-possible that p2 = −1. Therefore, the connected component of D1 in D meets theconnected component of D2 in D2. Thus, according to the induction hypothesis,this component contains a component Dj such that pj > 0.

Second case: δ1 = 1. In that case, δ2 is 1 or 2, depending on the unique componentof d.

Let rst suppose that δ2 = 1. Thus, d is transverse to S. Suppose rst that n ≥ 3.

• Suppose that p and q are both odd or even p1 =

1 if p and q are odd

0 else.

If p1 = 0, since the component of d is attached to D1 in D, the componentof D1 in D contains a component of d. So, the induction hypothesis showsthe proposition.

• Suppose that p is even and q is odd then p1 =

n−4

2 if n is evenn−3

2 else. These

numbers are both positive since n ≥ 3. If they happens to vanish, theargument of the previous case still works.

• Suppose that p is odd and q is even then p1 =

n−3

2 if n is oddn−2

2 elseis posi-

tive.


Suppose now that n = 2. Then p1 = 0. Since dE is attached to D1, the inductionhypothesis shows the lemma.

Let suppose now that δ2 = 2 and n ≥ 3. Then d is tangent to S.

• Suppose that p and q are both odd or even. Then p1 =

0 if p is even

1 else.

If p1 = 0 then the component D1 is attached to some Dn in D. We knowthat δn = 2. If δn+1 = 2 then pn 6= −1. If δn = 1 then, in view of theprocess of reduction of S, one has

q − p = (n− 1) p− (n− 2) q.

Thus, there exist ∆ such that p = ∆ (n− 1) and q = ∆n. Since both pand q are even, ∆ is even. However, one can see that the multiplicity ofSn is then ∆ which is even. Thus, one cannot have pn = −1. Finally, D1

is attached to Dn whose connected component in D contains a componentDj with pj > 0, according to the induction hypothesis.

• Suppose that p is even and q is odd. Then p1 =

n−2

2 if n is evenn−1

2 else.

Both numbers are strictly positive and the induction hypothesis proves theproposition.

• If p is odd and q is even. Then p1 =

n2 if n is evenn−1

2 else. This case is the

same as just above.

Suppose now that n = 2. Then p1 =

1 if p is odd

0 else. In the last case, in view of

the resolution, D1 is attached to D2. If δ3 = 2 then one cannot have p2 = −1. Ifδ3 = 1, then the multiplicity of S2 is p which is even. So p2 6= −1.

Third case: δ1 = 2. In that case, d has two transverse components. At least oneis attached to D1. Thus, if ever p1 = 0, then the component of D1 in D containsat least a component of d. So, by induction, the proposition is proved. If p1 > 0,the same argument works. The only case that remains is p1 = −1. It may happenonly in the two following cases:

• suppose that n = 3, δ2 = 1, p is odd and q is even. Following the resolutionof S1, D1 is attached to D3 in D. We know that δ3 = 2. Suppose thatδ4 = 1, then according to the Puiseux parametrization, one has

q − p = 2p− qwhich is impossible since p is odd. Thus, δ4 = 2 and, therefore, p3 cannotbe equal to −1. Now, in view of the induction hypothesis, the propositionis proved.

• Suppose that n = 2, δ2 = 1 and p is odd. In that case, D1 is attached toD2 in D. But since δ2 = 1, p2 cannot be equal to −1.

This completes the proof of the proposition.


Now, we introduce a foliation associated to Sd. Using the result of Lins-Neto in[14], we consider a foliation F [Sd] whose resolution has the same topology as the

resolution of Sd. For the sake of simplicity, we keep denoting by D =⋃Ni=1Di the

exceptional of its resolution. We require that

• F [Sd] is dicritical and smooth along DN .• If pi = −1 , then F [Sd] is dicritical and smooth along Di.• At each corner point of D that does not meet a dicritical component, F [Sd]admits a linear singularity written in some local coordinates (x, y)

(2.2) λxdy + ydx, λ /∈ Q−

where xy = 0 is a local equation of D.• For each Di with pi ≥ 0, F [Sd] admits pi more linear singularities attachedto the smooth part of D and written in some local coordinates (x, y)

(2.3) λxdy + ydx, λ /∈ Q−

where x = 0 is a local equation of D.The local analytic class of the singularities added above depends on the

value of λ which is called the Camacho-Sad index of the singularity s alongD. It is denoted by

λ = CSs (F [Sd] , D)

where s is the singularity.• Finally, for each component of dE attached toDj with pj ≥ 0, F [Sd] admitsone more linear singularity along Dj .

The above data must satisfy some compatibility conditions stated in the theoremof Lins-Neto : rst, two dicritical components cannot meet which is ensured bythe second property of Proposition 5. Second, the sum of the eigenvalues of thesingularities along a component Dj should be equal to the self intersection: for anyinvariant component Dj , one has∑

s∈Dj

CSs (F [Sd] , Dj) = −Dj ·Dj ,

a relation known as the Camacho-Sad relation [2]. The third property in Proposi-tion 5 allows us to choose the Camacho-Sad indexes of the linear singularities addedat (2.2) and (2.3) in order to ensure the Camacho-Sad relation for any componentDj .

A lot of foliations can be constructed as above. Indeed, we do not prescribe theway these local data are glued together. Hence, there is a big number of nonanalytically equivalent choices. However, all the constructed foliations share someproperties. In any case, F [Sd] is a dicritical foliation whose resolution has the sametopological type as the resolution of Sd. Its singularities are linearizable and thusF [Sd] is of second kind [15]. Moreover, the foliation F [Sd] is tangent to some curvetopologically equivalent to Sd. Finally, the algebraic multiplicity is the desired one.


Figure 2.2. Topology of F [Sd] for S =y5 = x13

and any di-

rection d.

Lemma 6. Regardless the foliation F [Sd] constructed as above, one has

ν (F [Sd]) =

[ν (Sd)

2

].

Proof. Following a formula in [13, 3], one has

ν (F [Sd]) =

N−1∑i=1

pie1i + δ1 − 1.

Now, writing the rst line of the relation (2.1) and since pN = 0 yields

N−1∑i=1

pie1i + δ1 − 1 =

[ν (S1)− δ1

2

]+ δ1 =

[ν (Sd)

2

].

2.2. Deformations of F [Sd].

2.2.1. Basic vector elds and deformations. Let ω be a germ of 1−form and X avector eld. The vector eld X is said to be basic for ω if and only if

LXω ∧ ω = d (ω (X)) ∧ ω − ω (X) dω = 0.

The following lemma is classical.

Lemma 7. Let X be a germ of basic vector eld for ω. Then for any t, the ow

at time t of X, e[t]X is an automorphism of ω, i.e.,(e[t]X

)∗ω ∧ ω = 0.


More generally, a germ of automorphism of ω is an automorphism φ such thatφ∗ω ∧ ω = 0. If φ is tangent to Id, then there exists a formal basic vector eld Xsuch that e[1]X = φ. In what follows, we will simply denote the ow at time 1 of Xby eX .

Thanks to basic automorphisms, we can describe a surgery construction that pro-duces many non-equivalent germs of foliations from a given one. Consider theresolution E : (M, D) →

(C2, 0

)of some singular foliation F at

(C2, 0

). For

any covering Uii∈I of a neighborhood of D in M and for any 2−intersectionUij = Ui ∩ Uj , we consider φij a basic automorphism of E∗F which is the iden-tity map along Uij ∩ D. We suppose that the family φiji,j satises the cocycle

relation: on any 3-intersection Uijk, one has

φij φjk φki = Id.

We construct a manifold with the following gluing

M [φij ] =∐i

Ui/x∼φij(x)

which is a neighborhood of some divisor isomorphic to D. This manifold is foliatedby a foliation obtained by gluing with the same collection of maps the restrictedfoliations

E∗F|Ui

i. The obtained foliated manifold is denoted by F [φij ] . The

fact that F [φij ] is actually a foliation that comes from the resolution of a con-

crete singular foliation of(C2, 0

)is a consequence of the Grauert contraction result

[9] and of the Hartgos extension result. For any family φii of germs of basicautomorphisms along Ui, one has the following isomorphism

F [φij ] ' F[φiφijφ

−1j

].

A foliation obtained by the above construction is said to be a basic surgery of F .Our goal is to study the basic surgeries of F [Sd] and in particular to prove thefollowing

Proposition 8. Any curve C topologically equivalent to Sd admits a 1-form ω ∈Ω1 (C) obtained from a basic surgery of F [Sd] .

2.2.2. The sheaf TSd. In the resolution E : (M, D) →(C2, 0

), let us consider the

sheaf TSd, with D as basis, of vector elds tangent to D and to SN = SE thatvanish along the strict transform dE .

For any divisor Σ =∑niΣi inM, we denote by Ω2 (Σ) the sheaf with D as basis,

of 2−forms ω such that

νΣi (ω) ≥ −ni.

Let F be a balanced equation of F [Sd] as dened in [3]. First, we prove the followingproposition.

Proposition 9. One has

H1(D,Ω2

(2 (F )

E − SEd +D))

= 0

where the divisor (F )Eis (F )

E= (F = 0)

E − (F =∞)E.


The proof is an induction on the length of the resolution E. The rst step is thefollowing lemma.

Lemma 10. Let us consider a germ of divisor Σ at the origin of(C2, 0

). Let E1 :

(M1, D1) →(C2, 0

)be the standard blowing-up of the origin. Then the following

are equivalent

• ν0 (Σ) ≥ n• The rst cohomology group of Ω2

(ΣE + nD1

)on D1 vanishes

(2.4) H1(D1,Ω

2(ΣE + nD1

))= 0.

Proof. Let φ be an equation of Σ. Consider the standard coordinates of the blowing-up together with its standard covering.

U1 :

y = y1x1

x = x1

U2 :

y = y2

x = y2x2

.

The global sections of Ω2 (Σ + nD) on each associated open sets are written

Ω2(ΣE + nD1

)(U1) =

f (x1, y1)

1

φ1xn1dx1 ∧ dy1

∣∣∣∣ f ∈ O (U1)

Ω2(ΣE + nD1

)(U2) =

g (x2, y2)

1

φ2yn2dx2 ∧ dy2

∣∣∣∣ g ∈ O (U2)

Ω2(ΣE + nD1

)(U1 ∩ U2) =

h (x1, y1)

1

φ1xn1dx1 ∧ dy1

∣∣∣∣h ∈ O (U1 ∩ U2)

where φ1 = φE1

xν1, φ2 = φE1

yν2and ν = ν0 (Σ). Since the covering U1, U2 is acyclic,

the dimension of the cohomology group (2.4) is the number of obstructions to thefollowing relation

h (x1, y1)1

φ1xn1dx1 ∧ dy1 = g (x2, y2)

1

φ2yn2dx2 ∧ dy2 − f (x1, y1)

1

φ1xn1dx1 ∧ dy1

which is equivalent to the cohomogical equation

(2.5) h (x1, y1) = −f (x1, y1)− 1

y−ν+n+11

g

(1

y1, y1x1

).

Let h = xi01 yj01 . Then h is an obstruction to (2.5) if and only if j0 < 0 and the

following system cannot be solved in (i, j) ∈ N2i0 = j

j0 = j − i− ν − n− 1⇐⇒

j = i0

i = i0 − j0 + ν − n− 1.

Thus, ν ≥ n if and only if there is no obstruction.

Now let us prove Proposition (9).

Proof. The proof of the proposition is an induction on the length of the resolutionof Sd. Let us write

E = E1 E2.


Let U1 be a neighborhood of D1 \ Sing (S2) and U2 a very small neighborhood ofSing (S2). We dened the following open sets

(2.6) U1 =(E2)−1

(U1) U2 =(E2)−1

(U2)

The system U1,U2 is an open covering of D. The associated Mayer-Vietoris

sequence for the sheaf Ω2(

2 (F )E − SEd +D

)is written

H0(U1,Ω

2(

2 (F )E − SEd +D

))⊕H0

(U2,Ω

2 (· · · ))→

→ H0(U1 ∩ U2,Ω

2 (· · · ))→ N → 0(2.7)

and

0→ N → H1(D,Ω2 (· · · )

)→(2.8)

→ H1(U1,Ω

2 (· · · ))⊕H1

(U2,Ω

2 (· · · )).

We are going to identify each term of the above exact sequences.

The manifold D1 \Sing (S2) is an open set of C. Thus, it is a Stein sub-manifold ofM. According to a theorem of Siu [18], it admits a system of Stein neighborhoodsinM. Since, the sheaf is coherent, its cohomology vanishes on U1 and in (2.8), thefollowing relation holds,

H1(U1,Ω

2(

2 (F )E − SEd +D

))= 0.

Let F2 be dened by the germ of foliation E∗1F [Sd] at Sing (S2). By construction,the foliation F2 let invariant S2. Let F2 be a balanced equation of F2. Let h be alocal equation of D1 at Sing (S2) . Three cases have to be considered

• If D1 and the component of D attached to D1 are invariant for F [Sd], thenF2 can be chosen so that

D∣∣U2

= D2 + (h)E2

(F2)E2

= (h)E2

+ (F )E∣∣∣U2

Thus if the direction d2 of S2 is chosen to be the local trace at Sing (S2) ofthe union of dE1 and D1, then the next equality is satised

2 (F )E − SEd +D

∣∣∣U2

= 2 (F2)E2

− SE2

2,d2 +D2

• If the component attached at D1 is not invariant, then the computationsabove still hold.

• Finally, if D1 is not invariant then F2 can be chosen so that

D∣∣U2

= D2 (F2)E2

= (F )E

Thus setting for the direction d2 of S2 the local trace at Sing (S2) of thesole dE1 yields

2 (F )E − SEd +D

∣∣∣U2

= 2 (F2)E2

− SE2

2,d2 +D2


In any case, applying inductively Proposition 9 to S2 and to the associated divisor

2 (F2)E2

− SE2

2,d2+D2 ensures that in (2.8)

H1(U2,Ω

2(

2 (F )E − SEd +D

))= H1

(U2,Ω

2(

2 (F2)E2

− SE2

2,d2 +D2))

= 0.

The map E2 induces the isomorphisms U1E2

−−→ U1 and U1 ∩ U2E2

−−→ U1 ∩ U2 fromwhich follow the isomorphisms in cohomology

H0(U1,Ω

2(

2 (F )E − SEd +D

))' H0

(U1,Ω

2(

2 (F )E1 − SE1

d +D1

))H0(U1 ∩ U2,Ω

2 (· · · ))' H0

(U1 ∩ U2,Ω

2 (· · · )).(2.9)

Let us prove that E2 induces also an isomorphism on the set of global sectionsalong U2 and U2, namely

(2.10) H0(U2,Ω

2(

2 (F )E − SEd +D

))E2

' H0(U2,Ω

2(

2 (F )E1 − SE1

d +D1

)).

By induction, it is enough to prove (2.10) when E2 is the simple blowing-up of

Sing (S2). If ω is a global section of Ω2(

2 (F )E − SEd +D

)on U2 then the push-

forward of ω by E2 can be extended analytically at Sing (S2) by Hartogs extension

result. It induces naturally a section of Ω2(

2 (F )E1 − SE1

d +D1

)on U2. To the

converse, let ω be a section of Ω2(

2 (F )E1 − SE1

d +D1

)on U2.

• If D1 is not dicritical for F [Sd] then ω is written

ω = hGdx ∧ dy

x,

where x is a local equation of D1, G is an adapted meromorphic function

whose divisor is 2 (F )E1−SE1

d and h is any holomorphic function. If δ2 = 1then the possible component of d is attached to D1 at a dierent point fromSing (S2). Thus the valuation of G is equal to

ν (G) = e2n − 2

n−1∑i=2

pie2i = e2n − 2

[e2n − 1

2

]− 2 ≥ −1

Now, after the blow E2 which is written in adapted coordinates E2 (x, t) =(x, tx) , the pull back of Ω is written

E2∗ω = h∗G∗dx ∧ dt.

Thus, the valuation of E2∗ω is at least −1. The exceptional divisor of E2

cannot be dicritical for F [Sd] since δ2 = 1. Therefore, E2∗ω is a section

of Ω2(

2 (F )E2

− SE2

d +D1 ∪D2

)along D1 ∪D2. An inductive argument

shows the same if E2 is the complete resolution of S2. If δ2 = 2 then one ofthe components of dE1 , say dE1

1 , is attached to Sing (S2). The componentdE1 can be attached to a dicritical component or not. In any case, thevaluation of G is at least

ν (G) ≥ e2n − 2

n−1∑i=2

pie2i − 1 = e2n − 2[e2n

2

]− 1


If the exceptional divisor of E2 is dicritical then e2n is odd and ν (G) ≥ 0.If not, ν (G) ≥ −1. Thus, regardless of the exceptional divisor of E2 being

dicritical or not, E2∗ω is a section of Ω2(

2 (F )E2

− SE2

d +D1 ∪D2

)along

D1 ∪D2.• if D1 is dicritical then δ2 = 1. Moreover, ω is written

ω = hGdx ∧ dy, E2∗ω = h∗G∗xdx ∧ dt

where

ν (G) + 1 = e2n −n−1∑i=2

pie2i + 1 = e2n − 1− 2

[e2n − 1

2

]≥ 0.

Hence, E2∗ω is a section of Ω2(

2 (F )E2

− SE2

d +D1 ∪D2

)along D1 ∪D2.

By induction on the length of E2, the isomorphism (2.10) is proved. Thus, theisomorphisms (2.4) and the exact sequence (2.7) identify N with the cohomologygroup

H1(D1,Ω

2(

2 (F )E1 − SE1

d +D1

)).

Let us prove that the latter vanishes. If p1 = −1, then D1 is dicritical and δ1 = 2and δ2 = 1. Therefore,

ν(

2 (F )E1 − SE1

d

)= e1n − 2

n−1∑i=2

pie1i = −1

since e1n is odd. If p1 6= −1, then

ν(

2 (F )E1 − SE1

d

)= e1n − 2

n−1∑i=1

pie1i − δ1 = e1n − δ1 − 2

[e1n − δ1

2

]− 2 ≤ −1.

Therefore, according to Lemma (10), N vanishes, which completes the proof ofProposition 9.

The operator of basic vector elds is dened by

B : X 7→ LXE∗ ω

F∧ E∗ ω

F∈ Ω2

where ω is any 1−form with an isolated singularity dening F [Sd] and F anybalanced equation of F [Sd].

Proposition 11. Let Bn (F [Sd]) be the sheaf dened by the kernel

Bn (F [Sd]) = ker(B|Mn·TSd

)where Mn is the sheaf of O−module generated by the functions E∗f , f ∈ (x, y)

n.

There is an exact sequence of sheaves

(2.11) 0→ Bn (F [Sd])→Mn · TSd →Mn · Ω2

(2 (F )

E − SEd +D)

In→ 0


where In is a sheaf whose support is contained in the singular locus of F [Sd].Moreover, one has

H1

D,Mn · Ω2(

2 (F )E − SEd +D

)In

= 0.

In particular, extracted from the long exact in cohomology associated to 2.11, there

is an exact sequence

(2.12) H1 (D,Bn (F [Sd]))→ H1 (D,Mn · TSd)→ 0

Proof. The rst part of the proposition is a computation in local coordinates. Wedescribe the image ofMn ·TSd by the operator B. Since, F [Sd] is of second kind [15], the multiplicties of F [Sd] and of the balanced equation F along any irreduciblecomponent Di of the exceptional divisor satisfy [3]

• νDi (F [Sd]) = νDi (E∗F ) if Di is dicritical• νDi (F [Sd]) = νDi(E

∗F ) + 1 else.

Let p be a regular point of D where F [Sd] is regular and tangent to exceptionaldivisor. In some local coordinates (x, y) around p, the pull-back E∗ ωF is written

E∗ω

F= u

dx

xwhere x is a local equation of D. Now, a local section X of Mn · TSd is written

X = xm(ax

∂

∂x+ b

∂

∂y

), a, b ∈ C x, y .

Therefore, applying the basic operator leads to

B (X) = xmu2 ∂a

∂y

dx ∧ dyx

which is a local section of Mn ·Ω2(

2 (F )E − SEd +D

). Since the equation ∂a

∂y = f

can be solved for any f , the operator B is onto. This property is true for any typeof regular points for F [Sd]. Now suppose that p is a singular point of F [Sd]. Byconstruction, it is linearizable. Let us x some coordinates (x, y) such that

E∗ω

F= u

(λdx

x+

dy

y

)and xy is a local equation of D. A local section of Mn · TSd is written

X = xkyl(ax

∂

∂x+ by

∂

∂y

).

Let us write E∗ ωF (X) = uxkyl (aλ+ b) = uG. Then

B (X) = u2

(x∂G

∂x− λy∂G

∂y

)dx

x∧ dy

y.

Writing G =∑i,j gijx

i+kyj+l yields

x∂G

∂x− λy∂G

∂y=∑i,j

gij ((i+ k)− λ (j + l))xi+kyj+l.


Therefore, B (X) lies in Mn · Ω2(

2 (F )E − SEd +D

). However, the image of B

depends on λ. Indeed, xing some coordinates (xp, yp) as above for each singularpoint, we introduce the sheaf In dened by the following properties

• if U does not meet any singular point of F [Sd] then In (U) = 0.• if U meets the singular points p1, · · · , pj then In (U) is the set of 2 - formsη dened in a neighborhood of U such that in the coordinates (xp, yp), it islocally written

η = u2p

∑

i, j ≥ 0(i+ k)− λp (j + l) = 0

gijxi+kp yj+lp

dxpxp∧ dypyp

.

for any gij . For instance, if λp /∈ Q, then for n = 0, the stack (I0)p is simply

the nite vector space C · u2pdxpxp∧ dyp

yp.

Doing so, the map B becomes naturally onto the quotient of the sequence (2.11).Moreover, the latter is still exact since, by construction, for any point p, the stack

(In)p is a supplementary subspace of the image of (B)p in(

Ω2(

2 (F )E − SEd +D

))p.

The sheaf Mn is generated by its global sections. Therefore, Proposition 9 ensuresthat

H1(D,Mn · Ω2

(2 (F )

E − SEd +D))

= 0.

The short exact sequence associated to the quotient by In induces a long exactsequence in cohomology that is written

· · · → 0→ H1

D,Mn · Ω2(

2 (F )E − SEd +D

)In

→ H2 (D, In)→ · · ·

Since the support of In contains only isolated points, its cohomology vanishes inrank 2 [7]. Therefore, the rst term in the sequence above vanishes also. Finally,the long exact sequence in cohomology associated to (2.11) proves the end of Propo-sition 11.

2.3. Deformations of F [Sd] . Proposition (11) can be expressed as follows: anyinnitesimal deformation of Sd tangent to D at order n can be followed by an inn-itesimal deformation of the foliation F [Sd] at the same level of tangency. Roughlyspeaking, the proof of Proposition 8 consists in an non-commutative analog. Actu-ally, let us consider the following sheaves of non-abelian groups

Denition 12. For any sub-sheaf I of the sheaf of tangent vector eld to Sd thatvanish along d and D, we consider

G (I)

the sheaf of non-abelian groups generated by the ows of vector elds in I.

The rst step of the proof is the following:


Proposition 13. Induced in non-abelian cohomology by the embedding G (B1 (F [Sd])) →G (M · TSd), the following sequence

H1 (D,G (B1 (F [Sd])))→ H1 (D,G (M · TSd))→ 0

is exact.

Proof. Let us consider a 1−cocycle φijij ∈ Z1 (D,G (M · TSd)) . By construction,

it is a ow

(2.13) φij = eXij

where Xijij ∈ Z1 (D,M · TSd) . By induction on n, we are going to prove that

there existBnijij∈ Z1 (D,B1 (F [Sd])), Xn

i i ∈ Z0 (D,M · TSd) andXnij

ij∈

Z1 (D,Mn · TSd) such that

(2.14) e−Xni φije

Xnj = eBnijeX

nij .

For n = 1, this is the relation (2.13). Now, suppose this is true for n. Accord-

ing to Proposition (11), there existsBnij

ij∈ Z1 (D,B1 (F [Sd])) and Y ni i ∈

Z0 (D,M · TSd) such that

Xnij = Y ni + Bnij − Y nj .

According to the formula of Campbell-Hausdor, taking the ow at time 1 yields

e−Yni e−X

ni φije

Xnj eYnj = e−Y

ni eB

nijeX

nijeY

nj

= eBnij

[e−B

nij , e−Y

ni

]e−Y

ni eX

nijeY

nj

= eBnij

[e−B

nij , e−Y

ni

]eB

nijeY

n+1ij

= eBnijeB

nij e−B

nij

[e−B

nij , e−Y

ni

]eB

nijeY

n+1ij︸︷︷︸

∈G(Mn+1·TSd)

= eBn+1ij eX

n+1ij ,

which ensures the property by induction. Taking n as big as necessary, the propo-sition is a consequence of the stability property proved in [4].

We can improve a bit the previous property taking advantage of the inductivestructure of the resolution of Sd.

Proposition 14. Let E : (M, D) →(C2, 0

)be the resolution of F [Sd]. Con-

sider the sheaf I · TSd, where I is the ideal of functions vanishing along D and

B0 (F [Sd]) = ker(B|I·TSd

). Then for every φijij ∈ Z1 (D,G (I · TSd)) there

exists a familyψkijijk = 0 . . . l of 1-cocycles in Z1 (D,G (B0 (F [Sd]))) such that

(2.15) M [φij ] 'M[ψ0ij

]· · ·[ψlij].

In particular,M [φij ] is the support of a foliation obtained by basic surgery of F [Sd]


Proof. The proof is an induction on the length of the resolution of Sd. Let us con-sider a 1-cocyle φijij in Z

1 (G (I · TSd)) . Let us considerφijijthe restriction

of the cocyle φijij to D2.We are going to apply inductively the property to S2,d2

for some adapted direction d2 of S2 as dened in the proof of Proposition 9. Ap-plying inductively Proposition 14 to

φijij

yields the existence of 0−cocycles inG(I · TS2

d2

)and of 1−cocycles G

(B0

(F[S2d2

]))such that

φij = φ1iψ

1ijφ

2iψ

2ij · · ·ψNij

(φNj)−1 (

φN−1j

)−1 · · ·(φ1j

)−1,

a relation that is equivalent to (2.15) forφijij. Now, consider the following

1−cocyle

φij =

φ12φ

1jφ

2j · · ·φNj for i = 1 and j = 2

Id else.

It belongs to Z1 (G (I · TSd)). Since M and I coincide along D1, it belongs alsoto Z1 (G (M · TSd)). Therefore, Proposition (13) yields a 0−cocycle and 1-cocyclerespectively in G (M · TSd) and G (B1 (F [Sd])) such that

φij = φiψijφ−1j .

In particular, if (i, j) 6= 2, then φ−1i φj = ψij . Therefore, for any (i, j) 6= (1, 2), one

can write

φij = φ1iψ

1ijφ

2iψ

2ij · · ·ψNij φiψijφ−1

j

(φNj)−1 (

φN−1j

)−1 · · ·(φ1j

)−1

and

φ12 = φ1ψ12φ−12

(φN2)−1 (

φN−12

)−1 · · ·(φ1

2

)−1

which is equivalent to (2.15) for φijij . The proposition is proved.

Finally, we can prove Theorem 8. Let E′ : (M′, D′) →(C2, 0

)be the resolution

of C. The curves C and Sd are topologically equivalent and irreducible. Thus, theexceptional divisors D and D

′are analytically equivalent. Following [4] section 3.2,

there exist a 1-cocycle φijij in G (I · TSd) such that

M′ 'M [φij ] .

According to Proposition (14), M′ is the support of a foliation obtained from abasic surgery of F [Sd] that lets invariant the curve C, which completes the proofof Proposition 8.

Among the consequences of Proposition 8, we are going to use the following one.If ν (Sd) is even then F [Sd] is not dicritical after one blowing-up since p1 6= −1.Therefore, as a corollary of Propostion 8, we obtain

Proposition 15. If ν (Sd) is even, then there exists a 1−form of multiplicity[ν(Sd)

2

]in Ω1 (Sd) which is not dicritical after one-blowing up.


2.4. Proof of Theorem 2. Let S be an irreducible germ of curve and let E :(M, D)→

(C2, 0

)be its minimal resolution. Let d be any direction for S. Suppose

that for a generic curve in the topological class of S, there exists a germ of 1-form

ω tangent to Sd of multiplicity ν <[ν(Sd)

2

]. If ν is supposed to be generically

as small as possible, the previous hypothesis is equivalent to its analog in family:there exists an analytic family of curves Sd (ε) in the topological class of Sd, locallyversal, with ε ∈

(CN , 0

)such that, for ε generic, there exists an analytic family of

1-forms ω (ε) tangent to Sd (ε) with

ν (ω (ε)) = ν <

[ν (Sd (ε))

2

]=

[ν (Sd)

2

].

For ε generic, we can also suppose that ω (ε) is equireducible [16]. Let

E (ε) : (M (ε) , D (ε))→(C2, 0

)be the equisingular family of minimal resolutions of the foliations F (ε) dened byω (ε) . In particular, E (ε) is also an equisingular family of resolutions of Sd (ε) . Forthe sake of simplicity, we still denote by Sd and E respectively the curve Sd (0) andthe resolution E (0) .

Let Tijij be a 1−cocycle in Z1 (M, TSd) . Let us consider the deformation ob-

tained by the gluing

M[e(t)Tij

].

Since the ow e(t)Tij lets globally invariant Sd, the manifold M[e(t)Tij

]admits

an invariant curve topologically equivalent to SEd . By versality, the so denedtopologically trivial deformation is equivalent to a deformation Sd (ε (t)) for someanalytic factorization ε (t) : (C, 0)→

(CN , 0

). The deformation Sd (ε (t)) is followed

by the deformation of foliations F (ε (t)). Therefore on the open setM (ε)∗which

is M (ε) deprived of the singular locus of E (ε)∗ F (ε) , the cocycle

e(t)Tij

ij

is

equivalent to a cocycle of basic automorphisms. Thus, there exist a 0−cocycle of

automorphism φi (t)i letting globally invariant S (ε (t))E(ε(t))d and D (ε (t)) and a

1−cocycle of basic automorphisms Bij (t)ij for F , such that on M (ε (t))∗, one

has

e(t)Tij = φi (t)Bij (t)φ−1j (t) .

Taking the derivative at t = 0 of the above expression yields to a cohomogicalrelation onM (0) =M.

(2.16) Tij = Ti + bij − Tj

where Ti is a 0−cocycle in TSd and bijij a 1−cocycle of with values in the

sub-sheaf of basic vector elds BF ⊂ TSd.

Let us denote by Ω the image sheaf of TSd by the basic operator B for F with agiven balanced equation F .


The following diagram

H1 (M∗, BF)

i

H1 (M, TSd)

j //

γ

((

H1 (M∗, TSd)

B

H1 (M∗,Ω)

is commutative. Since for any 1-cocycle Tijij ∈ Z1 (M, TSd), a relation such as

2.16 exists, the image of i contains the one of j. Therefore, γ is the zero map.

The sheaf Ω onM∗ can be described as follows

Ω = Ω2(

2 (F )E − SEd +

∑niDi

)where D =

∑Di and the ni's are some integers depending on F . This sheaf can be

extended analytically onM. The Mayer-Vietoris sequence applied to the coveringM∗,U ofM where U is an union of some small open balls around each singularityis written

· · · → H0 (M∗,Ω)⊕H0 (U ,Ω)δ−→ H0 (M∗ ∩ U ,Ω) →

→ H1 (M,Ω)→ H1 (M∗,Ω)⊕H1 (U ,Ω)→ · · ·

The Hartogs extension result ensures that δ is onto. Moreover, since U can besupposed to be Stein and Ω is coherent, we deduce the exact sequence

(2.17) 0→ H1 (M,Ω)→ H1 (M∗,Ω) .

Lemma 16. We have

H2 (M, BF) = 0.

Proof. Taking small ow-boxes on the regular part of E∗F , we can nd a niteStein covering Uαα∈I ∪ Uss∈Sing(E∗F) ofM such that

• for any s ∈ Sing (E∗F), Us is a very small neighborhood of s.• for any α ∈ I, Uα does not meet the singular locus of E∗F .• On any open set Uα with α ∈ I, as well as on any 2−intersection, thefoliation E∗F is analytically trivial . Recall that F is analytically trivial onU if there exists a map ψ : U → C2 that is a biholomorphism on its imagesuch that (

ψ−1)∗ F = dx = 0 .

Applying the Mayer-Vietoris sequence to the covering leads to the following longexact sequence(2.18)

· · · → ⊕α,βH1 (Uα ∩ Uβ , BF)→ H2 (M, BF)→→ ⊕α∈IH2 (Uα, BF)

⊕⊕s∈Sing(E∗F)H

2 (Us, BF)→ ⊕α,βH2 (Uα ∩ Uβ , BF)→ · · · .


Lemma. Let U a Stein open set in C2 foliatied by the foliation F given by dx.Then for any i ≥ 1,

Hi (U,BF) = 0.

Proof. Once the coordinates (x, y) are given, a basic vector eld X for F satises

LXdx ∧ dx = 0.

It is uniquely written X = a (x) ∂∂x + b (x, y) ∂

∂y . Thus, the sheaf BF is isomorphic

to the direct sum of sheaves O1∂∂x ⊕O2

∂∂y whose cohomology in rank greater than

1 is trivial on Stein open sets.

The foliation is analytically trivial on any 2-intersection Uα ∩ Uβ and on Uα withα ∈ I. Therefore, the lemma above ensures that.

⊕α,βHi (Uα ∩ Uβ , BF) = 0, i = 1, 2 ⊕α∈I H2 (Uα, BF) = 0.

Finally, (2.18) is written

H2 (M, BF) ' ⊕s∈Sing(E∗F)H2 (Us, BF) .

The open sets Us can be taken as small as needed. Thus, the inductive limit [7] onthe family of open sets containing the singular locus of E∗F is written

0 = ⊕s∈Sing(E∗F)H2 (s , BF) ' ⊕s∈Sing(E∗F) lim

Us→sH2 (Us, BF) ' H2 (M, BF) .

from which the lemma follows.

Now, let us consider Ωijij ∈ Z1 (M,Ω) . Choosing a covering such that B is onto

Ω (Uij) for any 2−intersection Uij , we can write

(2.19) Ωij = B (Xij) .

Now, applying the Czech operator ∆ yields

0 = ∆(Ωijij

)= ∆B

(Xijij

)= B∆

(Xijij

).

Hence, ∆(Xijij

)is 2- cocycle inBF . Lemma 16 ensures the existence of 1−cocycle

Bijij ∈ Z1 (M, BF) such that

∆(Xij −Bijij

)= 0.

Now, Bij being basic, Ωij is still written

Ωij = B (Xij −Bij) .Therefore, in the equation 2.19 we can suppose that Xijij lies in Z

1 (M, TSd).

According to the diagram 2.4, the image of Xijij inH1 (M∗,Ω) is zero. According

to (16), so is its image in H1 (M,Ω) which is in fact, the 1-cocycle Ωijij itself.Finally, it is proved that

H1 (M,Ω) = 0.

Now, let us consider E1 : (M1, D1)→(C2, 0

)the rst blowing-up in the resolution

E. The Mayer-Vietoris sequence of some adapted covering shows that

H1(M1,Ω

2(

2 (F )E1 − SE1

d + n1D1

))→ H1

(M,Ω2

(2 (F )

E − SEd +∑

niDi

))


and therefore,

(2.20) H1(M1,Ω

2(

2 (F )E1 − SE1

d + n1D1

))= 0.

We are going to prove that the latter equality leads to a contradiction with ν (F) <[ν(Sd)

2

].

We recall that F being a balanced equation of F [3], the next relation holds

ν (F) = ν (F )− 1 + τ (F)

where τ (F) is a positive integer called the tangency excess of F .

• Suppose that F is not dicritical after the rst blowing-up. A computationin coordinates ensures that n1 = 1 − 2τ (F) . However, if (2.20) is true,Lemma 10 shows that

2ν (F )− ν (Sd) ≥ n1 ⇐⇒ 2ν (F)− ν (Sd) ≥ −1.

But ν (F) ≤[ν(Sd)

2

]− 1 gives us

2ν (F)− ν (Sd) ≤ 2

[ν (Sd)

2

]− ν (Sd)− 2 < −1

which is a contradiction.• Suppose now that F is dicritical after the rst blowing-up. Then n1 =−2τ (F) . Again, Lemma 10 ensures that

2ν (F)− ν (Sd) ≥ −2.

If ν (F) ≤[ν(Sd)

2

]− 2 then we are led to a contradiction. Suppose that

ν (F) =[ν(Sd)

2

]− 1. If ν (Sd) is odd then

2ν (F)− ν (Sd) = 2

(ν (Sd)− 1

2− 1

)− ν (Sd) = −3,

which is still a contradiction. Suppose that ν (Sd) is even. Then, ν (F) =ν(Sd)

2 − 1. The multiplicity ν (F) being as small as possible in Ω1 (Sd), a

basis of Ω1 (Sd) can be written ω0, ω1 withν (Sd)

2− 1 = ν (ω0) ≤ ν (ω1) and ν (ω0) + ν (ω1) ≤ ν (Sd) .

Thus there are only three possibilities for ν (ω1) .

if ν (ω1) = ν(Sd)2 +1, then any 1-form ω of multiplicity ν(Sd)

2 in Ω1 (Sd)is written

ω = aω0 + bω1.

where a is a function of multiplicity 1 and b is any function. In par-ticular, its jet of smallest order is written

(a)1 · (ω0) ν(Sd)2

,

where (?)i stands for the jet of order i. Thus ω is dicritical after one

blowing-up. This would imply that any element of multiplicity ν(Sd)2


in the Saito module is dicritical after one blowing-up. This is a con-tradiction with Proposition 15.

if ν (ω1) = ν(Sd)2 or ν (ω1) = ν(Sd)

2 − 1 then using the criterion of Saitowe have

(ω0)ν(ω0) ∧ (ω1)ν(ω1) = 0.

Therefore, ω1 is dicritical after one blowing-up. If ν (ω1) = ν(Sd)2 then

any 1−form of multiplicity ν(Sd)2 is dicritical. That is impossible. If

ν (ω1) = ν(Sd)2 − 1, let us write

ω0 = P0ωr + · · ·ω1 = P1ωr + · · ·

where ωr = xdy − ydx. Consider ω in the module of Saito with mul-

tiplicity ν(Sd)2

ω = aω0 + bω1 = (aP0 + bP1)ωr + · · · .

If ν (a) = 0 or ν (b) = 0 then ν (ω) = ν(Sd)2 − 1 unless there ex-

ists a non vanishing constant C such that P1 = CP0. But in that caseω0, ω1 − Cω0 is still a basis of the module of Saito with ν (ω1 − Cω0) >ν(Sd)

2 − 1 which leads to a case already treated. Thus, ν (a) ≥ 1 andν (b) ≥ 1 and necessarily, ω is dicritical after one blowing-up. As

before, any 1-form of multiplicity ν(Sd)2 would be dicritical after one

blowing-up, which is impossible.

This completes the proof of the rst part of Theorem 2,

(2.21) minω∈Ω1(Sd)

ν (ω) =

[ν (Sd)

2

].

Let us prove now the existence of balanced basis for Ω1 (Sd) .

Let us suppose rst that ν (Sd) is even. Consider a basis ω1, ω2 of Ω1 (Sd). Ac-

cording to (2.21) there are some 1-forms with multiplicity ν(Sd)2 in Ω1 (Sd). Hence,

at least one of the forms in the basis, say ω1, has a multiplicity equal to ν(Sd)2 .

The multiplicity of ω2 is greater or equal to ν(Sd)2 . If it is equal, then the basis is

balanced. If not, ω1, ω1 + ω2 is still a basis and is balanced.

Suppose now that ν (Sd) is odd. If the direction of Sd is empty or contains one

component, let us consider S = Sd ∪ L where L is a smooth curve transverse tothe direction of Sd. Since the multiplicity of S is even, according to what has been

done above, the module Ω1(S)admits a balanced basis. Therefore there exists a

couple a 1−forms ω1, ω2 of multiplicity ν(Sd)+12 such that

ω1 ∧ ω2 = ulfdx ∧ dy, u (0) 6= 0.

where l is an irreducible equation of L. Now, according to (2.21), there exists ω

tangent to Sd such that ν (ω) = ν(Sd)−12 . The 1-form lω is tangent to S. Hence,

there exist two germs of functions a1 and a2 such that

lω = a1ω1 + a2ω2.


The functions a1 and a2 cannot both vanish. Suppose by symmetry that a1 does

not vanish, then lω, ω2 is a basis of Ω1(S).Thus

lω ∧ ω2 = vlfdx ∧ dy, v (0) 6= 0.

Dividing by l the above expression leads to the criterion of Saito for the balancedbasis ω, ω2.

If the direction of Sd contains two components L1 and L2, then let us consider

S = S ∪ L1. The module Ω1(S)admits a balanced basis ω1, ω2 with ν (ω1) =

ν (ω2) =

[ν(S)

2

]= ν(S)+1

2 . Now, there exist ω in Ω1 (Sd) with ν (ω) =[ν(Sd)

2

]=

ν(S)+12 . Since ω is also tangent to S ∪ L1, there exist two functions a1 and a2 such

that

ω = a1ω1 + a2ω2.

The functions a1 and a2 cannot both vanish so we can suppose that a1 (0) 6= 0.

The family ω, ω2 is still a basis of Ω1(S)that satises

ω ∧ ω2 = wfl1dx ∧ dy, w (0) 6= 0.

Thus, multiplying by l2 leads to

ω ∧ l2ω2 = wfl1l2dx ∧ dy, w (0) 6= 0

and ω, l2ω2 is a balanced basis of Ω1 (Sd).

This ends the proof of Theorem 2.

3. Generic dimension of the moduli space of an irreducible curve.

The proof of Theorem 1 relies on the following

Proposition 17. Let S be a curve such that Ω1 (S) admits a basis ω1, ω2 with

ν (ω1) + ν (ω2) = ν (S)

Then

dimCH1 (D1, TS) =

(ν1 − 1) (ν1 − 2)

2+

(ν2 − 1) (ν2 − 2)

2

with νi = ν (ωi).

Proof. Since ω1, ω2 is a basis of Ω1 (S), the criterion of Saito ensures that

(3.1) ω1 ∧ ω2 = ufdx ∧ dy.

for some unity u and some reduced equation f of S. Let us consider the standardcovering of D1 by two open sets U1 and U2 and two charts (x1, y1) and (x2, y2) with

y2 = y1x1 x2 =1

y1E1 (x1, y1) = (x1, y1x1) .


The pull-back of (3.1) by E1 is written

E∗1ω1 ∧ E∗1ω2 = E∗1uE∗1fx1dx1 ∧ dy1

xν1+ν21

E∗1ω1

xν11︸︷︷︸ω1

1

∧ E∗1ω2

xν21︸︷︷︸ω1

2

= xν1E∗1uE∗1f

xν1x1dx1 ∧ dy1.

Simpliying by xν = xν1+ν2 yields the relation

ω11 ∧ ω1

2 = E∗1ufx1dx1 ∧ dy1.

The two forms ω11 and ω1

2 are tangent to the exceptional divisor regardless of their

dicriticalness. Obviously, they are also tangent to f = 0. According to the Saitocriterion, at any point c of the exceptional divisor, the germ of

ω1

1 , ω12

at c is a

basis of the module Ω1((S ∪D1

)c

). Since, the covering U1, U2 is acyclic for the

coherent sheaf TS, one has

H1 (D1, TS) = H1 (U1, U2 , TS) =H0 (U1 ∩ U2, TS)

H0 (U1, TS)⊕H0 (U2, TS).

Now, the spaces of global sections on U1, U2 and the intersection can be describedas follows

H0 (U1 ∩ U2, TS) =φ12ω

11 + ψ12ω

12

∣∣φ12, ψ12 ∈ O (U1 ∩ U2)

H0 (U1, TS) =φ1ω

11 + ψ1ω

12

∣∣φ1, ψ1 ∈ O (U1)

H0 (U1, TS) =φ2ω

21 + ψ2ω

22

∣∣φ2, ψ2 ∈ O (U1 ∩ U2).

Thus, the cohomological equation is written

φ12ω11 + ψ12ω

12 = φ1ω

11 + ψ1ω

12 − φ2ω

21 + ψ2ω

22

= φ1ω11 + ψ1ω

12 − φ2y

−ν11 ω1

1 + ψ2y−ν21 ω1

2 .

Since,ω1

1 , ω12

is a basis of O-module, the above leads to the system

φ12 = φ1 − φ2y−ν11

ψ12 = ψ1 − ψ2y−ν21

.

Writing these equations using the Taylor expansion leads to the checked number ofobstructions.

Finally, the proof of the Theorem 1 relies on the remark that the generic dimensionof M (S) is equal to

dimCH1 (D,TSgen)

where Sgen is a generic point in M (S). The Mayer-Vietoris sequence associated tothe covering (2.6) and applied to the sheaf TSgen decomposes H1 (D,TSgen) alongthe resolution of Sgen

H1 (D,TSgen) ' H1 (D1, TSgen)⊕C H1(D2, TSgen

).

The curve Sgen admits a balanced basis according to Theorem 2. Hence, Theorem1 is an inductive application of Proposition 17 noticing that in that case

dimH1 (D1, TSgen) = σ2 (ν (Sgen)) .


Example 18. Let us consider the following Puiseux parametrization

S :

x = t8

y = t20 + t30 + t35.

The successive multiplicities νci

((E1 · · · Ei−1)

−1(S))are

8, 9, 5, 6, 5, 5, 3, ..

Thus the generic dimension of the moduli space is

σ (8) + σ (9) + σ (5) + σ (6) + σ (5) + σ (5) = 20

which is conrmed by the algorithm of Hefez and Hernandes.

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Documents

S S - univ-toulouse.fr · 2 DIMENSION OF THE MODULI ASPCE OF A CURVE IN THE COMPLEX PLANE. The rst computations of the generic dimension of M(S) go back to Zariski [19, 20]