Infinitesimal initial part of a singular foliation

This work provides a necessary and sufficient condition to assure that two generalized curve singular foliations have the same reduction of singularities and same Camacho-Sad indices at each infinitely near point.


INTRODUCTION
The germs of holomorphic foliations over (C 2 , 0) are dynamical objects much more general than curves or even levels of meromorphic functions.In (Camacho et al. 1984), Camacho, Lins Neto and Sad introduce a class of foliations that share the reduction of singularities with their curve of separatrices.A more accurate approximation to a foliation (that assures coherent linear holonomies) is to compare them with levels of multivalued functions (logarithmic foliations); this has been done in (Corral 2003) by adding a control of the Camacho-Sad indices.In this paper we show that the necessary and sufficient condition for two foliations to share reduction of singularities, separatrices and Camacho-Sad indices is to have the same initial parts up to blow-up.
Let M be an analytic complex manifold of dimension two and consider two germs of foliations F and G defined in a neighbourhood of a point p ∈ M. Let ω F , ω G ∈ 1 M, p be 1-forms defining F and G respectively in a neighbourhood of p.It is clear that if the n-jets of ω F and ω G coincide for n big enough, then the foliations F and G will share all the properties mentioned above.But this condition is not necessary as we show with examples.

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NURIA CORRAL with A, B ∈ O M, p and gcd(A, B) = 1.The multiplicity ν p (F ) of F at p is the minimum of the multiplicities ν p (A), ν p (B) (note that ν p ( ) is the vanishing order at p).The point p is called a singular point of F if ν p (F ) ≥ 1.
Let S be a germ of an irreducible analytic curve at p. We say that S is a separatrix of F through p if f divides ω ∧ d f , where f = 0 is a reduced equation of S. Denote by Sep p (F ) the set of separatrices of F through p.
Let π 1 : M 1 → (M, p) be the blow-up of M with center at p and let E 1 be the exceptional divisor 1 F and we say that π 1 is dicritical.Consider now a non-singular invariant curve S = (y = 0) of the foliation F through p.The multiplicity of F at p along S is given by (Camacho et al. 1984)).Notice that it coincides with the multiplicity of the restriction ξ | S = B(x, 0)∂/∂ x of the vector field ξ = B(x, y)∂/∂ x − A(x, y)∂/∂ y to the curve S. Its behaviour under blow-up is given by where S 1 is the strict transform of S by π 1 and p 1 = S 1 ∩ E 1 .Moreover, we have that The Camacho-Sad index I p (F , S) of F relative to S at p is given by where A(x, y) = ya(x, y) (see (Camacho and Sad 1982)).
It is important to notice that, if ω, ω ∈ 1 M, p are 1-forms defining F in a neighbourhood of p, then ω = h • ω where h ∈ O M, p is a unit.In particular, this implies that the jets of order ν p = ν p (F ) of ω and ω coincide up to multiplication by a constant.We can write ω = j≥ν p ω j , where the coefficients of ω j are homogeneous polynomials of degree j.Then the projective class [ω ν p ] of the ν p -jet of ω is well defined.We call  p (F ) = [ω ν p ] the initial part of F at p.
Let us write ω j = A j d x + B j dy with A j , B j ∈ O M, p and consider the homogeneous polynomial of degree ν p + 1 given by P ν p +1 = x A ν p + y B ν p .The tangent cone T C p (F ) of F at p is given by P ν p +1 (x, y) = 0.
If P ν p +1 ≡ 0, then T C p (F ) is a finite union of lines and we denote by p p (F ) the projective class of the polynomial P ν p +1 .Notice that two foliations with different initial parts, may have the same tangent cone.Moreover, the blow-up π 1 is non-dicritical if and only if P ν p +1 ≡ 0. Cienc (2009) 81 (4) Let us now recall the construction of the Newton polygon of a foliation.Taking local coordinates (x, y) in a neighbourhood of p ∈ M, a germ f ∈ O M, p of a function at p can be written as a convergent power series f = i+ j≥ν p ( f ) f i j x i y j .We denote ( f ; x, y) = {(i, j) : f i j = 0}.The Newton polygon N ( f ; x, y) is the convex hull of ( f ; x, y) + (R ≥0 ) 2 .If C is the germ of curve at p defined by f = 0, then N (C; x, y) = N ( f ; x, y).Consider now a germ of foliation F given by ω = A(x, y)d x + B(x, y)dy in a neighbourhood of p.The Newton polygon N (F ; x, y) of F is the convex hull of (ω; x, y) + (R ≥0 ) 2 where (ω; x, y) = (x A; x, y) ∪ (y B; x, y).

An Acad Bras
Finally, let us recall the desingularization process of a foliation.We say that p is a simple singularity of F if there are coordinates (x, y) centered at p so that F is defined by a 1-form of the type The minimal morphism π F , in the sense that it cannot be factorized by another morphism with the above properties, is called the minimal reduction of singularities of F .The centers of the blow-ups of any reduction of singularities of F are called infinitely near points of F. In particular, all the centers of the blow-ups to obtain π F and the final singularities in π −1 F ( p) are infinitely near points of F .This notion extends the well known notion of infinitely near points of a curve.

GENERALIZED CURVE FOLIATIONS
Let F = F(M, p) be the space of singular holomorphic foliations of (M, p).We denote by G the subspace of F composed by generalized curve foliations, that is, foliations without saddle-node singularities in their reduction of singularities (see (Camacho et al. 1984)).Given a germ C = ∪ r i=1 C i of analytic curve in (M, p), we denote by F C the subspace of F composed by foliations whose curve of separatrices is C, and we put G C = G ∩ F C .Notice that foliations in F C are non-dicritical.Particular elements of G C are the foliations given by d f = 0, where f = 0 is a reduced equation of C; let G C be one of such foliations.It is known (see (Camacho et al. 1984, Rouillé 1999)) that, for an element F ∈ G C , we have that: (i) The minimal reduction of singularities of F and C coincide.
for any q ∈ E 1 , where ( • ) q denotes the intersection multiplicity at q. PROOF OF LEMMA 1. Assertions 1 and 2 are a direct consequence of the above properties (ii) and (iii) applied at each infinitely near point of F. Let us prove assertion 3.
Let (x, y) be coordinates in a neighbourhood of p and take a 1-form ω = A(x, y)d x + B(x, y)dy defining F in a neighbourhood of p. Consider a reduced equation f = 0 of C and let G C be the foliation defined by d f = 0.If we denote ν = ν 0 (F ), then the multiplicity ν p (C) of C at p is equal to ν + 1.Therefore, we can write f = i≥ν+1 f i , A = i≥ν A i and B = i≥ν B i with f i , A i and B i being homogeneous polynomials of degree i.
Consider π 1 : M 1 → (M, p) the blow-up of p and denote by E 1 = π −1 1 ( p) the exceptional divisor.Let us prove that at each point q ∈ E 1 .Then the result follows using similar arguments.
Consequently, we only have to prove equality (3) at the singular points q 1 , q 2 , . . ., q t of π * 1 F in E 1 .We can assume that all the singular points belong to the first chart of E 1 and we take coordinates (x , y ) with E 1 = (x = 0) and π 1 (x , y ) = (x , x y ).In these coordinates, the foliation π * 1 G C is given by We can assume that the point q l is given by (0, a l ) in the coordinates (x , y ), for l = 1, . . ., t.In particular, we have that μ q l (π * 1 G C , E 1 ) = r l .In the coordinates (x , y ), the foliation π * 1 F is given by ω 1 = π * 1 ω/x ν = 0 with By definition, we have that μ q l (π * 1 F , E 1 ) = ord y=a l (P ν+1 (1, y)).Take (x l , y l ) coordinates centered at q l with x l = x and y l = y − a l .From expression (4), we deduce that Then the equality of the Newton polygons N (π * 1 G C ; x l , y l ) and N (π * 1 F ; x l , y l ) implies that the point (1, r l ) belongs to (ω 1 ; x l , y l ).Taking into account that we deduce that ord y=a l P ν+1 (1, y) ≥ r l ; ord y=a l B ν (1, y) ≥ r l − 1. (5) for each l = 1, . . ., t, and the following equality gives the result.
COROLLARY 1.Given two foliations F and G of G C , the polynomials p p (F ) and p p (G) coincide at each infinitely near point p of C.

LOGARITHMIC FOLIATIONS
Given a germ of plane curve with f i = 0 being a reduced equation of C i and λ i = 0. Notice that if η as above defines a logarithmic foliation in G C , then the foliation given by η + α = 0, with α holomorphic and dα = 0, is also logarithmic and belongs to G C .In fact, we can write η + α = r i=1 λ i d f * i / f * i for suitable equations f * i = 0 of C i .Thus, logarithmic foliations in G C are characterized by their exponent vector λ = (λ 1 , . . ., λ r ) ∈ P r −1 C .

NURIA CORRAL
We denote by L λ one of these foliations.Observe that not all the foliations defined by a 1-form of the type ( * ) belong to G C since they could be dicritical foliations.
Recall that a logarithmic foliation L is a logarithmic model of a foliation F if they have the same separatrices, the same reduction of singularities and the same Camacho-Sad indices at the final singularities after desingularization.The existence of logarithmic models for non-dicritical generalized curves was proved in (Corral 2003) and is unique once a reduced equation of the separatrices is fixed.Then, for each foliation F ∈ G C , we denote by λ(F ) the exponent vector of the logarithmic model of F .We denote by G C,λ the set of foliations F ∈ G C such that λ(F ) = λ.Thus, the set G C is equal to Notice that G C,λ = ∅ if and only if the foliation given by r i=1 λ i d f i / f i = 0 is non-dicritical.The goal of this article is to study the properties which characterize the foliations in one of such sets G C,λ .
Consider now a meromorphic 1-form η of the type ( * ) and assume that the foliation given by η = 0 belongs to G C .Thus all the foliations defined by η + α = 0, with α holomorphic, belong to G C,λ but in general they are not logarithmic.However, it is not true that all foliations of G C,λ are defined by a 1-form of the type η + α = 0 with α holomorphic.

INFINITESIMAL INITIAL PART
Given a non-dicritical foliation F ∈ F, we define the infinitesimal initial part J (F ) of F to be the family { q (F )} where q varies among the infinitely near points of F .We wonder under what conditions two foliations F and G have the same infinitesimal initial part.It is clear that having the same curve of separatrices and the same reduction of singularities are necessary conditions.But these conditions are not enough even if we work with generalized curve foliations.Notice also that a sufficient condition is that of F and G having the same n-jet at the point p, with n being big enough.However, this condition is not necessary as shown by d y 3 − x 11 = 0 and 11 − x 10 + yx 7 d x + 3 y 2 − x 8 dy = 0.Then, the result is THEOREM 1.Let F and G be two foliations in G C .The foliations F and G have the same infinitesimal initial part if and only if λ(F ) = λ(G).
PROOF.Take F and G two foliations in G C and assume that J (F ) = J (G).Consider the minimal reduction of singularities π C : M → (M, p) of C (notice that π C = π F = π G ).By hypothesis, the initial parts  q (F ) and  q (G) coincide at each point q ∈ π −1 C ( p).In particular, this implies that the Camacho-Sad indices I q (π * F , F) and I q (π * G, F) coincide for a component F ⊂ π −1 C ( p) through q and consequently λ(F ) = λ(G).
Reciprocally, take a foliation F ∈ G C,λ and let us show that  p (F ) =  p (L), where L = L λ is a logarithmic foliation in G C,λ .Put ν = ν p (F ) and assume that F and L are given by the holomorphic 1-forms ω F = 0 and ω L = 0 respectively, where p) composition of a finite number of blow-ups of points such that the strict transform π * F of F by π satisfies that -each irreducible component of the exceptional divisor D = π −1 ( p) is either invariant by π * F or transversal to π * F ; -all the singular points of π * F are simple and do not belong to a dicritical component of the exceptional divisor D.