Infinitesimal initial part of a singular foliation

Corral, Núria

doi:10.1590/s0001-37652009000400002

Cited by 2 publications

(3 citation statements)

References 4 publications

(7 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, as it is shown in [11], two non dicritical CGC sharing a complex logarithmic model (equivalently with the same set of separatrices and with the same system of indices) of multiplicity ν are defined by 1-forms with the same ν-jet when we fix the coordinates. In the real case these properties are not satisfied.…”

Section: Examplesmentioning

confidence: 98%

Real logarithmic models for real analytic foliations in the plane

Corral

Sanz

2011

Rev Mat Complut

Self Cite

View full text Add to dashboard Cite

Let S be a germ of a holomorphic curve at (C 2 , 0) with finitely many branches S 1 , . . . , S r and let I = (I 1 , . . . , I r ) ∈ C r . We show that there exists a nondicritical holomorphic foliation of logarithmic type at 0 ∈ C 2 whose set of separatrices is S and having index I i along S i in the sense of Lins Neto (Lecture Notes in Math. 1345Math. , 192-232, 1988 if the following (necessary) condition holds: after a reduction of singularities π : M → (C 2 , 0) of S, the vector I gives rise, by the usual rules of transformation of indices by blowing-ups, to systems of indices along components of the total transformS of S at points of the divisor E = π −1 (0) satisfying: (a) at any singular point ofS the two indices along the branches ofS do not belong to Q ≥0 and they are mutually inverse; (b) the sum of the indices along a component D of E for all points in D is equal to the self-intersection of D in M. This construction is used to show the existence of logarithmic models of real analytic foliations which are real generalized curves. Applications to real center-focus foliations are considered.

show abstract

Section: Examplesmentioning

confidence: 98%

Real logarithmic models for real analytic foliations in the plane

Corral

Sanz

2011

Rev Mat Complut

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, for generalized curve foliations we have that Lemma 2.2. [12] Assume that F is a non-dicritical generalized curve foliation. Let π : (X, P ) → (C 2 , 0) be a morphism composition of a finite number of punctual blow-ups and take an irreducible component E of the exceptional divisor π −1 (0) with P ∈ E. Then, the strict transforms π * F and π * G f satisfy that…”

Section: Local Invariants Of Foliations and Intersection Multiplicitiesmentioning

confidence: 99%

“…and ω E F by a constant. Moreover, if we consider (x l , y l ) coordinates centered at P E l with x l = x p and y l = y p − d E l , the equality of the Newton polygons [12], Lemma 1) and we can write…”

Section: Hence If E Is a Non-collinear Divisor We Have Thatmentioning

confidence: 99%

Jacobian curve of singular foliations

Corral¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Topological properties of the jacobian curve J F ,G of two foliations F and G are described in terms of invariants associated to the foliations. The main result gives a decomposition of the jacobian curve J F ,G which depends on how similar are the foliations F and G. The similarity between foliations is codified in terms of the Camacho-Sad indices of the foliations with the notion of collinear point or divisor. Our approach allows to recover the results concerning the factorization of the jacobian curve of two plane curves and of the polar curve of a curve or a foliation.

show abstract

Infinitesimal initial part of a singular foliation

Abstract: 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.

Cited by 2 publications

References 4 publications

Real logarithmic models for real analytic foliations in the plane

Real logarithmic models for real analytic foliations in the plane

Jacobian curve of singular foliations

Contact Info

Product

Resources

About