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Topological type of discriminants of some special families
(Springer Science+Business Media, 2021-09-08)
We will describe the topological type of the discriminant curve of the morphism $$(\ell , f)$$ ( ℓ , f ) , where $$\ell $$ ℓ is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.
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Characterization of second type plane foliations using Newton polygons
(Sciendo, 2022-05-01)
In this article we characterize the foliations that have the same Newton polygon that their union of formal separatrices, they are the foliations called of the second type. In the case of cuspidal foliations studied by Loray [ Lo ], we precise this characterization using the Poincaré-Hopf index. This index also characterizes the cuspidal foliations having the same process of singularity reduction that the union of its separatrices. Finally we give necessary and sufficient conditions when these cuspidal foliations are generalized curves, and a characterization when they have only one separatrix.
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Weierstrass 1-Forms and Nondicritical Generalized Curve Foliations
(World Scientific, 2023-05-01)
In this paper, we introduce a distinguished expression for a given 1-form with respect to a polynomial f ∈ C{x}[y], called Weierstrass form. We will use this form and the properties of plane analytical curves to give new characterizations of nondicritical generalized curve foliations.
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On Briançon–Skoda Theorem for Foliations
(Elsevier GmbH, 2023-12-01)
We generalize Mattei's result relative to the Briançon–Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve.
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On the Approximate Polar Curves of Foliations
(MDPI, 2023-02-01)
We present a decomposition theorem of the generic polar curves of a generalized curve foliation with only one separatrix and the Hamiltonian foliations defined by the approximate roots of the generatrix. This is a generalization to foliations of the decomposition theorem of approximate Jacobians given by García Barroso and Gwoździewicz for plane branches.
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On Milnor and Tjurina Numbers of Foliations
(Springer Science+Business Media, 2025-06-01)
We study the relationship between the Milnor and Tjurina numbers of a singular foliation F, in the complex plane, with respect to a balanced divisor of separatrices B for F. For that, we associate with F a new number called the χ-number and we prove that it is a C1 invariant for holomorphic foliations. We compute the polar excess number of F with respect to a balanced divisor of separatrices B for F, via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of B and the χ-number of F. On the other hand, we generalize, in the plane case and the formal context, the well-known result of Gómez-Mont given in the holomorphic context, which establishes the equality between the GSV-index of the foliation and the difference between the Tjurina number of the foliation and the Tjurina number of a set of separatrices of F. Finally, we state numerical relationships between some classic indices, as Baum–Bott, Camacho–Sad, and variational indices of a singular foliation and its Milnor and Tjurina numbers; and we obtain a bound for the sum of Milnor numbers of the local separatrices of a holomorphic foliation on the complex projective plane.
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