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Item type:Publication, 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. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, 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. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, 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.1 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, An upper bound for the GSV-index of a foliation(Springer Science+Business Media, 2025-04-01)Let F be a holomorphic foliation at p∈C2, and let B be a separatrix of F. We prove the following upper bound GSVp(F,B)≤4τp(F,B)-3μp(F,B), where GSVp(F,B) is the Gómez-Mont-Seade-Verjovsky index of the foliation F with respect to B, μp(F,B) is the multiplicity of F along B and τp(F,B) is the dimension of the quotient of C{x,y} by the ideal generated by the components of any 1-form defining F and any equation of B.4