arXiv subjects
Publications and source records attributed to M. Fernando Hernández Iglesias.
We study, for plane complex branches of genus one, the topological type of its generic polar curve, as a function of the semigroup of values and the Zariski invariant of the branch. We improve some results given by Casas-Alvero in 2023, since we filter the topological type fixed for the branch by the possible values of Zariski invariants.
We characterize nondicrital generalized curve foliations with fixed reduced separatrix. Moreover, we give suficient conditions when a plane analytic curve is its reduced separatrix. For that, we introduce a distinguished expression for a given 1-form, called {\it Weierstrass form}. Then, using Weierstrass forms, we characterize the nondicritical generalized curve foliations: first, for foliations with monomial separatrix using toric resolution; second, for foliations with reduced separatrix, using the $GSV$-index. In this last case the characterization, which is our main result, could be interpreted in function of a polar of the foliation and a polar of its reduced separatrix.
We will describe the topological type of the discriminant curve of the morphism $(\ell, f)$, where $\ell$ is a smooth curve and $f$ is an irreducible curve (branch) of multiplicity less than five or a branch 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.