Topological type of discriminants of some special families

TL;DR

This paper characterizes the topological type of discriminant curves for certain morphisms involving smooth and irreducible branches, showing it is determined by specific invariants like semigroup and Zariski invariant.

Contribution

It establishes a link between the topological type of discriminant curves and analytical invariants for special families of branches.

Findings

Topological type determined by semigroup and Zariski invariant.

Applicable to branches with multiplicity less than five.

Provides explicit criteria for discriminant curve classification.

Abstract

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.