We will describe the topological type of the discriminant curve of the morphism (, f ), where
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.
