Equisingularity in pencils of curves on germs of reduced complex surfaces

We study pencils of curves on a germ of complex reduced surface $(S,0)$ . These are families of curves parametrized by P¹ having 0 as the unique common point. We prove that for w∈ P¹ , the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or w is a limit value for the function $ f/g $ along the singular locus of $(S,0)$ , where f and g are generators of the pencil.