Curve singularities with one Puiseux pair and value sets of modules over their local rings

In this paper we characterize the value set $\Delta$ of the $R$-modules of the form $R+zR$ for the local ring $R$ associated to a germ $\xi$ of an irreducible plane curve singularity with one Puiseux pair. In the particular case of the module of K\"ahler differentials attached to $\xi$, we recover some results of Delorme. From our characterization of $\Delta$ we introduce a proper subset of semimodules over the value semigroup of the ring $R$. Moreover, we provide a geometric algorithm to construct all possible semimodules in this subset for a given value semigroup.