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

Abstract

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 Kähler 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.

v3: Theorem 3.4 in v2 has been eliminated as a result of a gap in the proof. Section 3.2 in v2 has been eliminated. The rest of main results remain unchanged. General redaction has been improved. v1: Comments are very welcome! 19 pages. 5 Figures