The aim of this article is to report on the authors' recent methods and results about moduli spaces for curve singularities and for modules over the local ring of a fixed curve singularity. We emphasize especially the general concept which lies behind these constructions. Therefore, the article might be useful to the reader who wishes to have the leading ideas and the main steps of the proofs explained without going into all the details. We also calculate explicit examples (for singularities and for modules) which illustrate the general theorems.
The construction of moduli spaces for certain objects means a geometric classification of the objects with respect to some equivalence relation. This is adequate, in particular, when it becomes too complicated to give a complete classification through (parametrized) normal forms. The basic concept, which stems from Mumford's Geometric Invariant Theory, is that of a coarse moduli space. Such a coarse moduli space for singularities respectively for modules over the local ring of a fixed singularity consists of a complex space or an algebraic variety M such that the points of M correspond in a unique way to equivalence classes of singularities or modules (with certain invariants fixed). Moreover, it is required that (flat) families of singularities or modules correspond to subvarieties of M. This means that the structure of M reflects neighbouring relations between objects which are given by small deformations.