The classical Noether-Lefschets Theorem states that for a sufficiently general surface S in P^3 the only algebraic curves lying on S are the complete intersections. In 2010 we proved an extension of this result to surfaces (and higher dimensional hypersurfaces in P^n) containing a fixed base locus. I will discuss this result and the describe how it can be applied together with tools from complex geometry and formal power series, to the study of class groups of local rings, in particular how they vary within an analytic isomorphism class. Among other things, we prove that any hypersurface singularity is analytically isomorphic to one whose local ring is a UFD and give a complete classification of the possible class groups for rational double point surface singularities.