Invariants of relatively generic structures on normal surface singularities

In the present article we work out a relative setup of generic structures on surface singularities. We fix an analytic type on a subgraph of a rational homology sphere resolution graph $\mathcal{T}$ and we choose a relatively generic normal surface singularity $\tX$ with resolution graph $\mathcal{T}$. We provide formulae for the geometric genus and the analytical Poincar\'e series of $\tX$. We determine the base point structure of natural line bundles on $\tX$ and give a lower bound on the multiplicity of $\tX$ which is expected to be sharp. We prove similar results about cohomology numbers of relatively generic line bundles on every singularity with rational homology sphere link.