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

Invariants of relatively generic structures on normal surface singularities