  We study the analytic and topological invariants associated with complex
normal surface singularities. Our goal is to provide topological formulae for
several discrete analytic invariants whenever the analytic structure is generic
(with respect to a fixed topological type), under the condition that the link
is a rational homology sphere. The list of analytic invariants include: the
geometric genus, the cohomology of certain natural line bundles, the cohomology
of their restrictions on effective cycles (supported on the exceptional curve
of a resolution), the cohomological cycle of natural line bundles, the
multivariable Hilbert and Poincar\'e series associated with the divisorial
filtration, the analytic semigroup, the maximal ideal cycle.
  The first part contains the definition of `generic structure' based on the
work of Laufer. The second technical ingredient is the Abel map introduced by
the authors.
  The results can be compared with certain parallel statements from the
Brill-Noether theory and from the theory of Abel map associated with projective
smooth curves, though the tools and machineries are very different.
