  Let $(X,o)$ be a complex normal surface singularity with rational homology
sphere link and let $\widetilde{X}$ be one of its good resolutions. Fix an
effective cycle $Z$ supported on the exceptional curve and also a possible
Chern class $l'\in H^2(\widetilde{X},\mathbb{Z})$. Define ${\rm Eca}^{l'}(Z)$
as the space of effective Cartier divisors on $Z$ and $c^{l'}(Z):{\rm
Eca}^{l'}(Z)\to {\rm Pic}^{l'}(Z)$, the corresponding Abel map. In this note we
provide two algorithms, which provide the dimension of the image of the Abel
map.
  Usually, $\dim {\rm Pic}^{l'}(Z)=p_g$, $\dim\,{\rm Im} (c^{l'}(Z))$ and ${\rm
codim}\,{\rm Im} (c^{l'}(Z))$ are not topological, they are in subtle
relationship with cohomologies of certain line bundles. However, we provide
combinatorial formulae for them whenever the analytic structure on
$\widetilde{X}$ is generic.
  The ${\rm codim}\,{\rm Im} (c^{l'}(Z))$ is related with
$\{h^1(\widetilde{X},\mathcal{L})\}_{\mathcal{L}\in {\rm Im} (c^{l'}(Z))}$; in
order to treat the `twisted' family $\{h^1(\widetilde{X},\mathcal{L}_0\otimes
\mathcal{L})\}_{\mathcal{L}\in {\rm Im} (c^{l'}(Z))}$ we need to elaborate a
generalization of the Picard group and of the Abel map. The above algorithms
are also generalized.
