The dimension of the image of the Abel map associated with normal surface singularities

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.