The Abel map for surface singularities I. Generalities and examples

Let $(X,o)$ be a complex normal surface singularity. We fix one of its good resolutions $\widetilde{X}\to X$, an effective cycle $Z$ supported on the reduced exceptional curve, and any possible (first Chern) class $l'\in H^2(\widetilde{X},\mathbb{Z})$. With these data we define the variety ${\rm ECa}^{l'}(Z)$ of those effective Cartier divisors $D$ supported on $Z$ which determine a line bundles $\mathcal{O}_Z(D)$ with first Chern class $l'$. Furthermore, we consider the affine space ${\rm Pic}^{l'}(Z)\subset H^1(\mathcal{O}_Z^*)$ of isomorphism classes of holomorphic line bundles with Chern class $l'$ and the Abel map $c^{l'}(Z):{\rm ECa}^{l'}(Z)\to {\rm Pic}^{l'}(Z)$. The present manuscript develops the major properties of this map, and links them with the determination of the cohomology groups $H^1(Z,\mathcal{L})$, where we might vary the analytic structure $(X,o)$ (supported on a fixed topological type/resolution graph) and we also vary the possible line bundles ${\mathcal{L}}\in {\rm Pic}^{l'}(Z)$. The case of generic line bundles of ${\rm Pic}^{l'}(Z)$ and generic line bundles of the image of the Abel map will have priority roles. Rewriting the Abel map via Laufer duality based on integration of forms on divisors, we can make explicit the Abel map and its tangent map. The case of superisolated and weighted homogeneous singularities are exemplified with several details. The theory has similar goals (but rather different techniques) as the theory of Abel map or Brill--Noether theory of reduced smooth projective curves.