  If $(\widetilde{X},E)\to (X,o)$ is the resolution of a complex normal surface
singularity and $c_1:{\rm Pic}(\widetilde{X})\to H^2(\widetilde{X},{\mathbb
Z})$ is the Chern class map, then ${\rm Pic}^{l'}(\widetilde{X}):=
c_1^{-1}(l')$ has a (Brill--Noether type) stratification $W_{l', k}:=
\{{\mathcal L}\in {\rm Pic}^{l'}(\widetilde{X})\,:\, h^1({\mathcal L})=k\}$. In
this note we determine it for elliptic singularities together with the
stratification according to the cycle of fixed components. For elliptic
singularities we also characterize the End Curve Condition and Weak End Curve
Condition in terms of the Abel map, we provide several characterization of
them, and finally we show that they are equivalent.
