The Abel map for surface singularities III. Elliptic germs

Abstract

If $(\tilde{X},E)\to (X,o)$ is the resolution of a complex normal surface singularity and $c_1:\mathrm{Pic}(\tilde{X})\to H^2(\tilde{X},\mathbb{Z})$ is the Chern class map, then ${\mathrm{Pic}}^{l^{\prime }}(\tilde{X}):=c_1^{-1}(l^{\prime })$ has a (Brill--Noether type) stratification $W_{l^{\prime },k}:=\{\mathcal{L}\in {\mathrm{Pic}}^{l^{\prime }}(\tilde{X}):h^1(\mathcal{L})=k\}$. 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.