# The Abel map for surface singularities III. Elliptic germs

## TL;DR

This paper studies the stratification of the Picard group for elliptic surface singularities, characterizes conditions related to the Abel map, and proves the equivalence of the End Curve and Weak End Curve Conditions.

## Contribution

It determines the stratification of the Picard group for elliptic singularities and characterizes the End Curve Conditions via the Abel map, establishing their equivalence.

## Key findings

- Stratification of Picard group for elliptic singularities determined.
- Characterizations of End Curve and Weak End Curve Conditions provided.
- Proved the equivalence of End Curve and Weak End Curve Conditions.

## Abstract

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.