The Abel map for surface singularities III: Elliptic germs

Abstract The present note is part of a series articles targeting the theory Abel maps associated with complex normal surface singularities rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, 2019; Nagy Adv 371:20, 2020; Pure Appl Q 16(4):1123–1146, 2020). Besides general theory, by study specific families we wish to show power this new method. Indeed, using applied for elliptic are able prove several key properties (e.g. statements next paragraph), which ‘old’ techniques were not reachable. If $$({\widetilde{X}},E)\rightarrow (X,o)$$ ( X ~ , E ) → o resolution singularity $$c_1:{\mathrm{Pic}}({\widetilde{X}})\rightarrow H^2({\widetilde{X}},{\mathbb {Z}})$$ c 1 : Pic H 2 Z Chern class map, then $${\mathrm{Pic}}^{l'}({\widetilde{X}}):= c_1^{-1}(l')$$ l ′ = - has (Brill–Noether type) stratification $$W_{l', k}:= \{{\mathcal {L}}\in {\mathrm{Pic}}^{l'}({\widetilde{X}})\,:\, h^1({\mathcal {L}})=k\}$$ W k { L ∈ h } . In determine it together according cycle fixed components. E.g., that closure any $$W(l',k)$$ an affine subspace. For also characterize End Curve Condition Weak terms provide characterization them, finally they equivalent.