The Abel map for surface singularities III: Elliptic germs

Abstract

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