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dc.contributor.authorNémethi A.en_US
dc.contributor.authorNagy J.en_US
dc.description.abstractAbstract. Let (X, o) be a complex normal surface singularity. We fix one of its good resolutions X → X, an effective cycle Z supported on the reduced exceptional curve, and any possible (first Chern) class l′ ∈ H 2 (X , Z). With these data we define the variety ECal′ (Z ) of those effective Cartier divisors D supported on Z which determine a line bundles OZ (D) with first Chern class l′. Furthermore, we consider the affine space Picl′ (Z) ⊂ H1(OZ∗ ) of isomorphism classes of holomorphic line bundles with Chern class l′ and the Abel map cl′ (Z) : ECal′ (Z) → Picl′ (Z). The present manuscript develops the major properties of this map, and links them with the determination of the cohomology groups H1(Z,L), where we might vary the analytic structure (X, o) (supported on a fixed topological type/resolution graph) and we also vary the possible line bundles L ∈ Picl′ (Z). The case of generic line bundles of Picl′ (Z) and generic line bundles of the image of the Abel map will have priority roles. Rewriting the Abel map via Laufer duality based on integration of forms on divisors, we can make explicit the Abel map and its tangent map. The case of superisolated and weighted homogeneous singularities are exemplified with several details. The theory has similar goals (but rather different techniques) as the theory of Abel map or Brill–Noether theory of reduced smooth projective curves.en_US
dc.publisherMathematische Annalenen_US
dc.subjectweighted homogeneous singularity.en_US
dc.subjectsuperisolated singularityen_US
dc.subjectsplice quotient singularityen_US
dc.subjectsurgery formulaeen_US
dc.subjectLaufer dualityen_US
dc.subjectBrill–Noether theoryen_US
dc.subjectPicard groupen_US
dc.subjecteffective Cartier divisoren_US
dc.subjectAbel mapen_US
dc.subjectnormal surface singularityen_US
dc.subjectresolution graphen_US
dc.subjectrational homology sphereen_US
dc.subjectnatural line bundleen_US
dc.subjectPoincar ́e seriesen_US
dc.titleThe Abel map for surface singularities I. Generalities and examplesen_US

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