The dimension of the image of the Abel map associated with normal surface singularities

Abstract

Let (X, o) be a complex normal surface singularity with rational homology sphere link and let Xe be one of its good resolutions. Fix an effective cycle Z supported on the exceptional curve and also a possible Chern class l ′ ∈ H2 (X, e Z). Define Ecal ′ (Z) as the space of effective Cartier divisors on Z and c l ′ (Z) : Ecal ′ (Z) → Picl ′ (Z), the corresponding Abel map. In this note we provide two algorithms, which provide the dimension of the image of the Abel map. Usually, dim Picl ′ (Z) = pg, dim Im(c l ′ (Z)) and codim Im(c l ′ (Z)) are not topological, they are in subtle relationship with cohomologies of certain line bundles. However, we provide combinatorial formulae for them whenever the analytic structure on Xe is generic. The codim Im(c l ′ (Z)) is related with {h 1 (X, e L)}L∈Im(c l ′ (Z)); in order to treat the ‘twisted’ family {h 1 (X, e L0 ⊗ L)}L∈Im(c l ′ (Z)) we need to elaborate a generalization of the Picard group and of the Abel map. The above algorithms are also generalized.