## 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.