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 widetilde{X} be one of its good resolutions. Consider an effective cycle Z supported on the exceptional curve and the isomorphism classes mathrm{Pic}(Z) of line bundles on Z. The set of possible values h^1(Z,mathcal {L}) for mathcal {L}in mathrm{Pic}(Z) can be understood in terms of the dimensions of the images of the Abel maps, as subspaces of mathrm{Pic}(Z). In this note we present two algorithms, which provide these dimensions. Usually, the dimension of mathrm{Pic}(Z) and of the dimension of the image of the Abel maps are not topological. However, we provide combinatorial formulae for them in terms of the resolution graph whenever the analytic structure on widetilde{X} is generic.