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

• János Nagy ^[1] ; András Némethi ^[2]
  1. [1] Hungarian Academy of Sciences

     Hungarian Academy of Sciences

     Hungría

     
  2. [2] University of Budapest, Hungary
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 3, 2022
• Idioma: inglés
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• Resumen

  • Let (X, o) be a complex normal surface singularity with rational homology sphere link and let X be one of its good resolutions. Consider an effective cycle Z supported on the exceptional curve and the isomorphism classes Pic(Z) of line bundles on Z.

    The set of possible values h1(Z,L) for L ∈ Pic(Z) can be understood in terms of the dimensions of the images of the Abel maps, as subspaces of Pic(Z). In this note we present two algorithms, which provide these dimensions. Usually, the dimension of 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 X is generic.