Let be a normal complex surface singularity. We define an invariant for in terms of pencils of compact complex curves. Similarly, for a pair of and (the maximal ideal of ), we define an invariant . We call (resp. ) the pencil genus of (resp. a pair of and ). In this paper, we give a method to construct pencils of compact complex curves by gluing a resolution space of and resolution spaces of some cyclic quotient singularities. Using this, we prove some formulae on and estimate . We also characterize Kodaira singularities in terms of .
Tadashi TOMARU. "Pencil genus for normal surface singularities." J. Math. Soc. Japan 59 (1) 35 - 80, January, 2007. https://doi.org/10.2969/jmsj/1180135500