Pencil genus for normal surface singularities

Abstract

Let $(X,o)$ be a normal complex surface singularity. We define an invariant $p_e(X,o)$ for $(X,o)$ in terms of pencils of compact complex curves. Similarly, for a pair of $(X,o)$ and $h\in {𝔪}_{X,o}$ (the maximal ideal of ${𝒪}_{X,o}$), we define an invariant $p_e(X,o,h)$. We call $p_e(X,o)$ (resp. $p_e(X,o,h)$) the pencil genus of $(X,o)$ (resp. a pair of $(X,o)$ and $h$). In this paper, we give a method to construct pencils of compact complex curves by gluing a resolution space of $(X,o)$ and resolution spaces of some cyclic quotient singularities. Using this, we prove some formulae on $p_e(X,o,h)$ and estimate $p_e(X,o)$. We also characterize Kodaira singularities in terms of $p_e(X,o,h)$.