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∈$¥mathfrak m$_X,o (the maximal ideal of $¥mathscr{O}$_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).