The possible values of geometric genera of normal surface singularities

Abstract

In this article we prove that the possible geometric genuses $p_g(\tX)$ corresponding to normal surface singularities $\tX$ with fixed negative definite resolution graph $\mathcal{T}$ form an interval of integers. Similarly let us have a resolution graph $\mathcal{T}$ and a fixed normal surface singularity $(X, 0)$ with resolution $\tX$ and resolution graph $\mathcal{T}$, furthermore consider a Chern class $l' \in L'$. We prove that the possible values of $h^1(\tX, \calL)$, where $\calL \in \pic^{l'}(\tX)$ form an interval of integers.