The possible values of geometric genera of normal surface singularities

In this article we prove that the possible geometric genuses $p_g(\text{tX})$ corresponding to normal surface singularities $\text{tX}$ 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 $\text{tX}$ and resolution graph $\mathcal{T}$, furthermore consider a Chern class $l^{\mathrm{\prime }}\in L^{\mathrm{\prime }}$. We prove that the possible values of $h^1(\text{tX},\text{calL})$, where $\text{calL}\in {\text{pic}}^{l^{\mathrm{\prime }}}(\text{tX})$ form an interval of integers.