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

The possible values of geometric genera of normal surface singularities