Hyperelliptic involutions on generic normal surface singularities

In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g>2$, then a generic smooth curve of genus $2$ is nonhyperelliptic. In this article we investigate the situation of normal surface singularities, so we fix a resolution graph $\mathcal{T}$ and a generic singularity with resolution $\text{tX}$ corresponding to it in the sense of \cite{NNII}. We consider an integer effective cycle $Z$ on the resolution $\text{tX}$ and investigate the existence of a complete linear series $g_2^1$ on it. The article has the main motivation that we will use heavily the results in it to compute the class of the image varieties of Abel maps in a following manuscript.