Abstract:

Xiao introduced a series of singularity indices to survey hyperelliptic fibrations. However, it remains unknown whether the second singularity index, $ s_2 $ , is non-negative. In this paper, I demonstrate a series of examples of degeneration of curves where $s_2$ tends to $-\infty$ as the genus $g$ grows. Moreover, I obtain a lower bound for $s_2$ for a given genus $g$ , thereby confirming that the index $s_2$ of fibrations for genus $g=2,3,4$ is non-negative.

Key words: algebraic surfaces, hyperelliptic fibrations, singularity indices