A note on singularity indices of hyperelliptic fibrations

Let f:X→B be a relatively minimal hyperelliptic fibration of genus g. For such a fibration f, Xiao introduced a series of singularity indices si(f) for 2≤i≤g+2. These indices provide an effective way to study the geometry of f. It is known that si(f)≥0 for i≥3, but it is not clear whether s2(f) is non-negative. In this note, we construct a sequence of hyperelliptic fibrations with s2(f)<0, where the genus g can be arbitrarily large.