Upper bounds on the genus of Albanese fibrations

Let S be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration f:\,S → B of genus g.We prove a quadratic upper bound on the genus g, i.e., g≤ h(χ(OS)), where h is a quadratic function. We also construct examples showing that the quadratic upper bounds can not be improved to the linear ones. In the special case when pg(S)=q(S)=1, we show that g≤ 14.