Marked Godeaux surfaces with special bicanonical fibers

In this paper we study marked numerical Godeaux surfaces with special bicanonical fibers. Based on the construction method of marked Godeaux surfaces in [12] we give a complete characterization for the existence of hyperelliptic bicanonical fibers and torsion fibers. Moreover, we describe how the families of Reid and Miyaoka with torsion $Z/3Z$ and $Z/5Z$ arise in our homological setting.