Marked Godeaux surfaces with special bicanonical fibers

Abstract

In this paper we study marked numerical Godeaux surfaces with special bicanonical fibers. Based on our construction method of marked Godeaux surfaces 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 $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/5\mathbb{Z}$ arise in our homological setting.

26 pages. The content of this paper was part of the original paper on arXiv:2009.05357 which we have divided into two parts