Marked Godeaux surfaces with special bicanonical fibers

TL;DR

This paper investigates marked numerical Godeaux surfaces with special bicanonical fibers, providing a complete characterization of hyperelliptic and torsion fibers and relating known families to their homological framework.

Contribution

It introduces a new construction method for marked Godeaux surfaces and characterizes the existence of specific fiber types, linking to Reid and Miyaoka families.

Findings

Complete characterization of hyperelliptic bicanonical fibers

Identification of torsion fibers with Z/3Z and Z/5Z

Connection of known families to the homological setting

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.