$\mathbb Z/2$-Godeaux surfaces

Abstract

We prove that the moduli space of numerical Godeaux surfaces with torsion group $\mathbb{Z}/2$ is irreducible and unirational of dimension 8. Moreover, we show that the topological fundamental group of these surfaces is also $\mathbb{Z}/2$. Our approach is based on the explicit construction of equations for all universal covers of numerical Godeaux surfaces with torsion group $\mathbb{Z}/2$.