On degenerations of $\mathbb{Z}/2$-Godeaux surfaces

Abstract

We compute equations for the Coughlan's family of Godeaux surfaces with torsion Z=2, which we call Z=2-Godeaux surfaces, and we show that it is (at most) 7 dimensional. We classify all non-rational KSBA degenerations W of Z=2-Godeaux surfaces with one Wahl singularity, showing that W is birational to particular either Enriques surfaces, or D 2;n elliptic surfaces, with n D 3; 4 or 6. We present examples for all possibilities in the first case, and for n D 3; 4 in the second.