On degenerations of \(\mathbb{Z} {/} 2\)-Godeaux surfaces (Q2104837)

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