Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)

We present a computational method for detecting highly singular members in families of algebraic varieties. Applying this approach to a family of numerical Godeaux surfaces, we obtain explicit examples with many singularities. In particular, we construct a Godeaux surface whose singular locus consists of two ${\mathsf{A}}_1$ and two ${\mathsf{A}}_3$ singularities. We show that this surface admits a $\mathbb{Z}\mathrm{/}2\times \mathbb{Z}\mathrm{/}4$ abelian cover which is a smooth minimal surface of general type with invariants $K^2=8$ and $p_g=0$, i.e. a fake quadric. Together with the result in the Appendix, this provides the first explicit construction of a fake quadric that does not arise as a quotient of a product of curves.