A new explicit fake quadric is constructed from a Z/2-Godeaux surface with two A1 and two A3 singularities, and shown to be the first example not arising as a quotient of a product of curves.
$\mathbb Z/2$-Godeaux surfaces
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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 $\mathbb{Z}/2$.
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Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)
A new explicit fake quadric is constructed from a Z/2-Godeaux surface with two A1 and two A3 singularities, and shown to be the first example not arising as a quotient of a product of curves.