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.
Narasimhan, Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995
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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.