Explicit Bicanonical Models of Eight Fake Quadrics

We compute explicit defining equations for eight fake quadrics arising as $\mathbb Z/2\times\mathbb Z/4$-covers of two singular $\mathbb Z/2$-Godeaux surfaces obtained in earlier work of the second author. Starting from explicit equations for the universal covers of the Godeaux surfaces, we reconstruct the relevant character eigenspaces and determine the homogeneous ideals of the bicanonical models of the eight fake quadrics in $\mathbb P^8$. All eight models are defined over $\mathbb Q$. We prove that the surfaces are pairwise non-isomorphic and rigid. Combined with the non-product result established in the earlier work, this gives the first explicit projective models of fake quadrics which are not isogenous to a product of curves.