The existence of surfaces of general type with the lowest possible values of the invariants, namely holomorphic Euler characterisitc $\chi=1$ and self-intersection of a canonical divisor $K^2=1$, are known to exist since Godeaux's construction in 1931. It has been shown that their torsion group can only be $\mathbb Z/n$ with $n=1,\ldots,5$. Reid constructed the moduli space for the cases $n=5,4,3.$ In this talk I will explain how my search for an explicit construction of a fake projective plane led us to the classification of $\mathbb Z/2$-Godeaux surfaces (the case $n=2$), and more recently to work on fake quadrics.
This is joint work with Eduardo Dias.
