The moduli space of \(\mathbb Q\)-homology projective planes with 5 quotient singular points
A normal complex projective surface \(S\) is called a \(\mathbb Q\)-homology projective plane it it has the same Betti numbers as \(\mathbb P^2\). If \(S\) is smooth then \(S\) is either isomorphic to \(\mathbb P^2\) or a so-called fake projective plane. It follows from the Bogomolov-Miyaoka-Yau inequality then if \(S\) has quotient singularities then there are at most five singular points. The author proves that if \(S\) has five quotient singularities then there are three \(A_1\)-singularities and two \(A_3\)-singularities. Moreover, the minimal resolution of singularities \(S'\) of \(S\) is an Enriques surface. The author shows that the moduli space of Enriques surfaces obtained in this way is zero-dimensional.
\(\mathbb Q\)-homology projective planes
scientific article; zbMATH DE number 5723516