Old and new examples of surfaces of general type with $p_g=0$

Vik. S. Kulikov

Izvestiya. Mathematics , Tome 68 (2004) no. 5, pp. 965-1008

Résumé

We investigate surfaces of general type with geometric genus $p_g=0$ which may be given as Galois coverings of the projective plane branched over an arrangement of lines with Galois group $G=(\mathbb Z/q\mathbb Z)^k$, where $k\geqslant 2$ and $q$ is a prime. Examples of such coverings include the classical Godeaux surface, Campedelli surfaces, Burniat surfaces, and a new surface $X$ with invariants $K_X^2=6$ and $(\mathbb Z/3\mathbb Z)^3\subset\operatorname{Tors}(X)$. We prove that the automorphism group of a generic surface of Campedelli type is isomorphic to $(\mathbb Z/2\mathbb Z)^3$. We describe the irreducible components of the moduli space containing the Burniat surfaces. We also show that the Burniat surface $S$ with $K_S^2=2$ has torsion group $\operatorname{Tors}(S)\simeq(\mathbb Z/2\mathbb Z)^3$ (and hence belongs to the family of Campedelli surfaces), that is, the corresponding statement in [9], [4], and [1, p. 237], about the torsion group of the Burniat surface $S$ with $K_S^2=2$ is not correct.