Paper detail

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

Surfaces of general type with geometric genus $p_g=0$, which can be given as Galois covering of the projective plane branched over an arrangement of lines with Galois group $G=(\mathbb Z/q\mathbb Z)^k$, where $k\geq 2$ and $q$ is a prime number, are investigated. The classical Godeaux surface, Campedelli surfaces, Burniat surfaces, and a new surface $X$ with $K_X^2=6$ and $(\mathbb Z/3\mathbb Z)^3\subset {Tors} (X)$ can be obtained as such coverings. It is proved that the group of automorphisms of a generic surface of the Campedelli type is isomorphic to $(\mathbb Z/2\mathbb Z)^3$. The irreducible components of the moduli space containing the Burniat surfaces are described. It is shown that the Burniat surface $S$ with $K_S^2=2$ has the torsion group ${Tors} (S)\simeq (\mathbb Z/2\mathbb Z)^3$, (therefore, it belongs to the family of the Campedelli surfaces), i.e., the corresponding statement in the papers of C. Peters "On certain examples of surfaces with $p_g=0$" in Nagoya Math. J. {\bf 66} (1977), and I. Dolgachev "Algebraic surfaces with $q=p_g=0$" in {\it Algebraic surfaces}, Liguori, Napoli (1977), and in the book of W. Barth, C. Peters, A. Van de Ven "Compact complex surfaces", p. 237, about the torsion group of the Burniat surface $S$ with $K_S^2=2$ is not correct.