Some new surfaces with \(p_ g=0\)
Let S be a complex projective nonsingular algebraic surface of general type with \(p_ g=0\), \(K^ 2=1\). Then, as is known, the torsion group \(Tors=Tors H_ 1(S,{\mathbb{Z}})\) is \({\mathbb{Z}}_ n\), \(1\leq n\leq 5\). A description of the corresponding moduli space for \(3\leq n\leq 5\) has been given by \textit{M. Reid} [J. Fac. Sci., Univ. Tokyo, Sect. I A 25, 75-92 (1978; Zbl 0399.14025)]. The remaining cases \(Tors=\{1\}\), \({\mathbb{Z}}_ 2\) also do occur, as appropriate examples show. In the paper under review the author constructs a 4 parameter family of surfaces as above with \(\pi_ 1={\mathbb{Z}}_ 2\). This is interesting since the only other example known with \(Tors={\mathbb{Z}}_ 2\), due to \textit{F. Oort} and \textit{C. Peters} [Indagationes Math. 43, 399-407 (1981; Zbl 0523.14025)] seems hard to deform. The surfaces are obtained by resolving the nodes of quotients Y/G, where Y is a complete intersection of 4 quadrics of \({\mathbb{P}}^ 6\) and G is a group of order 16. In particular, since G has a subgroup \({\mathbb{Z}}_ 8\) acting freely, Y/G is double covered by \(Y/{\mathbb{Z}}_ 8\), which is a Godeaux surface with \(p_ g=0\), \(K^ 2=2\), \(\pi_ 1={\mathbb{Z}}_ 8\). The author produces also two similar constructions of 4 parameter families of surfaces with \(\pi_ 1={\mathbb{Z}}_ 4\), double covered by some Godeaux-Reid surfaces.
scientific article; zbMATH DE number 3920665
scientific article