Complex surfaces of general type with \(K^{2} = 3,4\) and \(p_{g} = q = 0\) (Q2418701)

The autors of the article under review construct minimal complex surfaces of general type satisfying \(p_g = 0\) and \(K^2 = 3\) or \(4\) (and call those Keum-Naie Surfaces). The first constructions of such surfaces are due to \textit{J. Keum} [``Some new surfaces of general type with \(p_g=0\)'', unpublished manuscript (1988)] and \textit{D. Naie} [Math. Z. 215, No. 2, 269--280 (1994; Zbl 0791.14016)]; see also [\textit{I. Bauer} and \textit{F. Catanese}, Groups Geom. Dyn. 5, No. 2, 231--250 (2011; Zbl 1235.14033)] for a variation on the construction and a description of the corresponding moduli space. Both constructions, the original as well as the novel construction under review, proceed by taking double-covers of rather special Enriques surfaces containing eight disjoint \((-2)\)-curves, branched over carefully chosen configurations of curves. But whilst the Enriques surfaces employed by Keum and Naie arise as quotients of products of elliptic curves, the authors use a different construction: One obtains a \(K3\) surface by resolving the double-cover of \(\mathbb P^1\times\mathbb P^1\) branched over the union of eight fibres (four for either direction). This \(K3\) surface carries an involution which happens to identify pairs of the \(16\) \((-2)\)-curves, so that the quotient is an appropriate Enriques surface to begin with. The choice of the branched covering is well explained, but its details are of scope for this review. The example satisfying \(K^2=3\) contains a \((-3)\)-curve and the authors seem to suggest or expect a relationship between the contraction of that curve and the other example. However, they defer a closer investigation to future research.