The classification of minimal product-quotient surfaces with \(p_{g}=0\). (Q2840022)

A product-quotient surface is the minimal resolution \(S\) of the singularities of the quotient of a product \(C\times F\) of two complex curves by the action of a finite group \(G\) acting separately on the two factors. A systematic study of such quotient surfaces has been started in [The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871--1952), Torino, Italy, September 29--October 5, 2002. Torino: Università di Torino, Dipartimento di Matematica. 123--142 (2004; Zbl 1078.14051)] with the classification of the case where \(G\) is an abelian group acting freely and \(S\) has geometric genus \(p_g=0\). Then the assumption \(G\) abelian is dropped in [\textit{I. C. Bauer} et al., Pure Appl. Math. Q. 4, No. 2, 547--586 (2008; Zbl 1151.14027)]. The case where the surface has at most canonical singularities is classified in [Am. J. Math. 134, No. 4, 993--1049 (2012; Zbl 1258.14043)]. Finally in this paper under review the authors remove any restriction on the singularities of \(S\) and obtain a classification of product-quotient surfaces with \(p_g=0\) and self-intersection of the canonical divisor \(K_S^2>0\). In particular they prove that all but one of these surfaces are minimal. They also show that the Bloch conjecture holds for all surfaces. The calculations were performed using the computer algebra system Magma.