Involutions on surfaces of general type with \(p_{g}=0\). I: The composed case (Q2851095)

Let \(S\) be a smooth minimal surface of general type with geometric genus \(p_g=0\). If the self-intersection of the canonical divisor of \(S\) is \(K^2\geq 2\), then the image of the bicanonical map \(\phi_2\) of \(S\) is a surface ([\textit{G. Xiao}, Bull. Soc. Math. Fr. 113, 23--51 (1985; Zbl 0611.14031)]).NEWLINENEWLINELet \(d\) be the degree of \(\phi_2\). From results of \textit{M. Mendes Lopes} [Arch. Math. 69, No. 5, 435--440 (1997; Zbl 0921.14024)] and \textit{M. Mendes Lopes} and \textit{R. Pardini} [Bull. Lond. Math. Soc. 33, No. 3, 265--274 (2001; Zbl 1075.14514)], we know that \(K^2=9\Rightarrow d=1\), \(K^2\geq 7\Rightarrow d\leq 2\) and \(K^2\geq 5\Rightarrow d\leq 4\).NEWLINENEWLINEUnder the assumption \(d=2\), \textit{G. Xiao} [Am. J. Math. 112, No. 5, 713--736 (1990; Zbl 0722.14025)] considered the involution \(i\) induced by the bicanonical map and proved that if \(K^2\geq 5\), the quotient \(S/i\) is a rational surface.NEWLINENEWLINEIn this paper under review the author verifies that if \(K^2=5\), \(d=4\) and the bicanonical map of \(S\) is composed with an involution \(i\), then Xiao's result remains true, i.e. \(S/i\) is rational. The same happens for the case \(K^2=6\), from [\textit{M. Mendes Lopes} and \textit{R. Pardini}, Topology 40, No. 5, 977--991 (2001; Zbl 1072.14522)].