Numerical inequalities for surfaces with canonical map composed with a pencil
\textit{K. Konno} [Publ. Res. Inst. Math. Sci., Kyoto Univ. 30, No. 4, 671-688 (1994; Zbl 0841.14032)] proved the following result: If \(X\) is a minimal surface with canonical map composed with a pencil, then \(K^2_X\geq 4p_g-7\) and pointed out that the cases with \(K^2_X=4p_g-7\) can occur only for \(p_g\leq 4\). Of the three remaining cases \(p_g=2,3,4\), the case \(p_g=2\) is well known, it is the case of the surface with the quadricanonical map not birational [\textit{F. Enriques} and \textit{E. Bombieri}]. In the paper under review, the author proves that the remaining two cases \(p_g=3,4\) cannot occur. The case \(p_g=4\) was proposed as an open problem also by \textit{G. Xiao} [`Problem list' in: The 23rd Int. T. Taniguchi Symp. Katata, 36-41 (1988)].
scientific article; zbMATH DE number 1251206
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