A lower bound on the Euler-Poincaré characteristic of certain surfaces of general type with a linear pencil of hyperelliptic curves (Q2786452)

Let \(f:S\to C\) be a fibration from a surface \(S\) to a curve \(C\). We denote the genus of a general fiber of \(f\) by \(g(f)\). Let \(K_f\) be the relative canonical bundle \(K_S-f^*K_C\). We define the following numerical invariant with respect to \(f\): NEWLINE\[NEWLINE\chi_f:=\deg f_*K_f=\chi(\mathcal{O}_S)-(g(C)-1)(g(f)-1).NEWLINE\]NEWLINE Then we define \(\displaystyle\lambda(f):=K_f^2/\chi_f\). It has been shown in \textit{G. Xiao}'s paper [Math. Ann. 276, 449--466 (1987; Zbl 0596.14028)] that \(\displaystyle4-4/g(f)\leq\lambda(f)\leq 12\).NEWLINENEWLINEWe say that the fibration \(f\) is hyperelliptic if the general fibers of \(f\) are hyperelliptic curves (curves \(C\) with \(g(C)\geq 2\) and a flat morphism of degree 2 onto \(\mathbb{P}^1\)); \(f\) is \textit{relatively minimal} if \(X\) has no \((-1)\)-curves contained in fibers of \(f\).NEWLINENEWLINELet \(f:S\to C\) be a relatively minimal hyperelliptic fibration from a surface of general type \(S\) to a curve \(C\). Assume that \(f\) is not locally trivial. When \(\lambda(f)=4\) and \(g(f)\geq 4\), the author gives the best possible lower bound for \(\chi_f\) in terms of \(g(f)\) in his previous paper [\textit{H. Ishida}, Manuscr. Math. 118, No. 4, 467--483 (2005; Zbl 1092.14048)]. Now the paper under review gives the best possible lower bound for \(\chi_f\) in the case where \(C=\mathbb{P}^1\) and \(\lambda(f)\leq 4\). We denote that \(\Gamma(g(f))=3g(f)-9\) for \(g(f)\geq 6\) and \(\Gamma(g(f))= [3g(f)/2]\) for \(2\leq g(f)\leq 5\), then the main result of this paper is that \(\chi_f\geq \Gamma(g(f))\). Moreover for every \(g(f)\geq 2\), an example where \(\chi_f=\Gamma(g(f))\) is given. The author also points out that there is no upper bound for \(\chi_f\).