On the slope of hyperelliptic fibrations with positive relative irregularity (Q2833003)

Let \(f: S\to B\) be a fibration of curves of genus \(g\geq 2\), i.e., \(f\) is a proper surjective morphism from a smooth surface \(S\) to a smooth \(B\) with connected fibers, and the general fiber is a smooth curve of genus \(g\). The fibration \(f\) is assumed to be \textit{relatively minimal}, i.e. there is no \((-1)\)-curve contained in the fibers of \(f\). The fibration \(f\) is called \textit{locally trivial} if all its fibers are smooth and isomorphic to each other. The \textit{relative irregularity} \(q_f\) of \(f\) is defined to be \(q_f := q - b\), being \(b\) the genus of \(B\) and \(q\) the irregularity of \(S\).NEWLINENEWLINEIf \(f\) is not locally trivial, the \textit{slope} of \(f\) is defined to be \(\lambda_f:=\frac{K^2_f}{\chi_f}\), where \(K_f =K_S -f^* K_ B\) and \(\chi_f=\deg f_* \omega_{S/B}\). By Noether's formula \(0 <\lambda_f\leq 12\).NEWLINENEWLINEIn [J. Math. Soc. Japan 60, No. 1, 171--192 (2008; Zbl 1135.14016)] \textit{M. Á. Barja} and \textit{L. Stoppino} conjectured that if \(f: S\to B\) is a fibration of genus \(g\geq 2\) which is not locally trivial with \(q_f< g-1\), then \(\lambda_f \geq \frac{4(g-1)}{g-q_f}\).NEWLINENEWLINEIn the paper under review the authors prove the following lower bound for the slope \(\lambda_f\) of a hyperelliptic fibration \(f\), i.e. the general fiber of \(f\) is a hyperelliptic curve. Providing explicit examples they show that the bound is sharp.NEWLINENEWLINELet \(f: S\to B\) be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus \(g\). Then \(q_f\leq \frac{g+1}{2}\) and NEWLINE\[NEWLINE\lambda_f \geq \lambda_{g, q_f} =\begin{cases} 8-\dfrac{4(g+1)}{(q_f+1)(g-q_f)}, \text{ if } q_f\leq \dfrac{g-1}{2}\dfrac{8(g-1)}{g}, \text{ if }g\text{ is even and }\\ q_f=\dfrac{g}{2}; \text{ if }g\text{ is odd and } q_f=\dfrac{g+1}{2}.\end{cases} NEWLINE\]NEWLINENEWLINENEWLINEAs immediate consequence the conjecture of Barja and Stoppino holds in the hyperelliptic case.NEWLINENEWLINEIn [Math. Ann. 276, 449--466 (1987; Zbl 0596.14028)] \textit{G. Xiao} conjectured that ``for any locally non-trivial fibration \(f\) with \(\lambda_f<4\), \(f_*\omega_{S/B}\) is ample''. This conjecture was confirmed to be true by \textit{M. A. Barja} and \textit{F. Zucconi} [J. Math. Soc. Japan 52, No. 3, 633--635 (2000; Zbl 0974.14017)] in the non-hyperelliptic case. As corollary of their theorem, the authors show that Xiao's conjecture is true also if \(f\) is a hyperelliptic fibration.