On the slope of hyperelliptic fibrations with positive relative irregularity

Lu, Xin; Zuo, Kang

doi:10.1090/tran6682

On the slope of hyperelliptic fibrations with positive relative irregularity
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Trans. Amer. Math. Soc. 369 (2017), 909-934 Request permission

Abstract:

Let $f: S \to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $\lambda _f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $\lambda _f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $\lambda _f<4$.

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