## On the slope of hyperelliptic fibrations with positive relative irregularity

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## 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 $\lambda _f<4$.## References

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## Additional Information

**Xin Lu**- Affiliation: Department of Mathematics, East China Normal University, Shanghai 200241, People’s Republic of China
- Address at time of publication: Institut für Mathematik, Universität Mainz, 55099 Mainz, Germany
- Email: lvxinwillv@gmail.com
**Kang Zuo**- Affiliation: Institut für Mathematik, Universität Mainz, 55099 Mainz, Germany
- MR Author ID: 269893
- Email: zuok@uni-mainz.de
- Received by editor(s): March 9, 2014
- Received by editor(s) in revised form: December 7, 2014, January 19, 2015, and January 30, 2015
- Published electronically: May 2, 2016
- Additional Notes: This work was supported by SFB/Transregio 45 Periods, Moduli Spaces and Arithmetic of Algebraic Varieties of the DFG (Deutsche Forschungsgemeinschaft), partially supported by National Key Basic Research Program of China (Grant No. 2013CB834202), and also supported by NSFC
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 909-934 - MSC (2010): Primary 14D06, 14H10; Secondary 14D99, 14J29
- DOI: https://doi.org/10.1090/tran6682
- MathSciNet review: 3572259