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On the slope of hyperelliptic fibrations with positive relative irregularity


Authors: Xin Lu and Kang Zuo
Journal: Trans. Amer. Math. Soc. 369 (2017), 909-934
MSC (2010): Primary 14D06, 14H10; Secondary 14D99, 14J29
DOI: https://doi.org/10.1090/tran6682
Published electronically: May 2, 2016
MathSciNet review: 3572259
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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$.


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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
Email: zuok@uni-mainz.de

DOI: https://doi.org/10.1090/tran6682
Keywords: Fibrations, slope inequality, relative irregularity
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
Article copyright: © Copyright 2016 American Mathematical Society

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