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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The existence of hyperelliptic fibrations with slope four and high relative Euler-Poincaré characteristic


Author: Hirotaka Ishida
Journal: Proc. Amer. Math. Soc. 139 (2011), 1221-1235
MSC (2010): Primary 14D06; Secondary 14J29
Published electronically: November 4, 2010
MathSciNet review: 2748416
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Abstract: For any relatively minimal hyperelliptic fibration $ f$ with slope four, there exists the inequality with respect to the relative Euler-Poincaré characteristic $ \chi(f)$ of $ f$ and the genus $ g(f)$ of a fiber of $ f$. This inequality restricts the extent of pairs $ (g(f), \chi(f))$ for relatively minimal hyperelliptic fibrations $ f$ with slope four which exist. Hence, for given suitable integers $ g$ and $ z$, we consider the existence of a relatively minimal hyperelliptic fibration $ f$ with $ g(f)=g , \chi(f)=z$ and slope four. The main purpose in this paper, for any positive integer $ g$, is to prove that there exists a relatively minimal hyperelliptic fibration $ f$ with $ g(f)=g, \chi(f)\ge z(g)$ and slope four, where $ z(X)$ is a certain polynomial of degree two.


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Additional Information

Hirotaka Ishida
Affiliation: Ube National College of Technology, 2-14-1 Tokiwadai, Ube 755-8555, Yamaguchi, Japan
Email: ishida@ube-k.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10773-6
PII: S 0002-9939(2010)10773-6
Received by editor(s): February 17, 2009
Received by editor(s) in revised form: October 14, 2009, and April 20, 2010
Published electronically: November 4, 2010
Additional Notes: This research was partly supported by the research grant 19740022 from JSPS
Communicated by: Ted Chinburg
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.