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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  • 1. Arnaud Beauville, Surfaces algébriques complexes, Société Mathématique de France, Paris, 1978 (French). Avec une sommaire en anglais; Astérisque, No. 54. MR 0485887
  • 2. Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225
  • 3. E. Horikawa, On deformations of quintic surfaces, Invent Math. 31 (1975), 43-85.
  • 4. Eiji Horikawa, Algebraic surfaces of general type with small 𝐶²₁. I, Ann. of Math. (2) 104 (1976), no. 2, 357–387. MR 0424831
    Eiji Horikawa, Algebraic surfaces of general type with small 𝑐²₁. II, Invent. Math. 37 (1976), no. 2, 121–155. MR 0460340
    Eiji Horikawa, Algebraic surfaces of general type with small 𝑐²₁. III, Invent. Math. 47 (1978), no. 3, 209–248. MR 501370, 10.1007/BF01579212
    Eiji Horikawa, Algebraic surfaces of general type with small 𝑐²₁. IV, Invent. Math. 50 (1978/79), no. 2, 103–128. MR 517773, 10.1007/BF01390285
    Eiji Horikawa, Algebraic surfaces of general type with small 𝑐²₁. V, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 745–755 (1982). MR 656051
  • 5. Hirotaka Ishida, Bounds for the relative Euler-Poincaré characteristic of certain hyperelliptic fibrations, Manuscripta Math. 118 (2005), no. 4, 467–483. MR 2190108, 10.1007/s00229-005-0599-5
  • 6. Kazuhiro Konno, Clifford index and the slope of fibered surfaces, J. Algebraic Geom. 8 (1999), no. 2, 207–220. MR 1675150
  • 7. Ulf Persson, Double coverings and surfaces of general type, Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) Lecture Notes in Math., vol. 687, Springer, Berlin, 1978, pp. 168–195. MR 527234
  • 8. Ulf Persson, Chern invariants of surfaces of general type, Compositio Math. 43 (1981), no. 1, 3–58. MR 631426
  • 9. Gang Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), no. 3, 449–466. MR 875340, 10.1007/BF01450841

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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
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.