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

Ishida, Hirotaka

doi:10.1090/S0002-9939-2010-10773-6

The existence of hyperelliptic fibrations with slope four and high relative Euler-Poincaré characteristic
HTML articles powered by AMS MathViewer

by Hirotaka Ishida PDF

Proc. Amer. Math. Soc. 139 (2011), 1221-1235 Request permission

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.

References

Arnaud Beauville, Surfaces algébriques complexes, Astérisque, No. 54, Société Mathématique de France, Paris, 1978 (French). Avec une sommaire en anglais. MR 0485887
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, DOI 10.1007/978-3-642-57739-0
E. Horikawa, On deformations of quintic surfaces, Invent Math. 31 (1975), 43–85.
Eiji Horikawa, Algebraic surfaces of general type with small $C^{2}_{1}.$ I, Ann. of Math. (2) 104 (1976), no. 2, 357–387. MR 424831, DOI 10.2307/1971050
Hirotaka Ishida, Bounds for the relative Euler-Poincaré characteristic of certain hyperelliptic fibrations, Manuscripta Math. 118 (2005), no. 4, 467–483. MR 2190108, DOI 10.1007/s00229-005-0599-5
Kazuhiro Konno, Clifford index and the slope of fibered surfaces, J. Algebraic Geom. 8 (1999), no. 2, 207–220. MR 1675150
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
Ulf Persson, Chern invariants of surfaces of general type, Compositio Math. 43 (1981), no. 1, 3–58. MR 631426
Gang Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), no. 3, 449–466. MR 875340, DOI 10.1007/BF01450841