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On the slope of bielliptic fibrations

Author: Miguel A. Barja
Journal: Proc. Amer. Math. Soc. 129 (2001), 1899-1906
MSC (2000): Primary 14H10; Secondary 14J29
Published electronically: December 4, 2000
MathSciNet review: 1825895
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Abstract: Let $\pi :S\longrightarrow B$ be a bielliptic fibration. We prove $S$ is, up to base change, a rational double cover of an elliptic fibration and that $\pi $ is isotrivial provided it is smooth. Finally, we prove that the slope of $\pi $ is at least four provided the genus of the fibre is at least six.

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  • [1] E. Arbarello, M. Cornalba., P.A. Griffiths, J. Harris. Geometry of algebraic curves, vol I. Grund. Math. Wiss. 267. Springer-Verlag 1985. MR 86h:14019
  • [2] M.A. Barja, J.C. Naranjo. Extension of maps defined on many fibres. Collect. Math. 49, 2-3 (1998), 227-238. MR 2000e:14034
  • [3] W. Barth, C. Peters, A. Van de Ven. Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 4. Springer-Verlag (1984). MR 86c:32026
  • [4] Z. Chen. On the bound of the slope of a non-hyperelliptic fibration of genus 4. Intern. J. Math., vol 4, No.3 (1993), 367-378.
  • [5] R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics 52. Springer-Verlag (1977). MR 57:3116
  • [6] E. Horikawa. On deformation of Quintic Surfaces. Inventiones math. 31 (1975), 43-85.
  • [7] E. Horikawa. Notes on canonical surfaces. Tohoku Math. J. I 43 (1991), 141-148. MR 92e:14034
  • [8] E. Horikawa. Algebraic surfaces of general type with small $c^{2}_{1}$, V. J. Fac. Sci. Univ. Tokyo. Sect. A 28 (1981), 745-755. MR 84d:14019
  • [9] K. Konno. Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces. Ann. Sc. Norm. Pisa Ser. IV vol XX (1993), 575-595. MR 95b:14026
  • [10] K. Konno. A lower bound of the slope of trigonal fibrations. Intern. J. Math., vol 7, No.1 (1996), 19-27. MR 97d:14035
  • [11] K. Konno. Clifford index and the slope of fibred surfaces. J. Algebraic Geometry 8 (1999), 207-220. MR 2000e:14060
  • [12] S. Matsusaka. Some numerical invariants of hyperelliptic fibrations. J. Math. Kyoto Univ. 30-1 (1990), 33-57. MR 91a:14016
  • [13] U. Persson. Chern invariants of surfaces of general type. Compositio Math. 43 (1982), 3-58. MR 83b:14012
  • [14] M. Reid. Problems on pencils of small genus. Preprint.
  • [15] I.R. Shafarevich, et al. Algebraic Surfaces. Proc. Steklov Inst. Math. 75 (1965), AMS Translations, Providence R.I. (1967). MR 32:7557
  • [16] M. Teixidor. On translation invariance for $W^{r}_{d}$. J. reine angew. Math. 385 (1988), 10-23. MR 89a:14035
  • [17] G. Xiao. Surfaces fibrés en courbes de genre deux. Lect. Notes Math. 1137. Springer-Verlag 1985. MR 88a:14042
  • [18] G. Xiao. The fibrations of algebraic surfaces. Shangai Scientific & Technical Publishers 1992 (Chinese).
  • [19] G. Xiao. Fibered algebraic surfaces with low slope. Math. Ann. 276 (1987), 449-466. MR 88a:14046

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Additional Information

Miguel A. Barja
Affiliation: Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

Received by editor(s): December 12, 1997
Received by editor(s) in revised form: October 29, 1999
Published electronically: December 4, 2000
Additional Notes: Partially supported by CICYT PS93-0790 and HCM project n.ERBCHRXCT-940557.
Dedicated: A la memoria de Fernando
Communicated by: Ron Donagi
Article copyright: © Copyright 2000 American Mathematical Society

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