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On complex surfaces with 5 or 6 semistable singular fibers over ℙ1

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References

  1. Arakelov, Ju., S.: Families of algebraic curves with fixed degeneracy. Math. USSR Izvestija 5, 1277–1302 (1971)

    MATH  Google Scholar 

  2. Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Springer Verlag, 1984

  3. Beauville, A.: Le nombre minimum de fibres singulières d’un courbe stable sur ℙ1. In: Séminaire sur les pinceaux de courbes de genre au moins deux, (L. Szpiro, ed.), Astérisque 86, 1981, pp. 97–108

  4. Beauville, A.: Les familles stables de courbes elliptiques sur ℙ1 admettant quatre fibres singulières. C.R. Acad. Sci. Paris 294, 657–660 (1982)

    Google Scholar 

  5. Beauville, A.: L’inéqalité p g ≥ 2q-4 pour les surfaces de type général, Appendice à O. Debarre: Inéqalités numériques pour les surfaces de type général. Bull. Soc. Math. France 110 (3), 319–346 (1982)

    Google Scholar 

  6. Cornalba, M., Harris, J.: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. École Norm. Sup. (4) 21 (3), 455–475 (1988)

    Google Scholar 

  7. Debarre, O.: Inéqalités numériques pour les surfaces de type général. Bull. Soc. Math. France 110 (3), 319–346 (1982)

    MATH  Google Scholar 

  8. Horikawa, E.: Algebraic surfaces of general type with small c21, V. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (3), 745–755 (1981)

    MATH  Google Scholar 

  9. Liu, K.: Geometric height inequalities. Math. Res. Lett. 3 (5), 693–702 (1996)

    MATH  Google Scholar 

  10. Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127, 309–316 (1988)

    MathSciNet  MATH  Google Scholar 

  11. Tan, S.-L.: The minimal number of singular fibers of a semistable curve over ℙ1, J. Algebraic Geom. 4, 591–596 (1995)

    Google Scholar 

  12. Tan, S.-L.: Effective behavior of multiple linear systems. Asian J. Math. 8 (2), 287–304 (2004)

    MATH  Google Scholar 

  13. Viehweg, E.: Positivity of direct image sheaves and applications to families of higher dimensional manifolds. In: School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) ICTP Lect. Notes, 6, Int. Cent. Theoret. Phys., Trieste, 2001, pp. 249–284

  14. Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann. 276 (3), 449–466 (1987)

    MATH  Google Scholar 

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Correspondence to Sheng-Li Tan.

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The first author would like to thank the support of ‘‘DFG-Schwerpunktprogramm Globale Methoden in der Komplexen Geometrie’’ and the DFG-NSFC Chinese-German project ‘‘Komplexe Geometrie’’. He is also supported by the 973 Project Foundation and the Foundation of EMC for Doctoral Program. The third author is partially supported by Conacyt Grant 41459-F.

Acknowledgements. The first author would like to thank Prof. H. Esnault and Prof. E. Viehweg for their hospitality and useful discussions with Viehweg while he is visiting Universtät Essen. The third author would like to thank Prof. A. Beauville for useful comments. We all thank the referee for pointing out a mistake and providing us many valuable suggestions in rewriting the manuscript which make the paper readable.

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Tan, SL., Tu, Y. & Zamora, A. On complex surfaces with 5 or 6 semistable singular fibers over ℙ1. Math. Z. 249, 427–438 (2005). https://doi.org/10.1007/s00209-004-0706-4

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  • DOI: https://doi.org/10.1007/s00209-004-0706-4

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