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Note on families of semistable curves over ℙ1 with 4 singular fibers whose Jacobian are non-compact

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Let f: S → ℙ1 be a semistable family of curves of genus g ⩾ 2. We prove that if f admits exactly 5 singular fibers and 4 of them have non-compact Jacobian, then g = 2.

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References

  1. Beauville A. Le nombre de fibres singulieres d un courbe stable sur ℙ1. In: L. Szpiro, ed. Seminaire sur les pinceaux de courbes de genre au moins deux, Asterique, 1981, 86: 97–108

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

    MATH  MathSciNet  Google Scholar 

  3. Harris J, Morrison I. Moduli of Curves. GTM 187. New York: Springer-Verlag, 1998

    Google Scholar 

  4. Hartshorne R. Algebric Geometry. GTM 52. New York: Springer-Verlag, 1987

    Google Scholar 

  5. Kani E. Hurwitz spaces of genus 2 covers of an elliptic curve. Collect Math, 2003, 54: 1–51

    MATH  MathSciNet  Google Scholar 

  6. Kitagawa S, Konno K. Fibred rational surfaces with extremal Mordell-Weil lattices. Math Zeit, 2005, 251: 179–204

    Article  MATH  MathSciNet  Google Scholar 

  7. Kukulies S. On Shimura curves in the Schottky locus. J Algebraic Geom, 2010, 19: 371–397

    MATH  MathSciNet  Google Scholar 

  8. Mendes L M. Adjoint systems on surface. Bollettino UMI, 1995, 7: 169–170

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  10. Tan S L, Tu Y P, Yu F. On semistable families of curves over P1 with a small number of singular curves. Under Preparation

  11. Tan S L, Tu Y P, Zamora A G. On complex surfaces with 5 or 6 semistable singular fibers over ℙ1. Math Zeit, 2005, 249: 427–438

    Article  MATH  MathSciNet  Google Scholar 

  12. Tu Y P. Surfaces of Kodaira dimension zero with six semistable singular fibers over ℙ1. Math Zeit, 2007, 257: 1–5

    Article  MATH  Google Scholar 

  13. Viehweg E, Zuo K. A characterization of certain Shimura curves in the moduli stack of abelian varieties. J Diff Geom, 2004, 66: 233–287

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Fei Yu

  2. Department of Mathematics, East China Normal University, Shanghai, 200062, China

    Fei Yu

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  1. Fei Yu
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Correspondence to Fei Yu.

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Yu, F. Note on families of semistable curves over ℙ1 with 4 singular fibers whose Jacobian are non-compact. Sci. China Math. 53, 1711–1714 (2010). https://doi.org/10.1007/s11425-010-3116-6

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  • DOI: https://doi.org/10.1007/s11425-010-3116-6

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