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On the picard group of certain smooth surfaces in weighted projective spaces

Part of the Lecture Notes in Mathematics book series (LNM,volume 961)

Abstract

We consider a general member of a Lefschetz pencil of surfaces in weighted projective 3-spaces of type (1,1,a,b) where gad(a,b)=1. We show that such a surface either has Picard number equal to 1 or all of its 2-cohomolgy is algebraic.

Keywords

  • Singular Point
  • Toric Variety
  • Hodge Structure
  • Singular Locus
  • Picard Group

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References

  1. N. A'Campo, La fonction zêta d'une monodromie. Comment. Math. Helv. 50, 233–248 (1975).

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. V.I. Danilov, Newton polyhedron and vanishing cohomology. Functional analysis and its appl. 13, 32–47 (1979).

    CrossRef  MathSciNet  Google Scholar 

  3. P. Deligne, Théorie de Hodge II. Publ. Math. IHES 40, 5–57 (1971)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. P. Deligne & N. Katz, Groupes de monodromie en géométrie algébrique. SGA 7 II. Lecture Notes in Math. 340, Springer 1973.

    Google Scholar 

  5. P. Deligne, Proof of Zariski's conjecture, Sem. Bourbaki 1979'80.

    Google Scholar 

  6. C. Delorme, Espaces projectifs anisotropes. Bull. Soc. Math. France 103, 203–223 (1975).

    MathSciNet  MATH  Google Scholar 

  7. I. Dolgachev, Weighted projective varieties. Mimeographed notes. Moscow State University 1975/1976.

    Google Scholar 

  8. H.A. Hamm & Lê Dung Trang, Un théorème de Zariski du type de Lefschetz. Ann. Sc. ENS 6, 317–366 (1973).

    MATH  Google Scholar 

  9. R. Hartshorne, Equivalence relations on algebraic cycles. In: Algebraic Geometry, Arcata 1974. Proc. AMS Symp. Pure Math. Vol. XXIX, 129–164 (1975).

    CrossRef  MATH  Google Scholar 

  10. E.R. van Kampen, On the fundamental group of an algebraic curve. Amer. J. of Math. 55, 255–260 (1933).

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1982 Springer-Verlag

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Steenbrink, J. (1982). On the picard group of certain smooth surfaces in weighted projective spaces. In: Aroca, J.M., Buchweitz, R., Giusti, M., Merle, M. (eds) Algebraic Geometry. Lecture Notes in Mathematics, vol 961. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071290

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  • DOI: https://doi.org/10.1007/BFb0071290

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11969-2

  • Online ISBN: 978-3-540-39367-2

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