manuscripta mathematica

, Volume 84, Issue 1, pp 225–244

On the invariants of base changes of pencils of curves, I

  • Sheng-Li Tan
Article
  • 56 Downloads

Abstract

The main purpose of this paper is to prove the nonnegativity of the basic invariants of base changes of a surface fibration, which is conjectured by Xiao Gang. For this purpose we obtain some new inequalities between the invariants of the singularities ofzd=f(x, y).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ar] Arakelov, S. Ju.,Families of algebraic curves with fixed degeneracy, Math. USSR Izv.5 (1971), 1277–1302CrossRefGoogle Scholar
  2. [AW] Artin, M., Winters, G.,Degenerate fibres and stable reduction of curves, Topology10 (1971), 373–383MATHCrossRefMathSciNetGoogle Scholar
  3. [As] Ashikaga, T.,Normal two-dimensional hypersurface triple points and Horikawa type resolution, Tôhoku Math. J.44 (1992), 177–200.MATHMathSciNetGoogle Scholar
  4. [BPV] Barth, W., Peters, C., Van de Ven, A.,Compact Complex Surfaces, Berlin, Heidelberg, New York: Springer, 1984MATHGoogle Scholar
  5. [Be] Beauville, A.,L'inégualité p g ≥2q-4 pour les surfaces de type général, Bull. Sco. Math. France110 (1982), no. 3, 343–346Google Scholar
  6. [DM] Deligne, P., Mumford, D.,The irreducibility of the space of curves of given genus, Publ. IHES36 (1969), 75–109MATHMathSciNetGoogle Scholar
  7. [Du] Durfee, A. H.,The signature of smoothings of complex surface singularities, Math. Ann.232 (1978), 85–98MATHCrossRefMathSciNetGoogle Scholar
  8. [Ha] Hartshorne, R.,Algebraic Geometry, GTM 52, Springer-Verlag, 1977Google Scholar
  9. [Ho] Horikawa, E.,On deformations of quintic surfaces, Inv. Math.31 (1975), 43–85MATHCrossRefGoogle Scholar
  10. [HR] Hauser, H., Randell, R.,Report on the problem session, Singularities, (R. Randell, eds.), Contemp. Math. vol.90 (1989), pp. 119–134MathSciNetGoogle Scholar
  11. [La] Laufer, H. B.,On μ for surface singularities, Several Complex Variables, Part I (Wells, R. O., eds.), Proc. of Symposia in Pure Math., vol.30 Providence, Rhode Island: Amer. Math. Soc. (1977), pp. 45–49Google Scholar
  12. [Mi] Milnor, J.,Singular points of complex hypersurfaces, Ann. Math. Studies, vol.61, Princeton University Press, Princeton, N. J., 1968MATHGoogle Scholar
  13. [Pa] Parshin, A. N.,Algebraic curves over function fields I, Math. USSR Izv.2 (1968), 1145–1170MATHCrossRefGoogle Scholar
  14. [To] Tomari, M.,The inequality 8p g <μ for hypersurface two-dimensional isolated double points, Math. Nachr.164 (1993), 37–48MATHMathSciNetGoogle Scholar
  15. [X1] Xiao, G.,Problem list, In: Birational geometry of algebraic varieties: open problems. The 23rd International Symposium of the Taniguchi Foundation, (1988), pp. 36–41Google Scholar
  16. [X2] Xiao, G.,On the stable reduction of pencils of curves, Math. Z.203 (1990), 379–389MATHMathSciNetGoogle Scholar
  17. [X3] Xiao, G.,The fibrations of algebraic surfaces, Shanghai Scientific & Technical Publishers, 1992. (Chinese)Google Scholar
  18. [XY] Xu, Y.-J., Yau, S. S.-T.,A sharp estimate of the number of integral points in a tetrahedron, J. reine angew. Math.423 (1992), 199–219MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Sheng-Li Tan
    • 1
  1. 1.Department of MathematicsEast China Normal UniversityShanghaiP. R. of China

Personalised recommendations