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 ofz d=f(x, y).
Similar content being viewed by others
References
[Ar] Arakelov, S. Ju.,Families of algebraic curves with fixed degeneracy, Math. USSR Izv.5 (1971), 1277–1302
[AW] Artin, M., Winters, G.,Degenerate fibres and stable reduction of curves, Topology10 (1971), 373–383
[As] Ashikaga, T.,Normal two-dimensional hypersurface triple points and Horikawa type resolution, Tôhoku Math. J.44 (1992), 177–200.
[BPV] Barth, W., Peters, C., Van de Ven, A.,Compact Complex Surfaces, Berlin, Heidelberg, New York: Springer, 1984
[Be] Beauville, A.,L'inégualité p g ≥2q-4 pour les surfaces de type général, Bull. Sco. Math. France110 (1982), no. 3, 343–346
[DM] Deligne, P., Mumford, D.,The irreducibility of the space of curves of given genus, Publ. IHES36 (1969), 75–109
[Du] Durfee, A. H.,The signature of smoothings of complex surface singularities, Math. Ann.232 (1978), 85–98
[Ha] Hartshorne, R.,Algebraic Geometry, GTM 52, Springer-Verlag, 1977
[Ho] Horikawa, E.,On deformations of quintic surfaces, Inv. Math.31 (1975), 43–85
[HR] Hauser, H., Randell, R.,Report on the problem session, Singularities, (R. Randell, eds.), Contemp. Math. vol.90 (1989), pp. 119–134
[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. 45–49
[Mi] Milnor, J.,Singular points of complex hypersurfaces, Ann. Math. Studies, vol.61, Princeton University Press, Princeton, N. J., 1968
[Pa] Parshin, A. N.,Algebraic curves over function fields I, Math. USSR Izv.2 (1968), 1145–1170
[To] Tomari, M.,The inequality 8p g <μ for hypersurface two-dimensional isolated double points, Math. Nachr.164 (1993), 37–48
[X1] Xiao, G.,Problem list, In: Birational geometry of algebraic varieties: open problems. The 23rd International Symposium of the Taniguchi Foundation, (1988), pp. 36–41
[X2] Xiao, G.,On the stable reduction of pencils of curves, Math. Z.203 (1990), 379–389
[X3] Xiao, G.,The fibrations of algebraic surfaces, Shanghai Scientific & Technical Publishers, 1992. (Chinese)
[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), 199–219
Author information
Authors and Affiliations
Additional information
This work is supported by the National Natural Science Foundation of China and by the Science Foundation of the University Doctoral Program of CNEC. This paper is corrected while the author is visiting Max-Planck-Institut für Mathematik in Bonn.
Rights and permissions
About this article
Cite this article
Tan, SL. On the invariants of base changes of pencils of curves, I. Manuscripta Math 84, 225–244 (1994). https://doi.org/10.1007/BF02567455
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02567455