Geometriae Dedicata

, Volume 147, Issue 1, pp 323–355

Numerical properties of isotrivial fibrations



In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations \({\varphi : X \longrightarrow C}\), where X is a smooth, projective surface and C is a curve. In particular we prove that, if g(C) ≥ 1 and X is neither ruled nor isomorphic to a quasi-bundle, then \({K_X^2 \leq 8 \chi(\mathcal{O}_X)-2}\) ; this inequality is sharp and if equality holds then X is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that KX is ample, we obtain \({K_X^2 \leq 8\chi(\mathcal{O}_X)-5}\) and the inequality is also sharp. This improves previous results of Serrano and Tan.


Isotrivial fibrations Cyclic quotient singularities 

Mathematics Subject Classification (2000)

14J99 14J29 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauer I., Catanese F., Grunewald F.: The classification of surfaces with p g = q = 0 isogenous to a product of curves. Pure Appl. Math. Q. 4(2, part 1), 547–5861 (2008)MATHMathSciNetGoogle Scholar
  2. 2.
    Bauer, I., Catanese, F., Grunewald, F., Pignatelli, R.: Quotient of a product of curves by a finite group and their fundamental groups, e-print arXiv:0809.3420 (2008)Google Scholar
  3. 3.
    Barth W., Peters C., Van de Ven A.: Compact Complex Surfaces. Springer, Berlin (1984)MATHGoogle Scholar
  4. 4.
    Barlow R.: Zero-cycles on Mumford’s surface. Math. Proc. Camb. Philos. Soc. 126, 505–510 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Beauville A.: Complex Algebraic Surfaces. Cambridge University Press, Cambridge (1996)MATHCrossRefGoogle Scholar
  6. 6.
    Breuer T.: Characters and Automorphism Groups of Compact Riemann Surfaces. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  7. 7.
    Broughton S.A.: Classifying finite group actions on surfaces of low genus. J. Pure Appl. Algebra 69, 233–270 (1990)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Catanese F.: Fibred surfaces, varieties isogenous to a product and related moduli spaces. Am. J. Math. 122, 1–44 (2000)MATHMathSciNetGoogle Scholar
  9. 9.
    Carnovale G., Polizzi F.: The classification of surfaces of general type with p g = q = 1 isogenous to a product. Adv. Geom. 9, 233–256 (2009)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Conway J.H., Curtis R.T., Parker R.A., Wilson R.A.: Atlas of Finite Groups. Oxford University Press, Oxford (1985)MATHGoogle Scholar
  11. 11.
    Debarre O.: Inegalités numériques pour les surfaces de type générale. Bull. Soc. Math. de France 110, 319–346 (1982)MATHMathSciNetGoogle Scholar
  12. 12.
    Farkas H.M., Kra I.: Riemann Surfaces, Graduate Texts in Mathematics. Vol. 71, 2nd edn. Springer, Berlin (1992)Google Scholar
  13. 13.
    Freitag E.: Uber die Struktur der Funktionenkörper zu hyperabelschen Gruppen I. J. Reine. Angew. Math. 247, 97–117 (1971)MATHMathSciNetGoogle Scholar
  14. 14.
    The GAP Group. GAP—Groups, Algorithms, and Programming, Version 4.4. (2006)
  15. 15.
    Harvey W.J.: On the branch loci in Teichmüller space. Trans. Am. Math. Soc. 153, 387–399 (1971)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jones G.A., Singerman D.: Complex Functions. Cambridge University Press, Cambridge (1987)MATHGoogle Scholar
  17. 17.
    Laufer H.B.: Normal Two-Dimensional Singularities, Annals of Mathematics Studies. Princeton University Press, Princeton (1971)Google Scholar
  18. 18.
    Mistretta E., Polizzi F.: Standard isotrivial fibrations with p g = q = 1.  II. J. Pure Appl. Algebra 214, 344–369 (2010)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Némethi, A., Popescu-Pampu, P.: On the Milnor fibers of cyclic quotient singularities, e-print arXiv:0805.3449v1 (2008)Google Scholar
  20. 20.
    Ogg A.P.: On pencils of curves of genus 2. Topology 5, 355–362 (1966)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Orlik P., Wagreich P.: Algebraic surfaces with k*-action. Acta Math. 138, 43–81 (1977)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Polizzi F.: Standard isotrivial fibrations with p g = q = 1. J. Algebra 321, 1600–1631 (2009)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Riemenschneider O.: Deformationen von quotientensingularitäten (nach Zyklischen Gruppen). Math. Ann. 209, 211–248 (1974)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Serrano, F.: Fibrations on algebraic surfaces. In: Lanteri, A., Palleschi, M., Struppa, D.C. (eds.) Geometry of Complex Projective Varieties (Cetraro 1990), pp. 291–300. Mediterranean Press (1993)Google Scholar
  25. 25.
    Serrano F.: Elliptic surfaces with an ample divisor of genus two. Pac. J. Math. 152, 187–199 (1992)MATHMathSciNetGoogle Scholar
  26. 26.
    Serrano F.: Isotrivial fibred surfaces. Annali di Matematica pura e applicata CLXXI, 63–81 (1996)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Tan S.L.: On the invariant of base changes of pencils of curves, II. Math. Z. 222, 655–676 (1996)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità della CalabriaArcavacata di Rende (CS)Italy

Personalised recommendations