Skip to main content
Log in

Involutions on surfaces withp g =q = 1

  • Published:
Collectanea mathematica Aims and scope Submit manuscript

Abstract

In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general typeS withp g =q = 1 having an involutioni such thatS/i is a nonruled surface and such that the bicanonical map ofS is not composed withi. A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type withp g =q = 1 andK 2 = 7 having birational bicanonical map is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Barth, C. Peters, and A. Van deVen,Compact Complex Surfaces, SpringerVerlag, Berlin, 1984.

    MATH  Google Scholar 

  2. A. Beauville, Surfaces algébriques complexes,Astérisque 54 (1978), 172.

    MathSciNet  Google Scholar 

  3. G. Borrelli, The classification of surfaces of general type with nonbirational bicanonical map,J. Algebraic Geom. 16 (2007), 625–669.

    MATH  MathSciNet  Google Scholar 

  4. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I, The user language,J. Symbolic Comput. 24 (1997), 235–265.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Calabri, C. Ciliberto, and M. Mendes Lopes, Numerical Godeaux surfaces with an involution,Trans. Amer. Math. Soc. 359 (2007), 1605–1632.

    Article  MATH  MathSciNet  Google Scholar 

  6. F. Catanese, On a class of surfaces of general type,Algebraic Surfaces, CIME, Liguori 16 (1981), 269–284.

    Google Scholar 

  7. F. Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces,Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97–120, Contemp. Math.241, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  8. F. Catanese and C. Ciliberto, Surfaces withp g =q = 1,Problems in the theory of surfaces and their classification (Cortona, 1988), 49–79, Sympos. Math.32, Academic Press, London, 1991.

    Google Scholar 

  9. F. Catanese and C. Ciliberto, Symmetric products of elliptic curves and surfaces of general type withp g =q = 1,J. Algebraic Geom. 2 (1993), 389–411.

    MATH  MathSciNet  Google Scholar 

  10. F. Catanese and R. Pignatelli, Fibrations of low genus, I,Ann. Sci. École Norm. Sup. (4) 39 (2006), 1011–1049.

    MATH  MathSciNet  Google Scholar 

  11. C. Ciliberto and M. Mendes Lopes, On surfaces withp g =q = 2 and nonbirational bicanonical maps,Adv. Geom. 2 (2002), 281–300.

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Esnault and E. Viehweg,Lectures on Vanishing Theorems, DMV Seminar,20 Birkhäuser Verlag, Basel, 1992.

    MATH  Google Scholar 

  13. Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants,Math. Ann. 268 (1984), 159–171.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Pardini, Abelian covers of algebraic varieties,J. Reine Angew. Math. 417 (1991), 191–213.

    MATH  MathSciNet  Google Scholar 

  15. R. Pignatelli, Some (big) irreducible components of the moduli space of minimal surfaces of general type withp g =q = 1 andK 2 = 4,Rend. Lincei Mat. Appl., to appear.

  16. F. Polizzi, Surfaces of general type withp g =q = 1,K 2 = 8 and bicanonical map of degree 2,Trans. Amer. Math. Soc. 358 (2006), 759–798.

    Article  MATH  MathSciNet  Google Scholar 

  17. F. Polizzi, On surfaces of general type withp g =q = 1 isogenous to a product of curves,Comm. Algebra 36 (2008), 2023–2053.

    Article  MATH  MathSciNet  Google Scholar 

  18. F. Polizzi, Standard isotrivial fibrations withp g =q = 1,J. Algebra 321 (2009), 1600–1631.

    Article  MATH  MathSciNet  Google Scholar 

  19. C. Rito, On equations of double planes withp g =q = 1,Math. Comput., to appear.

  20. C. Rito, On surfaces withp g =q = 1 and nonruled bicanonial involution,Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 81–102.

    MATH  MathSciNet  Google Scholar 

  21. F. Sakai, Semistable curves on algebraic surfaces and logarithmic pluricanonical maps,Math. Ann. 254 (1980), 89–120.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carlos Rito.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rito, C. Involutions on surfaces withp g =q = 1. Collect. Math. 61, 81–106 (2010). https://doi.org/10.1007/BF03191228

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03191228

Keywords

MSC2000

Navigation