Skip to main content
SpringerLink
Log in

Surfaces of general type with p g =0, K 2=6 and non birational bicanonical map

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

Let S be a minimal surface of general type with p g =0 and K 2=6, such that its bicanonical map is not birational. The map φ is a morphism of degree ≤ 4 onto a surface. The case of degφ = 4 is completely classified in [Topology, 40 (5) (2001), 977–991] and the present paper completes the characterization of these surfaces. It is proven that the degree of φ cannot be equal to 3, and the geometry of surfaces with degφ = 2 is analysed in detail. The last section contains three examples of such surfaces, two of which appear to be new.

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

Access this article

Log in via an institution

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. Beauville, A.: L’application canonique pour les surfaces de type général. Invent. Math. 55, 121–140 (1979)

    MathSciNet  MATH  Google Scholar 

  2. Beauville, A.: L’inégalité p g ≥ 2q-4 pour les surfaces de type général. Bull. Soc. Math. de France 110, 343–346 (1982)

    MATH  Google Scholar 

  3. Catanese, F.: Singular bidouble covers and the construction of interesting algebraic surfaces. In: Algebraic geometry: Hirzebruch 70, A.M.S. Contemporary Mathematics, 241, 1999, pp. 97–120

  4. Ciliberto, C.: Alcuni aspetti della classificazione delle famiglie di curve in uno spazio proiettivo. Boll. Un. Mat. Ital. A (7) 10(3), 491–535 (1996)

    Google Scholar 

  5. Dolgachev, I., Mendes Lopes, M., Pardini, R.: Rational surfaces with many nodes. Compositio Math. 132, 349–363 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Halanay, E.: Normal surfaces of degree 8 in ℙn. Ann. Univ. Ferrara Sez. VII (N.S.) 41, 131–145 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Inoue, M.: Some new surfaces of general type. Tokyo J. Math. 17, 295–319 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Ionescu, P.: Embedded projective varieties of small invariants. In: Algebraic geometry, Bucharest 1982, Lecture Notes in Math., 1056, Springer, Berlin, 1984, pp. 142–186

  9. Mendes Lopes, M.: Adjoint systems on surfaces. Boll. Un. Mat. Ital. A (7) 10, 169–179 (1996)

    Google Scholar 

  10. Mendes Lopes, M.: The degree of the generators of the canonical ring of surfaces of general type with p g =0. Arch. Math. 69, 435–440 (1997)

    Article  MATH  Google Scholar 

  11. Mendes Lopes, M., Pardini, R.: A connected component of the moduli space of surfaces of general type with p g =0. Topology 40(5), 977–991 (2001)

    Article  MATH  Google Scholar 

  12. Mendes Lopes, M., Pardini, R.: The bicanonical map of surfaces with p g = 0 and K 2 ≥ 7. Bull. London Math. Soc. 33, 265–274 (2001)

    MATH  Google Scholar 

  13. Mendes Lopes, M., Pardini, R.: The bicanonical map of surfaces with p g = 0 and K 2≥7, II. Bull. London Math. Soc. 35, 337–343 (2003)

    Article  MATH  Google Scholar 

  14. Pardini, R.: Abelian covers of algebraic varieties. J. reine angew. Math. 417, 191–213 (1991)

    MathSciNet  MATH  Google Scholar 

  15. Pardini, R.: The classification of double planes of general type with p g = 0 and K 2=8. J. Algebra, 259, 95–118 (2003)

    Google Scholar 

  16. Peters, C.: On certain examples of surfaces with p g = 0 due to Burniat. Nagoya Math. J. 166, 109–119 (1977)

    Google Scholar 

  17. Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127, 309–316 (1988)

    MathSciNet  MATH  Google Scholar 

  18. Xiao, G.: Finitude de l’application bicanonique des surfaces de type général. Bull. Soc. Math. France 113, 23–51 (1985)

    MathSciNet  MATH  Google Scholar 

  19. Xiao, G.: Degree of the bicanonical map of a surface of general type. Am. J. Math. 112(5), 713–737 (1990)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáatica, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Margarida Mendes Lopes

  2. Dipartimento di Matematica, Universitàa di Pisa, Via Buonarroti 2, 56127, Pisa, Italy

    Rita Pardini

Authors
  1. Margarida Mendes Lopes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rita Pardini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Margarida Mendes Lopes.

Additional information

Mathematics Subject Classification (2000): 14J29

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mendes Lopes, M., Pardini, R. Surfaces of general type with p g =0, K 2=6 and non birational bicanonical map. Math. Ann. 329, 535–552 (2004). https://doi.org/10.1007/s00208-004-0524-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-004-0524-3

Keywords

Search

Navigation