The Fundamental Group and Torsion Group of Beauville Surfaces

  • Ingrid BauerEmail author
  • Fabrizio Catanese
  • Davide Frapporti
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 123)


We give a survey on the fundamental group of surfaces isogenous to a higher product. If the surfaces are regular, e.g. if they are Beauville surfaces, the first homology group is a finite group. We present a MAGMA script which calculates the first homology groups of regular surfaces isogenous to a product.


Beauville surface Fundamental group 


  1. 1.
    I. Bauer, Some new surfaces with \(p_g = q = 0\), in The Fano Conference, Turin University, Torino, pp. 123–142 (2004)Google Scholar
  2. 2.
    I. Bauer, F. Catanese, F. Grunewald, Beauville surfaces without real structures, Geometric Methods in Algebra and Number Theory, volume 235 of Progress in Mathematics (Birkhäuser, Boston, 2005), pp. 1–42CrossRefGoogle Scholar
  3. 3.
    I. Bauer, F. Catanese, F. Grunewald, The classification of surfaces with \(p_g=q=0\) isogenous to a product of curves. Pure Appl. Math. Q. 4(2), 547–586 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    I. Bauer, F. Catanese, F. Grunewald, R. Pignatelli, Quotients of products of curves, new surfaces with \(p_g=0\) and their fundamental groups. Am. J. Math. 134(4), 993–1049 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    I. Bauer, F. Catanese, R. Pignatelli, Surfaces of general type with geometric genus zero: a survey, in Complex and Differential Geometry. Springer Proceedings in Mathematics, vol. 8 (Springer, Berlin, 2011), pp. 1–48Google Scholar
  6. 6.
    I. Bauer, R. Pignatelli, The classification of minimal product-quotient surfaces with \(p_g = 0\). Math. Comput. 81(280), 2389–2418 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    I. Bauer, R. Pignatelli, Product-quotient surfaces: new invariants and algorithms (2013). arXiv:1308.5508
  8. 8.
    W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I. The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997)Google Scholar
  9. 9.
    F. Catanese, Everywhere nonreduced moduli spaces. Invent. Math. 98(2), 293–310 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    F. Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces. Am. J. Math. 122(1), 1–44 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    G. Carnovale, F. Polizzi, The classification of surfaces with \(p_g = q = 1\) isogenous to a product of curves. Adv. Geom. 9(2), 233–256 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    T. Dedieu, F. Perroni, The fundamental group of a quotient of a product of curves. J. Group Theory 15(3), 439–453 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    D. Frapporti, R. Pignatelli, Mixed quasi-étale quotients with arbitrary singularities. Glasg. Math. J. 57(1), 143–165 (2015)Google Scholar
  14. 14.
    D. Frapporti, Mixed quasi-étale surfaces, new surfaces of general type with \(p_g=0\) and their fundamental group. Collect. Math. 64(3), 293–311 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    S. Garion, On Beauville structures for PSL (2,q) (2010). arXiv:1003.2792
  16. 16.
    C. Gleißner, in The classification of regular surfaces isogenous to a product of curves with \(\chi ({\cal {O}}_S) = 2\). ed. by I. Bauer, S. Garion, A. Vdovina. Beauville Surfaces and Groups (Springer, Cham, 2015)Google Scholar
  17. 17.
    E. Mistretta, F. Polizzi, Standard isotrivial fibrations with \(p_g=q=1\) II. J. Pure Appl. Algebra 214(4), 344–369 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    M. Penegini, The classification of isotrivially fibred surfaces with \(p_g=q=2\). Collect. Math. 62(3), 239–274 (2011) (With an appendix by Sönke Rollenske)Google Scholar
  19. 19.
    F. Polizzi, On surfaces of general type with \(p_g=q=1\) isogenous to a product of curves. Commun. Algebra 36(6), 2023–2053 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    F. Polizzi, Standard isotrivial fibrations with \(p_g =q = 1\). J. Algebra 321(6), 1600–1631 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    F. Serrano, Isotrivial fibred surfaces. Ann. Mat. Pura Appl. 171(4), 63–81 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    T.I. Shabalin, Homology of some surfaces with \(p_g = q = 0\) isogenous to a product. Izv. RAN. Ser. Mat. 78(6), 211–221 (2014)Google Scholar
  23. 23.
    F. Zucconi, Surfaces with \(p_g = q = 2\) and an irrational pencil. Can. J. Math. 55(3), 649–672 (2003)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ingrid Bauer
    • 1
    Email author
  • Fabrizio Catanese
    • 1
  • Davide Frapporti
    • 1
  1. 1.Lehrstuhl Mathematik VIIIMathematisches Institut der Universität Bayreuth, NW IIBayreuthGermany

Personalised recommendations