Abstract
Let S be a Campedelli surface (a minimal surface of general type with p g = 0, K 2 = 2), and \({\pi\colon Y\to S}\) an etale cover of degree 8. We prove that the canonical model \({\overline {Y}}\) of Y is a complete intersection of four quadrics \({\overline {Y}=Q_{1}\cap Q_{2}\cap Q_{3}\cap Q_{4}\subset\mathbb{P}^{6}}\) . As a consequence, Y is the universal cover of S, the covering group G = Gal(Y/S) is the topological fundamental group π 1 S and G cannot be the dihedral group D 4 of order 8.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barlow R.N.: Some new surfaces with p g = 0. Duke Math. J. 51, 889–904 (1984)
Beauville A.: L’application canonique pour les surfaces de type général. Inv. Math. 55, 121–140 (1979)
Beauville, A.: Calabi-Yau threefold with non-abelian fundamental group. In: New Trends in Algebraic Geometry, pp. 13–17. Warwick (1996)
Ciliberto C., Mendes Lopes M., Pardini R.: Surfaces with K 2 < 3χ and finite fundamental group. Math. Res. Lett. 14, 1081–1098 (2007)
Fujita T.: On the structure of polarized varieties with Δ–genus zero. J. Fac. Sci. Univ. Tokyo 22, 103–115 (1975)
Fujita, T.: On the structure of polarized varieties of total deficiency one I, II and III. J. Math. Soc. Japan 32:709–725 (1980), ibid. 33:415–434 (1981), ibid. 36:75–89 (1984)
Fujita, T.: Projective varieties of Δ–genus one. In: “Algebraic and Topological Theories”, to the memory of T. Miyata, pp. 149–175. Kinokuniya (1985)
Konno K.: On the Quadric Hull of a Canonical Surface, Algebraic Geometry, pp. 217–235. de Gruyter, Berlin (2002)
Mendes Lopes M., Pardini R.: On the algebraic fundamental group of surfaces with K 2 ≤ 3χ. J. Differ. Geom. 77, 188–199 (2007)
Mendes Lopes M., Pardini R.: Numerical Campedelli surfaces with fundamental group of order 9. J.E.M.S. 10, 457–476 (2008)
Miyaoka, Y.: On numerical Campedelli surfaces. In: Complex Analysis and Algebraic Geometry, Papers dedicated to K. Kodaira, pp. 113–118. Iwanami Shoten, Tokyo (1977)
Naie D.: Numerical Campedelli surfaces cannot have the symmetric group as the algebraic fundamental group. J. Lond. Math. Soc. 59, 813–827 (1999)
Reid, M.: Surfaces with p g = 0, \({K^{2}_{S}=2}\). Preprint available at http://www.warwick.ac.uk/staff/Miles.Reid/surf/
Reid, M.: π 1 for Surfaces with small K 2. In: Algebraic Geometry. Springer LNM 732, pp. 534–544, Copenhagen (1978)
Reid, M.: Young person’s guide to canonical singularities. In: Algebraic Geometry (Bowdoin, 1985) Proc. Sympos. Pure Math. 46 (Part 1), 345–414 (1987)
Reid, M.: Quadrics through a canonical surface. In: Algebraic Geometry (L’Aquila, 1988), Springer LNM 1417, pp. 191–213, Berlin (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is a member of the Centre for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Lisboa. The second is a member of G.N.S.A.G.A.–I.N.d.A.M.
Rights and permissions
About this article
Cite this article
Mendes Lopes, M., Pardini, R. & Reid, M. Campedelli surfaces with fundamental group of order 8. Geom Dedicata 139, 49–55 (2009). https://doi.org/10.1007/s10711-008-9317-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-008-9317-2