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New canonical triple covers of surfaces


Author: Carlos Rito
Journal: Proc. Amer. Math. Soc. 143 (2015), 4647-4653
MSC (2010): Primary 14J29
DOI: https://doi.org/10.1090/S0002-9939-2015-12599-3
Published electronically: March 31, 2015
MathSciNet review: 3391024
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct a surface of general type with canonical map of degree $ 12$ which factors as a triple cover and a bidouble cover of $ \mathbb{P}^2$. We also show the existence of a smooth surface with $ q=0,$ $ \chi =13$ and $ K^2=9\chi $ such that its canonical map is either of degree $ 3$ onto a surface of general type or of degree $ 9$ onto a rational surface.


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  • [1] W. Barth, A quintic surface with 15 three-divisible cusps, Preprint, Erlangen, 2000.
  • [2] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225 (2004m:14070)
  • [3] Arnaud Beauville, L'application canonique pour les surfaces de type général, Invent. Math. 55 (1979), no. 2, 121-140 (French). MR 553705 (81m:14025), https://doi.org/10.1007/BF01390086
  • [4] Donald I. Cartwright and Tim Steger, Enumeration of the 50 fake projective planes, C. R. Math. Acad. Sci. Paris 348 (2010), no. 1-2, 11-13 (English, with English and French summaries). MR 2586735, https://doi.org/10.1016/j.crma.2009.11.016
  • [5] Fabrizio Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 97-120. MR 1718139 (2000j:14061), https://doi.org/10.1090/conm/241/03630
  • [6] Ciro Ciliberto, Rita Pardini, and Francesca Tovena, Prym varieties and the canonical map of surfaces of general type, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 4, 905-938. MR 1822412 (2001m:14058)
  • [7] Rong Du and Yun Gao, Canonical maps of surfaces defined by abelian covers, Asian J. Math. 18 (2014), no. 2, 219-228. MR 3217634
  • [8] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.6.5, 2013.
  • [9] Jonghae Keum, Quotients of fake projective planes, Geom. Topol. 12 (2008), no. 4, 2497-2515. MR 2443971 (2009g:14042), https://doi.org/10.2140/gt.2008.12.2497
  • [10] Kazuhiro Konno, On the irregularity of special non-canonical surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), no. 4, 671-688. MR 1308962 (95j:14049), https://doi.org/10.2977/prims/1195165794
  • [11] Rick Miranda, Triple covers in algebraic geometry, Amer. J. Math. 107 (1985), no. 5, 1123-1158. MR 805807 (86k:14008), https://doi.org/10.2307/2374349
  • [12] Rita Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191-213. MR 1103912 (92g:14012), https://doi.org/10.1515/crll.1991.417.191
  • [13] Rita Pardini, Canonical images of surfaces, J. Reine Angew. Math. 417 (1991), 215-219. MR 1103913 (92j:14050), https://doi.org/10.1515/crll.1991.417.215
  • [14] Ulf Persson, Double coverings and surfaces of general type, Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) Lecture Notes in Math., vol. 687, Springer, Berlin, 1978, pp. 168-195. MR 527234 (80h:14017)
  • [15] Gopal Prasad and Sai-Kee Yeung, Fake projective planes, Invent. Math. 168 (2007), no. 2, 321-370. MR 2289867 (2008h:14038), https://doi.org/10.1007/s00222-007-0034-5
  • [16] C. Rito, Surfaces with $ p_g=0,$ $ k^2=3$ and 5-torsion, arXiv:1310.4071 [math.AG], 2013.
  • [17] Sheng Li Tan, Galois triple covers of surfaces, Sci. China Ser. A 34 (1991), no. 8, 935-942. MR 1150664 (93e:14021)
  • [18] Sheng Li Tan, Surfaces whose canonical maps are of odd degrees, Math. Ann. 292 (1992), no. 1, 13-29. MR 1141782 (93b:14065), https://doi.org/10.1007/BF01444606
  • [19] Sheng Li Tan, Cusps on some algebraic surfaces and plane curves, Complex Analysis, Complex Geometry and Related Topics - Namba, vol. 60, 2003, pp. 2003, 106-121.
  • [20] G. van der Geer and D. Zagier, The Hilbert modular group for the field $ {\bf Q}(\surd 13)$, Invent. Math. 42 (1977), 93-133. MR 0485704 (58 #5526)
  • [21] Francesco Zucconi, Numerical inequalities for surfaces with canonical map composed with a pencil, Indag. Math. (N.S.) 9 (1998), no. 3, 459-476. MR 1692141 (2000c:14053), https://doi.org/10.1016/S0019-3577(98)80027-X

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Additional Information

Carlos Rito
Affiliation: Universidade de Trás-os-Montes e Alto Douro, UTAD, Quinta de Prados, 5000-801 Vila Real, Portugal, www.utad.pt
Address at time of publication: Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, Apartado 1013, 4169-007 Porto, Portugal, www.fc.up.pt
Email: crito@fc.up.pt

DOI: https://doi.org/10.1090/S0002-9939-2015-12599-3
Received by editor(s): November 4, 2013
Received by editor(s) in revised form: July 31, 2014
Published electronically: March 31, 2015
Additional Notes: The author wishes to thank Margarida Mendes Lopes, Sai-Kee Yeung, Gopal Prasad, Donald Cartwright, Tim Steger and especially Amir Dzambic and Rita Pardini for useful correspondence. The author is a member of the Center for Mathematics of the University of Porto. This research was partially supported by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT–Fundação para a Ciência e a Tecnologia under the projects PEst–C/MAT/UI0144/2013 and PTDC/MAT-GEO/0675/2012.
Communicated by: Lev Borisov
Article copyright: © Copyright 2015 American Mathematical Society

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