Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Dihedral coverings of algebraic surfaces and their application


Author: Hiro-o Tokunaga
Journal: Trans. Amer. Math. Soc. 352 (2000), 4007-4017
MSC (2000): Primary 14E20; Secondary 14E15
Published electronically: March 15, 2000
MathSciNet review: 1675238
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Abstract:

In this article, we study dihedral coverings of algebraic surfaces branched along curves with at most simple singularities. A criterion for a reduced curve to be the branch locus of some dihedral covering is given. As an application we have the following:

Let $B$ be a reduced plane curve of even degree $d$ having only $a$ nodes and $b$ cusps. If $2a + 6b > 2d^2 - 6d + 6$, then $\pi_1(\mathbf{P}^2 \setminus B)$ is non-abelian.

Note that Nori's result implies that $\pi_1(\mathbf{P}^2 \setminus B)$ is abelian, provided that $2a + 6b < d^2$.


References [Enhancements On Off] (What's this?)

  • [A] Enrique Artal-Bartolo, Sur les couples de Zariski, J. Algebraic Geom. 3 (1994), no. 2, 223–247 (French). MR 1257321
  • [BPV] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574
  • [B1] Egbert Brieskorn, Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen, Math. Ann. 166 (1966), 76–102 (German). MR 0206973
  • [B2] Egbert Brieskorn, Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann. 178 (1968), 255–270 (German). MR 0233819
  • [C] F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), no. 2, 483–515. MR 755236
  • [E] Wolfgang Ebeling, Lattices and codes, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1994. A course partially based on lectures by F. Hirzebruch. MR 1280458
  • [H] Eiji Horikawa, On deformations of quintic surfaces, Proc. Japan Acad. 49 (1973), 377–379. MR 0330173
  • [MP] Rick Miranda and Ulf Persson, Configurations of 𝐼_{𝑛} fibers on elliptic 𝐾3 surfaces, Math. Z. 201 (1989), no. 3, 339–361. MR 999732, 10.1007/BF01214900
  • [Ni] V.V. Nikulin: Integral symmetric bilinear forms and some of their applications, Math. USSR Izv. 14 103-167 (1980).
  • [No] Madhav V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 305–344. MR 732347
  • [O] M. Oka: Geometry of cuspidal sextics and their dual curves, preprint.
  • [S] Tetsuji Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240. MR 1081832
  • [T1] Michel Vaquié, Irrégularité des revêtements cycliques des surfaces projectives non singulières, Amer. J. Math. 114 (1992), no. 6, 1187–1199 (French). MR 1198299, 10.2307/2374758
  • [T2] H. Tokunaga: Dihedral coverings of $\mathbf P^{2}$ branched along quintic curves, preprint.
  • [T3] Hiro-o Tokunaga, A remark on E. Artal-Bartolo’s paper: “On Zariski pairs” [J. Algebraic Geom. 3 (1994), no. 2, 223–247; MR1257321 (94m:14033)], Kodai Math. J. 19 (1996), no. 2, 207–217. MR 1397422, 10.2996/kmj/1138043600
  • [T4] Hiro-o Tokunaga, Dihedral coverings branched along maximizing sextics, Math. Ann. 308 (1997), no. 4, 633–648. MR 1464914, 10.1007/s002080050094
  • [T5] Hiro-o Tokunaga, Some examples of Zariski pairs arising from certain elliptic 𝐾3 surfaces, Math. Z. 227 (1998), no. 3, 465–477. MR 1612673, 10.1007/PL00004386
  • [X] Gang Xiao, Galois covers between 𝐾3 surfaces, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 73–88 (English, with English and French summaries). MR 1385511
  • [Z] O. Zariski: On the problem of existence of algebraic functions of two variables possessing a given branch curves, Amer. J. Math. 51, 305-328 (1929).

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Additional Information

Hiro-o Tokunaga
Affiliation: Department of Mathematics and Information Science, Kochi University, Kochi 780-8520, Japan
Address at time of publication: Department of Mathematics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397 Japan
Email: tokunagamath.kochi-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02524-1
Received by editor(s): June 20, 1998
Published electronically: March 15, 2000
Additional Notes: This research is partly supported by the Grant-in-Aid for Encouragement of Young Scientists 09740031 from the Ministry of Education, Science and Culture.
Article copyright: © Copyright 2000 American Mathematical Society