Dihedral coverings of algebraic surfaces and their application

Tokunaga, Hiro-o

doi:10.1090/S0002-9947-00-02524-1

Dihedral coverings of algebraic surfaces and their application
HTML articles powered by AMS MathViewer

by Hiro-o Tokunaga PDF

Trans. Amer. Math. Soc. 352 (2000), 4007-4017 Request permission

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

Enrique Artal-Bartolo, Sur les couples de Zariski, J. Algebraic Geom. 3 (1994), no. 2, 223–247 (French). MR 1257321
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, DOI 10.1007/978-3-642-96754-2
Egbert Brieskorn, Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen, Math. Ann. 166 (1966), 76–102 (German). MR 206973, DOI 10.1007/BF01361440
Egbert Brieskorn, Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann. 178 (1968), 255–270 (German). MR 233819, DOI 10.1007/BF01352140
F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), no. 2, 483–515. MR 755236
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, DOI 10.1007/978-3-322-96879-1
Eiji Horikawa, On deformations of quintic surfaces, Proc. Japan Acad. 49 (1973), 377–379. MR 330173
Rick Miranda and Ulf Persson, Configurations of $\textrm {I}_n$ fibers on elliptic $K3$ surfaces, Math. Z. 201 (1989), no. 3, 339–361. MR 999732, DOI 10.1007/BF01214900
V.V. Nikulin: Integral symmetric bilinear forms and some of their applications, Math. USSR Izv. 14 103-167 (1980).
Madhav V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 305–344. MR 732347
M. Oka: Geometry of cuspidal sextics and their dual curves, preprint.
Tetsuji Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240. MR 1081832
Michel Vaquié, Irrégularité des revêtements cycliques des surfaces projectives non singulières, Amer. J. Math. 114 (1992), no. 6, 1187–1199 (French). MR 1198299, DOI 10.2307/2374758
H. Tokunaga: Dihedral coverings of $\mathbb {P} ^{2}$ branched along quintic curves, preprint.
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, DOI 10.2996/kmj/1138043600
Hiro-o Tokunaga, Dihedral coverings branched along maximizing sextics, Math. Ann. 308 (1997), no. 4, 633–648. MR 1464914, DOI 10.1007/s002080050094
Hiro-o Tokunaga, Some examples of Zariski pairs arising from certain elliptic $K3$ surfaces, Math. Z. 227 (1998), no. 3, 465–477. MR 1612673, DOI 10.1007/PL00004386
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
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).