Abstract
We show that the fundamental group of the complement of any irreducible tame torus sextics in ℙ2 is isomorphic to ℤ2 * ℤ3 except one class. The exceptional class has the configuration of the singularities {C 3,9, 3A2} and the fundamental group is bigger than ℤ2 * ℤ3. In fact, the Alexander polynomial is given by (t 2 −t+1)2. For the proof, we first reduce the assertion to maximal curves and then we compute the fundamental groups for maximal tame torus curves.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Artal and J. Carmona, Zariski pairs, fundamental groups and Alexander polynomials, J. Math. Soc. Japan, 50 (1998), 521–543.
J. I. Cogolludo, Fundamental group for some cuspidal curves, Bull. London Math. Soc., 31 (1999), 136–142.
R.H. Crowell and R.H. Fox, Introduction to Knot Theory, Ginn and Co. (1963).
A. I. Degtyarev, Alexander polynomial of a curve of degree six, J. Knot Theory and its Ramifications, 3 (1994), 439–454.
E. Fadell and J. van Buskirk, The Braid group of E 2 and S 2, Duke Math. J., 29 (1962), 243–258.
A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J., 49 (1982), 833–851.
W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Dover Publications, Inc., New York (1976).
M. Oka, On the fundamental group of the complement of a reducible curve in P 2, J. London Math. Soc., 2 (1976), 239–252.
M. Oka, Geometry of plane curves via toroidal resolution, in Algebraic Geometry and Singularities, ed. by A. Campilo, Progress in Math. 134, 1996, 95–118.
M. Oka, Flex curves and their applications, Geometriae Dedicata, 75 (1999), 67–100.
M. Oka, Geometry of cuspidal sextics and their dual curves, in Advanced Studies in Pure Math. vol. 29, 245–277, 2000, Singularities and arrangements, Sapporo-Tokyo 1998.
D.T. Pho, Classification of singularities on torus curves of type (2, 3), Kodai Math. J., 24 (2001), 259–284.
H. Tokunaga, Galois covers for S4 and A4 and their applications, Preprint 2000.
A. Muhammed Uludag, Groupes fondamentaux d’une famille de courbes rationnelles cuspidales, Doctor thesis at University of Grenoble I (2000).
O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math.,51 (1929), 305–328.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Basel AG
About this chapter
Cite this chapter
Oka, M., Pho, D.T. (2002). Fundamental Group of Sextics of Torus Type. In: Libgober, A., Tibăr, M. (eds) Trends in Singularities. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8161-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8161-6_7
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9461-6
Online ISBN: 978-3-0348-8161-6
eBook Packages: Springer Book Archive