Irreducible plane sextics with large fundamental groups
Alex DEGTYAREV
Source: J. Math. Soc. Japan Volume 61, Number 4
(2009), 1131-1169.
Abstract
We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1257520503
Digital Object Identifier: doi:10.2969/jmsj/06141131
Zentralblatt MATH identifier: 05651147
Mathematical Reviews number (MathSciNet): MR2588507
References
V. I. Arnol'd, A. N. Varchenko and S. M. Guseĭn-Zade, Singularities of differentiable maps, I, The classification of critical points, caustics and wave fronts, Nauka, Moscow, 1982 (Russian); English translation: Monographs in Mathematics, 82, Birkhäuser Boston Inc., Boston, MA, 1985.
Mathematical Reviews (MathSciNet): MR777682
J. I. Cogolludo, Fundamental group for some cuspidal curves, Bull. London Math. Soc., 31 (1999), 136–142.
Mathematical Reviews (MathSciNet): MR1664168
Zentralblatt MATH: 0928.14021
Digital Object Identifier: doi:10.1112/S0024609398005323
A. Degtyarev, Isotopy classification of complex plane projective curves of degree 5, Algebra i Analis, 1 (1989), 78–101 (Russian); English translation in Leningrad Math. J., 1 (1990), 881–904.
Mathematical Reviews (MathSciNet): MR1027461
A. Degtyarev, Alexander polynomial of a curve of degree six, J. Knot Theory Ramifications, 3 (1994), 439–454.
Mathematical Reviews (MathSciNet): MR1304394
Zentralblatt MATH: 0848.57006
Digital Object Identifier: doi:10.1142/S0218216594000320
A. Degtyarev, Quintics in $\mbi{C}\mathrm{p}^{2}$ with nonabelian fundamental group, Algebra i Analis, 11 (1999), 130–151; (Russian); English translation in Leningrad Math. J., 11 (2000), 809–826.
Mathematical Reviews (MathSciNet): MR1734350
A. Degtyarev, On deformations of singular plane sextics, J. Algebraic Geom., 17 (2008), 101–135.
Mathematical Reviews (MathSciNet): MR2357681
Zentralblatt MATH: 1131.14040
A. Degtyarev, Oka's conjecture on irreducible plane sextics, J. London Math. Soc., 78 (2008), 329–351.
Mathematical Reviews (MathSciNet): MR2439628
Zentralblatt MATH: 1158.14026
Digital Object Identifier: doi:10.1112/jlms/jdn029
A. Degtyarev, Oka's conjecture on irreducible plane sextics. II, J. Knot Theory Ramifications., to appear.
arXiv: math.AG/0702546
Mathematical Reviews (MathSciNet): MR2554335
Zentralblatt MATH: 1174.14027
Digital Object Identifier: doi:10.1142/S0218216509007348
A. Degtyarev, Zariski $k$-plets via dessins d'enfants, Comment. Math. Helv., 84 (2009), 639–671.
Mathematical Reviews (MathSciNet): MR2507257
Digital Object Identifier: doi:10.4171/CMH/176
A. Degtyarev, On irreducible sextics with non-abelian fundamental group, Proceedings of Niigata–Toyama Conferences 2007, Adv. Stud. Pure Math., arXiv:0711.3070, to appear.
Mathematical Reviews (MathSciNet): MR2604077
Zentralblatt MATH: 1193.14037
A. Degtyarev, Fundamental groups of symmetric sextics, J. Math. Kyoto Univ., 48 (2008), 765–792.
Mathematical Reviews (MathSciNet): MR2513586
Zentralblatt MATH: 1172.14020
Project Euclid: euclid.kjm/1250271318
C. Eyral and M. Oka, Fundamental groups of the complements of certain plane non-tame torus sextics, Topology Appl., 153 (2006), 1705–1721.
Mathematical Reviews (MathSciNet): MR2227024
Zentralblatt MATH: 1096.14021
Digital Object Identifier: doi:10.1016/j.topol.2005.06.006
E. R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math., 55 (1933), 255–260.
Mathematical Reviews (MathSciNet): MR1506962
Digital Object Identifier: doi:10.2307/2371128
JSTOR: links.jstor.org
W. Magnus, Braid groups: A survey, Lecture Notes in Math., 372, Springer, 1974, pp. 463–487.
Mathematical Reviews (MathSciNet): MR353290
Zentralblatt MATH: 0286.20039
W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Presentations of groups in terms of generators and relations, Second revised edition, Dover Publications, Inc., New York, 1976.
Mathematical Reviews (MathSciNet): MR422434
A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J., 49 (1982), 833–851.
Mathematical Reviews (MathSciNet): MR683005
Zentralblatt MATH: 0524.14026
Digital Object Identifier: doi:10.1215/S0012-7094-82-04941-9
Project Euclid: euclid.dmj/1077315533
A. Libgober, Alexander modules of plane algebraic curves, Contemporary Math., 20 (1983), 231–247.
Mathematical Reviews (MathSciNet): MR718145
Zentralblatt MATH: 0523.14014
V. V. Nikulin, Integer quadratic forms and some of their geometrical applications, Izv. Akad. Nauk SSSR, Ser. Mat., 43 (1979), 111–177 (Russian); English translation in Math. USSR–Izv., 14 (1980), 103–167.
Mathematical Reviews (MathSciNet): MR525944
M. V. Nori, Zariski conjecture and related problems, Ann. Sci. École Norm. Sup., 4 série, 16 (1983), 305–344.
Mathematical Reviews (MathSciNet): MR732347
Zentralblatt MATH: 0527.14016
M. Oka, Alexander polynomial of sextics, J. Knot Theory Ramifications, 12 (2003), 619–636.
Mathematical Reviews (MathSciNet): MR1999635
Zentralblatt MATH: 1053.32019
Digital Object Identifier: doi:10.1142/S0218216503002676
M. Oka, Zariski pairs on sextics, I, Vietnam J. Math., 33 (2005), Special Issue, 81–92.
Mathematical Reviews (MathSciNet): MR2227705
Zentralblatt MATH: 1117.14030
M. Oka and D. T. Pho, Classification of sextics of torus type, Tokyo J. Math., 25 (2002), 399–433.
Mathematical Reviews (MathSciNet): MR1948673
Zentralblatt MATH: 1062.14036
Digital Object Identifier: doi:10.3836/tjm/1244208862
Project Euclid: euclid.tjm/1244208862
M. Oka and D. T. Pho, Fundamental group of sextics of torus type, Trends in singularities, Trends Math., Birkhäuser, Basel, 2002, pp. 151–180.
Mathematical Reviews (MathSciNet): MR1900785
Zentralblatt MATH: 1054.14022
A. Özgüner, Classical Zariski pairs with nodes, M.Sc. thesis, Bilkent University, 2007.
H. Tokunaga, Irreducible plane curves with the Albanese dimension 2, Proc. Amer. Math. Soc., 127 (1999), 1935–1940.
Mathematical Reviews (MathSciNet): MR1637444
Zentralblatt MATH: 0917.14015
Digital Object Identifier: doi:10.1090/S0002-9939-99-05116-3
JSTOR: links.jstor.org
H. Tokunaga, A note on triple covers of $\mbi{P}^{2}$, to appear.
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.
Mathematical Reviews (MathSciNet): MR1506719
Digital Object Identifier: doi:10.2307/2370712
JSTOR: links.jstor.org
O. Zariski, The topological discriminant group of a Riemann surface of genus $p$, Amer. J. Math., 59 (1937), 335–358.
Mathematical Reviews (MathSciNet): MR1507244
Digital Object Identifier: doi:10.2307/2371416
JSTOR: links.jstor.org
J.-G. Yang, Sextic curves with simple singularities, Tohoku Math. J. (2), 48 (1996), 203–227.
Mathematical Reviews (MathSciNet): MR1387816
Digital Object Identifier: doi:10.2748/tmj/1178225377
Project Euclid: euclid.tmj/1178225377
Journal of the Mathematical Society of Japan
