Publication: Effective invariants of braid monodromy
Loading...
Full text at PDC
Publication Date
2007
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
Abstract
In this paper we construct new invariants of algebraic curves based on (not necessarily generic) braid monodromies. Such invariants are effective in the sense that their computation allows for the study of Zariski pairs of plane curves. Moreover, the Zariski pairs found in this work correspond to curves having conjugate equations in a number field, and hence are not distinguishable
by means of computing algebraic coverings. We prove that the embeddings of the curves in the plane are not homeomorphic. We also apply these results to the classification problem of elliptic surfaces.
Description
UCM subjects
Unesco subjects
Keywords
Citation
H. Abelson, Topologically distinct conjugate varieties with finite fundamental group, Topology 13 (1974), 161–176.
E. Artal, J. Carmona, and J.I. Cogolludo, On sextic curves with big Milnor number, Trends in Singularities (A.Libgober and M. Tibar, eds.), Trends in Mathematics, Birkhauser Verlag Basel/Switzerland, 2002, pp. 1–29.
Braid monodromy and topology of plane curves, Duke Math. J.118 (2003), no. 2,261–278.
E. Artal, J. Carmona, J.I. Cogolludo, and H. Tokunaga, Sextics with singular points in special position, J. Knot Theory Ramifications 10 (2001), no. 4, 547–578.
J. S. Birman, Braids, links, and mapping class groups,Princeton University Press, Princeton,N.J., 1974, Annals of Mathematics Studies, No. 82.
E. Brieskorn, Automorphic sets and braids and singularities, Braids (Santa Cruz, CA, 1986),Amer. Math. Soc., Providence, RI, 1988, pp. 45–115.
J. Carmona, Monodromıa de trenzas de curvas algebraicas planas, Ph.D. thesis, Universidad de Zaragoza, 2003.
A.I. Degtyar¨ev, Isotopic classification of complex plane projective curves of degree 5,Leningrad Math. J. 1 (1990), no. 4, 881–904.
M. Fukae, Monodromies of rational elliptic surfaces and extremal elliptic K3 surfaces,Preprint available at arXiv:math.AG/0205062.
The GAP Group, Aachen, St Andrews, GAP – Groups,Algorithms, and Programming,Version 4.2, 2000, (http://www-gap.dcs.st-and.ac.uk/~gap).
V. Kharlamov and V. Kulikov, Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curves, C. R. Acad. Sci. Paris S´er. I Math. 333 (2001),no. 9, 855–859.
A. Libgober, Invariants of plane algebraic curves via representations of the braid groups,Invent. Math. 95 (1989), no. 1, 25–30.
Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), Kluwer Acad. Publ., Dordrecht,2001,pp. 215–254.
R. Miranda and U. Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986),537–558. MR0867347 (88a:14044)
U. Persson,Double sextics and singular K-3 surfaces,Algebraic geometry, Sitges (Barcelona),1983,Lecture Notes in Math., vol. 1124,Springer, Berlin, 1985,pp. 262–328.
J. P. Serre, Exemples de varietes projectives conjugu´ees non homeomorphes, C. R. Acad. Sci.Paris Ser. I Math. 258 (1964), 4194–4196.
I. Shimada and D.-Q. Zhang, Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces, Nagoya Math. J. 161 (2001), 23–54.
J.-G. Yang, Sextic curves with simple singularities, Tohoku Math. J. (2) 48 (1996), no. 2,203–227.