The Orevkov invariant of an affine plane curve

Neumann, Walter; Norbury, Paul

doi:10.1090/S0002-9947-02-03094-5

The Orevkov invariant of an affine plane curve
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by Walter D. Neumann and Paul Norbury PDF

Trans. Amer. Math. Soc. 355 (2003), 519-538 Request permission

Abstract:

We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.

References

Shreeram S. Abhyankar, On the semigroup of a meromorphic curve. I, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 249–414. MR 578864
S. A. Broughton, On the topology of polynomial hypersurfaces, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 167–178. MR 713056
Pierre Deligne, Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après W. Fulton), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 1–10 (French). MR 636513
A. Dimca and A. Nemethi, On the monodromy of complex polynomials, Duke Math. J. 108 (2001), 199-209.
David Eisenbud and Walter Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985. MR 817982
William Fulton, On the fundamental group of the complement of a node curve, Ann. of Math. (2) 111 (1980), no. 2, 407–409. MR 569076, DOI 10.2307/1971204
Shulim Kaliman, Rational polynomials with a $\textbf {C}^*$-fiber, Pacific J. Math. 174 (1996), no. 1, 141–194. MR 1398374, DOI 10.2140/pjm.1996.174.141
A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49 (1982), no. 4, 833–851. MR 683005, DOI 10.1215/S0012-7094-82-04941-9
Walter D. Neumann, Complex algebraic plane curves via their links at infinity, Invent. Math. 98 (1989), no. 3, 445–489. MR 1022302, DOI 10.1007/BF01393832
J.-P. Brasselet, Lê Dũng Tráng, and J. Seade, Euler obstruction and indices of vector fields, Topology 39 (2000), no. 6, 1193–1208. MR 1783853, DOI 10.1016/S0040-9383(99)00009-9
Walter D. Neumann and Paul Norbury, Unfolding polynomial maps at infinity, Math. Ann. 318 (2000), no. 1, 149–180. MR 1785580, DOI 10.1007/s002080000117
Walter D. Neumann and Paul Norbury, Vanishing cycles and monodromy of complex polynomials, Duke Math. J. 101 (2000), no. 3, 487–497. MR 1740685, DOI 10.1215/S0012-7094-00-10134-2
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
Madhav V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 305–344. MR 732347, DOI 10.24033/asens.1450
Mutsuo Oka, Two transforms of plane curves and their fundamental groups, J. Math. Sci. Univ. Tokyo 3 (1996), no. 2, 399–443. MR 1424436
S. Yu. Orevkov, The fundamental group of the complement of a plane algebraic curve, Mat. Sb. (N.S.) 137(179) (1988), no. 2, 260–270, 272 (Russian); English transl., Math. USSR-Sb. 65 (1990), no. 1, 267–277. MR 971697, DOI 10.1070/SM1990v065n01ABEH001310
S. Yu. Orevkov, The commutant of the fundamental group of the complement of a plane algebraic curve, Uspekhi Mat. Nauk 45 (1990), no. 1(271), 183–184 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 1, 221–222. MR 1050939, DOI 10.1070/RM1990v045n01ABEH002331
Peter Russell, Good and bad field generators, J. Math. Kyoto Univ. 17 (1977), no. 2, 319–331. MR 444627, DOI 10.1215/kjm/1250522770
Avinash Sathaye and Jon Stenerson, Plane polynomial curves, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 121–142. MR 1272025
Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1