Available in electronic format
Available in print format
Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
     

The Orevkov invariant of an affine plane curve

Author(s): Walter D. Neumann; Paul Norbury
Journal: Trans. Amer. Math. Soc. 355 (2003), 519-538.
MSC (2000): Primary 14H30, 14R10, 57M25
Posted: October 1, 2002
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

1.
S. Abhyankar, On the semigroup of a meromorphic curve, Proc. Internat. Sympos. Algebraic Geometry, Kyoto (1977), 249-414. MR 83h:14020

2.
S. A. Broughton, On the topology of polynomial hypersurfaces, Proc. Amer. Math. Soc. Sympos. Pure Math. 40 (1983), 167-178. MR 85d:14033

3.
Pierre Deligne, Le groupe du complement d'une courbe plane n'ayant que des points ordinaires est abelien (d'apres W. Fulton), Seminaire Bourbaki, Lecture Notes in Math. 842, Springer Verlag (1981), 1-10. MR 83f:14026

4.
A. Dimca and A. Nemethi, On the monodromy of complex polynomials, Duke Math. J. 108 (2001), 199-209.

5.
D. Eisenbud and W. D. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Ann. Math. Stud. 110, Princeton Univ. Press (1985). MR 87g:57007

6.
W. Fulton, On the fundamental group of the complement of a node curve, Ann. of Math. 111 (1980), 407-409. MR 82e:14035

7.
S. Kaliman, Rational polynomials with a ${\mathbb C}^*$-fiber, Pacific J. Math. 174 (1996), 141-194. MR 97h:14026
8.
A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49 (1982), 833-851. MR 84g:14030

9.
W. D. Neumann, Complex algebraic curves via their links at infinity, Invent. Math. 3 (1989), 445-489. MR 91c:57014

10.
W. D. Neumann, Irregular links at infinity of complex affine plane curves, Quarterly J. Math. 50 (1999), 301-320. MR 2001i:32047

11.
W. D. Neumann and P. Norbury, Vanishing cycles and monodromy of complex polynomials, Duke Math. J. 101 (2000), 487-497. MR 2002d:32048

12.
W. D. Neumann and P. Norbury, Unfolding polynomial maps at infinity, Math. Ann. 318 (2000), 149-180. MR 2001j:32028

13.
W. D. Neumann and L. Rudolph, Unfoldings in knot theory, Math. Ann. 278 (1987), 409-439 and Corrigendum 282 (1988), 349-351. MR 89j:57017a,b

14.
Madhav V. Nori, Zariski's conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), 305-344. MR 86d:14027

15.
Mutsuo Oka, Two transforms of plane curves and their fundamental groups, J. Math. Sci. Univ. Tokyo 3 (1996), 399-433. MR 97j:14030

16.
S. Yu. Orevkov, The fundamental group of the complement of a plane algebraic curve, Mat. Sb. 137 (179) (1988), 260-270; English transl., Math. USSR Sb. 65 (1990), 267-277. MR 90e:14028

17.
S. Yu. Orevkov, The commutant of the fundamental group of the complement of a plane algebraic curve, Russian Math. Surveys 45 (1990), 221-222. MR 91g:14020

18.
P. Russell, Good and bad field generators, J. Math. Kyoto Univ. 17 (1977), 319-331. MR 56:2977

19.
A. Sathaye and J. Stenerson, On plane polynomial curves, Algebraic geometry and its applications, C.L. Bajaj, Ed., Springer (1994), 121-142. MR 95a:14032

20.
J. P. Serre, Algebraic groups and class fields, Graduate Texts in Math. 117, Springer-Verlag (1988). MR 88i:14041

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14H30, 14R10, 57M25

Retrieve articles in all Journals with MSC (2000): 14H30, 14R10, 57M25


Additional Information:

Walter D. Neumann
Affiliation: Department of Mathematics, Barnard College, Columbia University, New York, New York 10027
Email: neumann@math.columbia.edu

Paul Norbury
Affiliation: Department of Pure Mathematics, Adelaide University, Adelaide, Australia 5005
Address at time of publication: Department of Mathematics, Melbourne University, Parkville, Australia, 3052
Email: pnorbury@maths.adelaide.edu.au

DOI: 10.1090/S0002-9947-02-03094-5
PII: S 0002-9947(02)03094-5
Received by editor(s): November 17, 2001
Posted: October 1, 2002
Additional Notes: Supported under NSF grant no. DMS-0083097
Copyright of article: Copyright 2002, American Mathematical Society


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google