Complex algebraic plane curves via the poincaré-Hopf formula. I. Parametric lines

Pacific Journal of Mathematics

Volumn 229, Issue 2, 2007, Pages 307-338

  Borodzik, Maciej ^a   Zoładek, Henryk ^a  

^a UNIVERSITY OF WARSAW   (Poland)

Author keywords

Affine algebraic curve; Index of a vector field; Puiseux expansion

Indexed keywords

EID: 51249125093     PISSN: 00308730     EISSN: None     Source Type: Journal    
DOI: 10.2140/pjm.2007.229.307     Document Type: Article

References (31)

1. S. S. Abhyankar and T. T. Moh, " Newton-Puiseux expansion and generalized Tschirnhausen transformation, I, II" , J. Reine Angew. Math. 260 (1973), 47-83 and 261 (1973), 29-54. MR 49 #2724 Zbl 0272.12102
2. S. S. Abhyankar and T. T. Moh, " Embeddings of the line in the plane" , J. Reine Angew. Math. 276 (1975), 148-166. MR 52 #407 Zbl 0332.14004
3. N. A'Campo and M. Oka, " Geometry of plane curves via Tschirnhausen resolution tower" , Osaka J. Math. 33:4 (1996), 1003-1033. MR 98d:14038 Zbl 0904.14014
4. M. Borodzik and H. Żoła{ogonek}dek, " Complex algebraic plane curves via Poincaré-Hopf formula, II: Annuli" , preprint, University of Warsaw, 2005.
5. D. Eisenbud and W. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies 110, Princeton University Press, Princeton, NJ, 1985. MR 87g:57007 Zbl 0628.57002
6. H. Flenner and M. Zaidenberg, " On a class of rational cuspidal plane curves" , Manuscripta Math. 89:4 (1996), 439-459. MR 97e:14041 Zbl 0868.14014
7. G.-M. Greuel, C. Lossen, and E. Shustin, " Castelnuovo function, zero-dimensional schemes and singular plane curves" , J. Algebraic Geom. 9:1 (2000), 663-710. MR 2001g: 14045 Zbl 1037.14009
8. P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. MR 80b:14001 Zbl 0408.14001
9. R. V. Gurjar and M. Miyanishi, " On contractible curves in the complex affine plane" , Tohoku Math. J. (2) 48:3 (1996), 459-469. MR 97h:14052 Zbl 0876.14018
10. *-fiber" , Pacific J. Math. 174:1 (1996), 141-194. MR 97h:14026 Zbl 0868.32010
11. S. Kleiman and R. Piene, " Enumerating singular curves on surfaces" , pp. 209-238 in Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math. 241, Amer. Math. Soc., Providence, RI, 1999. MR 2001c:14079 Zbl 0953.14031
12. P. B. Kronheimer and T. S. Mrowka, " Gauge theory for embedded surfaces. I" , Topology 32:4 (1993), 773-826. MR 94k:57048 Zbl 0799.57007
13. A. Lins Neto, " Algebraic solutions of polynomial differential equations and foliations in dimension two" , pp. 192-232 in Holomorphic dynamics (Mexico, 1986), Lecture Notes in Math. 1345, Springer, Berlin, 1988. MR 90c:58142 Zbl 0677.58036
14. T. Matsuoka and F. Sakai, " The degree of rational cuspidal curves" , Math. Ann. 285:2 (1989), 233-247. MR 91a:14014 Zbl 0661.14023
15. J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies 61, Princeton University Press, Princeton, N.J., 1968. MR 39 #969 Zbl 0184.48405
16. M. Miyanishi, Open algebraic surfaces, CRM Monograph Series 12, American Mathematical Society, Providence, RI, 2001. MR 2002e:14101 Zbl 0964.14030
17. Y. Nakazawa and M. a. Oka, " Smooth plane curves with one place at infinity" , J. Math. Soc. Japan 49:4 (1997), 663-687. MR 98i:14032 Zbl 0903.14007
18. W. D. Neumann, " Complex algebraic plane curves via their links at infinity" , Invent. Math. 98:3 (1989), 445-489. MR 91c:57014 Zbl 0734.57011
19. W. D. Neumann and P. Norbury, " Rational polynomials of simple type" , Pacific J. Math. 204:1 (2002), 177-207. MR 2003f:14010 Zbl 1055.14062
20. S. Y. Orevkov, " On rational cuspidal curves, I: Sharp estimate for degree via multiplicities" , Math. Ann. 324:4 (2002), 657-673. MR 2004b:14048 Zbl 1014.14010
21. L. Rudolph, " Embeddings of the line in the plane" , J. Reine Angew. Math. 337 (1982), 113-118. MR 84h:14019 Zbl 0486.14003
22. L. Rudolph, " Some knot theory of complex plane curves" , Enseign. Math. (2) 29:3-4 (1983), 185-208. New version (2001) posted on arXiv. MR 85m:57016 Zbl 0566.14015 math.GT/0106058
23. A. Sathaye and J. Stenerson, " Plane polynomial curves" pp. 121-142 in Algebraic geometry and its applications (West Lafayette, IN, 1990), edited by C. L. Bajaj, Springer, New York, 1994. MR 95a:14032 Zbl 0809.14022
24. J.-P. Serre, Groupes algébriques et corps de classes, Hermann, Paris, 1959. MR 21 #1973 Zbl 0097.35604
25. 2" , J. Math. Soc. Japan 26 (1974), 241-257. MR 49 #3188 Zbl 0276.14001
26. 2" , Proc. Japan Acad. Ser. A Math. Sci. 54:6 (1978), 157-162. MR 58 #16683 Zbl 0404.14009
27. H. Yoshihara, " Rational curve with one cusp" , Proc. Amer. Math. Soc. 89:1 (1983), 24-26. MR 84j:14037 Zbl 0579.14024
28. H. Yoshihara, " Rational curve with one cusp, II" , Proc. Amer. Math. Soc. 100:3 (1987), 405-406. MR 88g:14029 Zbl 0632.14025
29. 2 is equivalent to a quasihomogeneous curve" , Dokl. Akad. Nauk SSSR 271:5 (1983), 1048-1052. In Russian. MR 85i:14018
30. M. G. Zaidenberg and S. Y. Orevkov, " On rigid rational cuspidal plane curves" , Uspekhi Mat. Nauk 51:1(307) (1996), 149-150. In Russian; translated in Russ. Math. Surv. 51:1 (1996), 179-180. MR 97d:14037 Zbl 0564.14014
31. H. Żoła{ogonek}dek, " A new topological proof of the Abhyankar-Moh theorem" , Math. Z. 244:4 (2003), 689-695. MR 2004j:14069 Zbl 1050.14058