Number of singular points of an annulus in ℂ 2

Maciej Borodzik[1]; Henryk Zołądek[2]

  • [1] University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warsaw (Poland)
  • [2] Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 4, page 1539-1555
  • ISSN: 0373-0956

Abstract

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Using BMY inequality and a Milnor number bound we prove that any algebraic annulus ℂ * in ℂ 2 with no self-intersections can have at most three cuspidal singularities.

How to cite

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Borodzik, Maciej, and Zołądek, Henryk. "Number of singular points of an annulus in $\mathbb{C}^2$." Annales de l’institut Fourier 61.4 (2011): 1539-1555. <http://eudml.org/doc/219700>.

@article{Borodzik2011,
abstract = {Using BMY inequality and a Milnor number bound we prove that any algebraic annulus $\{\mathbb\{C\}\}^\{*\}$ in $\{\mathbb\{C\}\}^2$ with no self-intersections can have at most three cuspidal singularities.},
affiliation = {University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warsaw (Poland); Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland},
author = {Borodzik, Maciej, Zołądek, Henryk},
journal = {Annales de l’institut Fourier},
keywords = {Annulus; cuspidal singular point; codimension; algebraic curve; annulus; singularity; Milnor number},
language = {eng},
number = {4},
pages = {1539-1555},
publisher = {Association des Annales de l’institut Fourier},
title = {Number of singular points of an annulus in $\mathbb\{C\}^2$},
url = {http://eudml.org/doc/219700},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Borodzik, Maciej
AU - Zołądek, Henryk
TI - Number of singular points of an annulus in $\mathbb{C}^2$
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 4
SP - 1539
EP - 1555
AB - Using BMY inequality and a Milnor number bound we prove that any algebraic annulus ${\mathbb{C}}^{*}$ in ${\mathbb{C}}^2$ with no self-intersections can have at most three cuspidal singularities.
LA - eng
KW - Annulus; cuspidal singular point; codimension; algebraic curve; annulus; singularity; Milnor number
UR - http://eudml.org/doc/219700
ER -

References

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  1. Shreeram S. Abhyankar, Tzuong Tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148-166 Zbl0332.14004MR379502
  2. Maciej Borodzik, Number of singular points of a genus g curve with one place at infinity 
  3. Maciej Borodzik, Henryk Żołądek, Complex algebraic plane curves via the Poincaré-Hopf formula. I. Parametric lines, Pacific J. Math. 229 (2007), 307-338 Zbl1153.14026MR2276513
  4. Maciej Borodzik, Henryk Żołądek, Complex algebraic plane curves via Poincaré-Hopf formula. III. Codimension bounds, J. Math. Kyoto Univ. 48 (2008), 529-570 Zbl1174.14028MR2511050
  5. Maciej Borodzik, Henryk Żołądek, Complex algebraic plane curves via Poincaré-Hopf formula. II. Annuli, Israel J. Math. 175 (2010), 301-347 Zbl1202.14031MR2607548
  6. Pierrette Cassou-Nogues, Mariusz Koras, Peter Russell, Closed embeddings of ℂ * in ℂ 2 . I, J. Algebra 322 (2009), 2950-3002 Zbl1218.14019MR2567406
  7. Hubert Flenner, M. G. Zaĭdenberg, Q -acyclic surfaces and their deformations, Classification of algebraic varieties (L’Aquila, 1992) 162 (1994), 143-208, Amer. Math. Soc., Providence, RI Zbl0838.14027MR1272698
  8. S. Yu. Orevkov, On rational cuspidal curves. I. Sharp estimate for degree via multiplicities, Math. Ann. 324 (2002), 657-673 Zbl1014.14010MR1942244
  9. Masakazu Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace C 2 , J. Math. Soc. Japan 26 (1974), 241-257 Zbl0276.14001MR338423
  10. Isao Wakabayashi, On the logarithmic Kodaira dimension of the complement of a curve in P 2 , Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), 157-162 Zbl0404.14009MR498590
  11. M. G. Zaĭdenberg, V. Ya. Lin, An irreducible, simply connected algebraic curve in ℂ 2 is equivalent to a quasihomogeneous curve, Dokl. Akad. Nauk SSSR 271 (1983), 1048-1052 Zbl0564.14014MR722017
  12. M. G. Zaĭdenberg, V. Ya. Lin, Linear and complex analysis problem book, 1043 (1984), 662-663, Springer-Verlag, Berlin 
  13. M. G. Zaĭdenberg, S. Yu. Orevkov, Some estimates for plane cuspidal curves, (1993) 
  14. M. G. Zaĭdenberg, S. Yu. Orevkov, On the number of singular points of complex plane curves, (1995) 

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