Final forms for a three-dimensional vector field under blowing-up

Felipe Cano

Annales de l'institut Fourier (1987)

  • Volume: 37, Issue: 2, page 151-193
  • ISSN: 0373-0956

Abstract

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We study the final situations which may be obtained for a singular vector field by permissible blowing-ups of the ambient space (in dimension three). These situations are preserved by permissible blowing-ups and its structure is simple from the view-point of the integral branches. Technically, we take a logarithmic approach, by marking in each step the exceptional divisor of the transformation.

How to cite

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Cano, Felipe. "Final forms for a three-dimensional vector field under blowing-up." Annales de l'institut Fourier 37.2 (1987): 151-193. <http://eudml.org/doc/74750>.

@article{Cano1987,
abstract = {We study the final situations which may be obtained for a singular vector field by permissible blowing-ups of the ambient space (in dimension three). These situations are preserved by permissible blowing-ups and its structure is simple from the view-point of the integral branches. Technically, we take a logarithmic approach, by marking in each step the exceptional divisor of the transformation.},
author = {Cano, Felipe},
journal = {Annales de l'institut Fourier},
keywords = {vector fields singularities; desingularization; blowing-ups},
language = {eng},
number = {2},
pages = {151-193},
publisher = {Association des Annales de l'Institut Fourier},
title = {Final forms for a three-dimensional vector field under blowing-up},
url = {http://eudml.org/doc/74750},
volume = {37},
year = {1987},
}

TY - JOUR
AU - Cano, Felipe
TI - Final forms for a three-dimensional vector field under blowing-up
JO - Annales de l'institut Fourier
PY - 1987
PB - Association des Annales de l'Institut Fourier
VL - 37
IS - 2
SP - 151
EP - 193
AB - We study the final situations which may be obtained for a singular vector field by permissible blowing-ups of the ambient space (in dimension three). These situations are preserved by permissible blowing-ups and its structure is simple from the view-point of the integral branches. Technically, we take a logarithmic approach, by marking in each step the exceptional divisor of the transformation.
LA - eng
KW - vector fields singularities; desingularization; blowing-ups
UR - http://eudml.org/doc/74750
ER -

References

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  1. [1] S. S. ABHYANKAR, Desingularization of plane curves, Proc. Arcata 1981, A.M.S., Vol. 40, part 1, pp. 1-46. Zbl0521.14005MR85d:14024
  2. [2] CAMACHO-LINS NETO-SAD, Topological invariants and equidesingularization for holomorphic vector fields, J. Diff. Geom., 20 (1984), 143-174. Zbl0576.32020MR86d:58080
  3. [3] CAMACHO-SAD, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math., 115 (1982), 579-595. Zbl0503.32007MR83m:58062
  4. [4] F. CANO, Transformaciones cuadráticas y clasificación de las curvas integrales de un campo de vectores, P. Sec. Mat. Univ. Vall., 6 (1983), 1-24. 
  5. [5] F. CANO, Desingularization of plane vector fields, Transac. of the A.M.S., Vol. 296, N 1. 83/93 (1986). Zbl0612.14011MR87j:14009
  6. [6] F. CANO, Games of desingularization for a three-dimensional field, to appear in Springer Lecture Notes. 
  7. [7] F. CANO, Local and global results on the desingularization of three-dimensional vector fields, to appear in Asterisque. Zbl0645.14005
  8. [8] F. CANO, Ramas integrales de ciertos campos de vectores, Proc. GMEL. Coimbra, (1985), 4 pp. 
  9. [9] V. COSSART, Forme normale pour une fonction en caractéristique positive et dimension trois, to appear in Travaux en Cours, Hermann. Zbl0621.14015
  10. [10] D. CERVEAU and G. MATTEI, Formes holomorphes intégrables singulières, Astérisque, 97 (1982). Zbl0545.32006
  11. [11] H. HIRONAKA, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math., 79 (1964), 109-326. Zbl0122.38603MR33 #7333
  12. [12] A. SEIDENBERG, Reduction of the singularities of Ady = Bdx, Am. J. of Math. (1968), 248-269. Zbl0159.33303MR36 #3762

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