Final forms for a three-dimensional vector field under blowing-up
Cano, Felipe
Annales de l'Institut Fourier, Tome 37 (1987), p. 151-193 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

On étudie les situations finales que l’on peut obtenir pour un champ de vecteurs singulier en éclatant l’espace ambiant avec des centres permis (en dimension trois). Ces situations sont préservées par des éclatements permis et elles ont une structure simple du point de vue des courbes intégrales. Techniquement, on adopte une vision logarithmique, en signalant dans chaque étape le diviseur exceptionnel de la transformation.

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.

@article{AIF_1987__37_2_151_0,
     author = {Cano, Felipe},
     title = {Final forms for a three-dimensional vector field under blowing-up},
     journal = {Annales de l'Institut Fourier},
     volume = {37},
     year = {1987},
     pages = {151-193},
     doi = {10.5802/aif.1091},
     mrnumber = {88j:58105},
     zbl = {0607.58027},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1987__37_2_151_0}
}

Cano, Felipe. Final forms for a three-dimensional vector field under blowing-up. Annales de l'Institut Fourier, Tome 37 (1987) pp. 151-193. doi : 10.5802/aif.1091. http://gdmltest.u-ga.fr/item/AIF_1987__37_2_151_0/

[1] S. S. Abhyankar, Desingularization of plane curves, Proc. Arcata 1981, A.M.S., Vol. 40, part 1, pp. 1-46. | MR 85d:14024 | Zbl 0521.14005

[2] Camacho-Lins Neto-Sad, Topological invariants and equidesingularization for holomorphic vector fields, J. Diff. Geom., 20 (1984), 143-174. | MR 86d:58080 | Zbl 0576.32020

[3] Camacho-Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math., 115 (1982), 579-595. | MR 83m:58062 | Zbl 0503.32007

[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] F. Cano, Desingularization of plane vector fields, Transac. of the A.M.S., Vol. 296, N 1. 83/93 (1986). | MR 87j:14009 | Zbl 0612.14011

[6] F. Cano, Games of desingularization for a three-dimensional field, to appear in Springer Lecture Notes.

[7] F. Cano, Local and global results on the desingularization of three-dimensional vector fields, to appear in Asterisque. | Zbl 0645.14005

[8] F. Cano, Ramas integrales de ciertos campos de vectores, Proc. GMEL. Coimbra, (1985), 4 pp.

[9] V. Cossart, Forme normale pour une fonction en caractéristique positive et dimension trois, to appear in Travaux en Cours, Hermann. | Zbl 0621.14015

[10] D. Cerveau and G. Mattei, Formes holomorphes intégrables singulières, Astérisque, 97 (1982). | Zbl 0545.32006

[11] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math., 79 (1964), 109-326. | MR 33 #7333 | Zbl 0122.38603

[12] A. Seidenberg, Reduction of the singularities of Ady = Bdx, Am. J. of Math. (1968), 248-269. | MR 36 #3762 | Zbl 0159.33303