## Final forms for a three-dimensional vector field under blowing-up.(English)Zbl 0607.58027

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). The final situations generalize the simple singularities for the two dimensional case, i.e. the linear part of the vector field has two distinct eigenvalues $$a\neq b\neq 0$$ with a/b not a strictly positive rational number. In the simple corners, the only integral branches are the two components of the exceptional divisor, and in the other simple points, there is exactly one integral branch not contained in the exceptional divisor. In dimension three we have a similar behaviour, hence a description of the integral branches once a final situation is reached. These final situations are also persistent under permissible blowing-ups, like in the two dimensional case. Technically, we take a logarithmic approach, by marking in each step the exceptional divisor of the transformation and we use numerical invariants in order to control the algorithms for reaching the final forms.

### MSC:

 37G99 Local and nonlocal bifurcation theory for dynamical systems 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory

### Keywords:

vector fields singularities; desingularization; blowing-ups
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### References:

 [1] S. S. ABHYANKAR, Desingularization of plane curves, Proc. Arcata 1981, A.M.S., Vol. 40, part 1, pp. 1-46. · Zbl 0521.14005 [2] CAMACHO-LINS NETO-SAD, Topological invariants and equidesingularization for holomorphic vector fields, J. Diff. Geom., 20 (1984), 143-174. · Zbl 0576.32020 [3] CAMACHO-SAD, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math., 115 (1982), 579-595. · 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). · 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. · Zbl 0122.38603 [12] A. SEIDENBERG, Reduction of the singularities of ady = bdx, Am. J. of Math. (1968), 248-269. · Zbl 0159.33303
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