Desingularizations of plane vector fields

Cano, F.

doi:10.1090/S0002-9947-1986-0837799-3

Desingularizations of plane vector fields

Author: F. Cano
Journal: Trans. Amer. Math. Soc. 296 (1986), 83-93
MSC: Primary 14D05; Secondary 14B05, 32B30
DOI: https://doi.org/10.1090/S0002-9947-1986-0837799-3
MathSciNet review: 837799
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Abstract: The singularities of a plane vector field can be reduced under quadratic blowing ups. We describe a control method for the singularities of the vector field which works for ground fields of any characteristic and which has no essential obstruction for generalizing to higher dimensional cases.

[1] F. Cano, Teoria de distribuciones sobre variedades algebraicas, Colecc. de Mon. del Instituto Jorge Juan, C.S.I.C., Madrid, 1983.
[2] Jean Giraud, Forme normale d’une fonction sur une surface de caractéristique positive, Bull. Soc. Math. France 111 (1983), no. 2, 109–124 (French, with English summary). MR 734215
[3] -, Condition de Jung pour les revêtements radiciels de hauteur , Proc. Algebraic Geometry, Tokyo/Kyoto 1982, Lecture Notes in Math., vol. 1016, Springer-Verlag, 1983, pp. 313-333.
[4] H. Hironaka, Desingularization of excellent surfaces, Advanced Science Seminar in Algebraic Geometry (Summer 1967), Mimeographed notes by B. Benett, Bowdoin College.
[5] Heisuke Hironaka, Characteristic polyhedra of singularities, J. Math. Kyoto Univ. 7 (1967), 251–293. MR 225779, https://doi.org/10.1215/kjm/1250524227
[6] A. Seidenberg, Reduction of singularities of the differential equation 𝐴𝑑𝑦=𝐵𝑑𝑥, Amer. J. Math. 90 (1968), 248–269. MR 220710, https://doi.org/10.2307/2373435
[7] Arno van den Essen, Reduction of singularities of the differential equation 𝐴𝑑𝑦=𝐵𝑑𝑥, Équations différentielles et systèmes de Pfaff dans le champ complexe (Sem., Inst. Rech. Math. Avancée, Strasbourg, 1975) Lecture Notes in Math., vol. 712, Springer, Berlin, 1979, pp. 44–59. MR 548142