Holomorphic foliations by curves on 3 with non-isolated singularities
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 2, pp. 297-321.

Let be a holomorphic foliation by curves on 3 . We treat the case where the set Sing() consists of disjoint regular curves and some isolated points outside of them. In this situation, using Baum-Bott’s formula and Porteuos’theorem, we determine the number of isolated singularities, counted with multiplicities, in terms of the degree of , the multiplicity of along the curves and the degree and genus of the curves.

Soit un feuilletage holomorphe de dimension 1 dans 3 . Nous considérons le cas où l’ensemble Sing() est formé par des courbes lisses et disjointes et quelques points isolés en dehors de ces courbes. Dans cette situation, en employant la formule de Baum-Bott et le théorème de Porteous, nous déterminons le nombre de singularités isolées, comptées avec multiplicités, en fonction du degré de , de la multiplicité de le long des courbes et du degré et du genre des courbes.

DOI: 10.5802/afst.1123
Nonato Costa, Gilcione 1

1 Departamento de Matemática - ICEX - UFMG. Cep 30123-970 - Belo Horizonte, Brazil.
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Nonato Costa, Gilcione. Holomorphic foliations by curves on $\mathbb{P}^3$ with non-isolated singularities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 2, pp. 297-321. doi : 10.5802/afst.1123. http://archive.numdam.org/articles/10.5802/afst.1123/

[1] Baum, P.; Bott, R. On the zeros of meromorphic vector-fields, Essays on Topology and Related topics (Mémoires dédiés à Georges de Rham), Springer-Verlag, Berlin, 1970, pp. 29-47 | MR | Zbl

[2] Bott, R.; Tu, L. W. Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Volume 82, Springer, 1982 | MR | Zbl

[3] Fulton, W. Intersection Theory, Springer-Verlag, Berlin Heidelberg, 1984 | MR | Zbl

[4] Gómez-Mont, X. Holomorphic foliations in ruled surfaces, Trans. American Mathematical Society, Volume 312 (1989), pp. 179-201 | MR | Zbl

[5] Griffiths, P.; Harris, J. Principles of Algebraic Geometry, John Wiley & Sons, Inc., 1994 | MR | Zbl

[6] Hartshorne, R. Algebraic Geometry, Springer-Verlag, New York Inc, 1977 | MR | Zbl

[7] Porteous, I. R. Blowing up Chern class, Proc. Cambridge Phil. Soc., Volume 56 (1960), pp. 118-124 | MR | Zbl

[8] Sancho, F. Number of singularities of a foliation on n , Proceedings of the American Mathematical Society, Volume 130 (2001), pp. 69-72 | MR | Zbl

[9] Suwa, T. Indices of vector fields and residues of singular holomorphic foliation, Hermann, 1998 | MR | Zbl

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