Holomorphic foliations by curves on $\mathbb{P}^3$ with non-isolated singularities

Nonato Costa, Gilcione

doi:10.5802/afst.1123

Holomorphic foliations by curves on

ℙ^{3}

with non-isolated singularities

Nonato Costa, Gilcione ¹

¹ Departamento de Matemática - ICEX - UFMG. Cep 30123-970 - Belo Horizonte, Brazil.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 2, pp. 297-321.

Résumé
Abstract

Soit $ℱ$ un feuilletage holomorphe de dimension $1$ dans $ℙ^{3}$ . Nous considérons le cas où l’ensemble $Sing (ℱ)$ 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 $ℱ$ , de la multiplicité de $ℱ$ le long des courbes et du degré et du genre des courbes.

Let $ℱ$ be a holomorphic foliation by curves on $ℙ^{3}$ . We treat the case where the set $Sing (ℱ)$ 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 $ℱ$ , the multiplicity of $ℱ$ along the curves and the degree and genus of the curves.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/afst.1123

Affiliations des auteurs :

Nonato Costa, Gilcione ¹

¹ Departamento de Matemática - ICEX - UFMG. Cep 30123-970 - Belo Horizonte, Brazil.

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     author = {Nonato Costa, Gilcione},
     title = {Holomorphic foliations by curves on $\mathbb{P}^3$ with non-isolated singularities},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {297--321},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 15},
     number = {2},
     year = {2006},
     doi = {10.5802/afst.1123},
     zbl = {1129.32018},
     mrnumber = {2244219},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/afst.1123/}
}

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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, Série 6, Tome 15 (2006) no. 2, pp. 297-321. doi : 10.5802/afst.1123. https://www.numdam.org/articles/10.5802/afst.1123/

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