Algebraic Geometry/Ordinary Differential Equations
Special subschemes of the scheme of singularities of a plane foliation
Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 581-585.

From the fact that a foliation by curves of degree greater than one, with isolated singularities, in the complex projective plane P2 is uniquely determined by its subscheme of singular points (the singular subscheme of the foliation), we pose the problem of existence of proper closed subschemes Z of the singular subscheme which still determine the foliation in a unique way. We prove the existence of such subschemes Z for foliations with reduced singular subscheme. Unlike the degree degZ of such subschemes is not sharp for the posed problem, we show that it is so in the sense that Z defines the so-called polar net of the foliation.

Du fait qu'un feuilletage en courbes de degré strictement supérieur à un, avec singularités isolées, dans le plan projectif complexe P2 est uniquement déterminé par son sous-schéma des points singuliers (le sous-schéma singulier du feuilletage), nous posons le problème de l'existence de sous-schémas fermés propres Z du sous-schéma singulier qui déterminent encore le feuilletage d'une manière unique. Nous démontrons l'existence d'un sous-schéma Z pour les feuilletages avec un sous-schéma singulier réduit. Si le degré degZ de tels sous-schémas n'est pas optimal pour le problème posé, nous montrons qu'il en est ainsi dans le sens où Z définit ce qu'on appelle le réseau polaire du feuilletage.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.03.018
Campillo, Antonio 1; Olivares, Jorge 2

1 Departmento de Álgebra, Geometría y Topología, Universidad de Valladolid, 47005 Valladolid, Spain
2 Centro de Investigación en Matemáticas, A.C. A.P. 402, Guanajuato 36000, Mexico
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Campillo, Antonio; Olivares, Jorge. Special subschemes of the scheme of singularities of a plane foliation. Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 581-585. doi : 10.1016/j.crma.2007.03.018. http://www.numdam.org/articles/10.1016/j.crma.2007.03.018/

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