Special subschemes of the scheme of singularities of a plane foliation

Abstract: From the fact that a foliation by curves of degree greater than one, with isolated singularities, in the complex projective plane 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. To cite this article: A. Campillo, J. Olivares, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]