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 P 2 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 deg Z 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).