Foliations on ℂℙ2 of degree 2 with degenerate singularities

Abstract

In this paper we construct non-singular, locally-closed, algebraic varieties which are sets of foliations on ℂℙ2 of degree 2 with a certain degenerate singularity. We obtain the dimension and closure of these varieties. To do that we construct a stratification (based on GIT, see [7]) of the space of foliations with respect to the action by change of coordinates. We prove that the set of unstable foliations has two irreducible components. We have the following corollary: a foliation of degree 2 defined by a pencil of conics is unstable if and only if the pencil is unstable. Finallywe give another proof of the fact that there are only 4 foliations of degree 2 with a unique singular point (see [5]).