About the stability between a foliation of degree two and the pencil of conics that de?nes it

Abstract

In this paper, we study foliations on the projective plane of degree two which have a ?rst integral with degree two. Such ?rst integrals de?ne a pencil of conics.
The Hilbert-Mumford criterion is a powerful tool of the Geometric Invariant Theory. An application of this theory is the characterizarion of the instability of the space of foliations of degree two, with respect to the action by a change of coordinates, and the characterization of the stability of pencils of conics, given by Alcántara.
The aim of the paper is to give another proof of the fact that a foliation of degree two de?ned by a pencil of conics is unstable if, and only if, the pencil is unstable.