Invariant theory of foliations of the projective plane

We study the invariant theory of singular foliations of the projective plane. Our first main result is that a foliation of degree m>1 is not stable only if it has singularities in dimension 1 or contains an isolated singular point with multiplicity at least (m^2-1)/(2m+1). Our second main result is the construction of an invariant map from the space of foliations of degree m to that of curves of degree m^2+m-2. We describe this map explicitly in case m=2.