Invariant theory of foliations of the projective plane

TL;DR

This paper investigates the invariant properties of singular foliations on the projective plane, establishing stability criteria based on singularity dimensions and multiplicities, and constructs an explicit invariant map relating foliations to algebraic curves.

Contribution

It provides new stability conditions for foliations and explicitly constructs an invariant map to algebraic curves, advancing the understanding of foliation invariants.

Findings

Foliations of degree m>1 are unstable if they have 1-dimensional singularities or a singular point with high multiplicity.

An explicit invariant map from foliations to algebraic curves of degree m^2+m-2 is constructed.

The map is described explicitly for the case m=2.

Abstract

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.