The Good Quotient of the Semi-Stable Foliations of $${\mathbb{CP}}^2$$ of Degree 1

Abstract

Let F1 = PH 0 (CP 2 , T CP 2 ) be the space of foliations of CP 2 of degree 1, i.e., the projective space of vector fields of CP 2 . We consider the linear action P GL(3, C) × F1 → F1, (g, X) → gX = DgX • (g −1 ). We prove that the Good Quotient of the semi-stable points for this action is CP 1 .

Invariant theory of foliations of the projective plane

“…We will deal only with one-dimensional singular foliations of P 2 k from now on, and will thus drop the adjective "singular." Taking duals in the Euler sequence (1), we obtain the exact sequence…”

“…It was using this criterion that Goméz-Mont and Kempf [8] have shown that a foliation whose all singular points have Milnor number 1 is stable, that is, corresponds to a stable point of F m . And Alcántara [1], [2] has characterized the semi-stable foliations of degrees 1 and 2.…”

“…(In fact, they showed the same result holds for singular foliations of higher dimension spaces as well.) Only recently, Alcántara [1], [2] has characterized the semi-stable foliations of degree 1 and 2, and studied their quotient spaces.…”