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

Abstract.

Let ${\mathcal{F}}_1 = {\mathbb{P}}H^{0}({\mathbb{CP}}^{2}, {\mathcal{T}} {\mathbb{CP}}^{2})$ be the space of foliations of ${\mathbb{CP}}^{2}$ of degree 1, i.e., the projective space of vector fields of ${\mathbb{CP}}^{2}$. We consider the linear action $PGL(3,{\mathbb{C}})\times {\mathcal{F}}_{1}\rightarrow {\mathcal{F}}_{1},(g, X) \mapsto gX = DgX \circ (g^{-1})$. We prove that the Good Quotient of the semi-stable points for this action is $\mathbb{CP}^{1}$.