On the Quotient of Projective Frame Space and the Desargues Theorem

We consider an $n$-dimensional projective space $\mathbb{P}_n$ ($n\geq2$) and a fixed point $A$ on it. Let $F(\mathbb{P}_n)$ be the manifold of all the projective frames of $\mathbb{P}_n$ having $A$ as their first vertice. We define the action of stabilizer G of $A$ in the projective group $GP(n)$ in a natural way. The Lie group epimorphism $\beta\colon G\to GL(V)$ acts as follows $g\mapsto d_A g$ where $V=T_A \mathbb{P}_n$. We study the geometry of orbit space $\Phi(\mathbb{P}_n)$ of the space $F(\mathbb{P}_n)$ under the action of the kernel $H= ker\beta$ of the epimorphism $\beta$. By applying some $n$-dimensional version of the Desargues theorem we could get a purely geometrical description of such $H$-orbits