  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
