Let $SL\_{n+1}$ act on $\mathbb{P}^n$ in the natural way. Suppose I take two linear subspaces $\mathbb{P}^m$ and $\mathbb{P}^{n-m}$, with $m < n$, that intersect in one point. Is the action of $SL\_{n+1}$ transitive on the set of such couples of linear subspaces?



Action of $SL_{n+1}$ on couples of linear spaces in $\mathbb{P}^n$.
Sure it does. Denote your variety by $X$. Consider this action on the corresponding vector space $V$. Then your projective subspaces correspond to linear subspaces $V\_1$ and $V\_2$ of dimension $m+1$ and $n-m+1$ that intersect transversely. There is an obvious projection from the variety $Y$ of all frames in $V$ to $X$ (take the linear hull of the first $m+1$ and the last $n-m+1$ vectors). Now, up to a multiplication of all the vectors in the frame by a common scalar, $SL\_{n+1}$ acts transitively on $Y$. Thus, it acts transitively on $X$.


