Let $U\subset \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be the Zariksi open set of ordered quadruple of distinct points in the projective line. The quotient of $U$ by the projective transformation group $PSL(2)$ can be identified to $\mathbb{P}^1$ by cross-ratio.

Motivated by a paper of Fock and Goncharov (arXiv:math/0311149), I want to understand the configuration space of flags.

**Question:** Let $\mathscr{F}$ be the variety of flags in $\mathbb{P}^2$.Let $n\geq 3$. Is there a Zariski open set $U\subset\mathscr{F}^{(n)}$ like above, consisting of $n$ flags "in general position" in some sense, such that the quotient of $PSL(3)$-action on $U$ is a projective variety? And what is the quotient?

For those who know a little geometric invariant theory, I could have just asked "How to describle (semi-)stable points of the $PSL(3)$-action on $\mathscr{F}^{(n)}$, and what is the quotient?"