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?"



Configuration space of flags
There are many GIT quotients, since to define one requires a choice of $G$-line bundle, so a pair of naturals for each $F$. 


There's an obvious democratic choice -- $(a,b) = (1,1)$ for every $F$ -- but I think this will in general lead to properly semistable reduction, which is a little icky. (Instead of just taking the quotient of an open set, further identifications are required.)


I will be lazy and put all the burden on the first three: $(2n,2n), (4n-1,3n), (n,1)$, and $(1,1)$ for each of the rest. I think the effect on the stable set is the following. The first two flags must be in general position w.r.t. one another, so we can use $PGL(3)$ to move them to the base and anti-base flag, respectively. This leaves over the diagonal matrices $T$ acting on $F^{n-2}$. The line in the third flag shouldn't be in the $xy$, $xz$, or $yz$ coordinate planes. That defines an open set in $F$ whose $T$-quotient turns out to be ${\mathbb P}^1$. So the final quotient space is ${\mathbb P}^1 \times F^{n-3}$.


For more about these undemocratic GIT quotients, see the purple book [Guillemin-Lerman-Sternberg].


