It is well known that the set of all polygons with consecutive side lengths $l_1, \dots, l_n$ in $\mathbb{R}^3$, considered up to rigid motions, is a compact complex manifold. Of course, I am assuming, for the sake of simplicity that there are no "straight line" polygons i.e., the length vector $L:= (l_1,\dots, l_n)$ is generic. I will denote this space by $M_L$.

Consider configurations of $n$ not necessarily distinct points $p_i\in \mathbb{P}^1$. Each point $p_i$ is assigned the weight $l_i$. A configuration is called stable if the sum of weights of equal points is less than the total weight $\sum l_j$. The moduli space of stable points on $\mathbb{P}^1$ is defined as the stable configurations modulo projective automorphisms.

It was shown by Kapovich and Millson (and also by Klyachko) that $M_L$ is isomorphic to the moduli space of stable points on $\mathbb{P}^1$.

Before I ask my questions let me recall the definition of one more moduli space. The moduli space of $n$ distinct labeled points on $\mathbb{P}^1$ is obtained from $(\mathbb{P}^1)^n$ by removing the diagonals $z_i = z_j$ for $i\neq j$ and factoring out by the $PSL(2, \mathbb{C})$ action. This space, denoted $M_{0,n}$, has a smooth compactification denoted $M_n$. The way I understand the construction of $M_n$ is as follows: first realize that $M_{0,n}$ is the complement of the braid arrangement in $\mathbb{P}^{n-1}$. Then one reaches $M_n$ by successively blowing along certain intersections.

Now my questions:

- For any (generic) choice of length vector one can observe that $M_{0, n} \subset M_L$ as the set of all those polygons for which no to directions are same (i.e., stable configurations where are all points are distinct). In view of this, can one say that $M_L$ is a smooth compactification of $M_{0,n}$?
- Is there some nice map from $M_n$ to $M_L$ (or in the other direction) which is an isomorphism on $M_{0,n}$? I have read the following description of $M_n$ in several papers: '' when two points in $M_{0,n}$ come close too each other instead of colliding a new $\mathbb{P}^1$ pops out and these points move on it". I don't understand what this means mathematically but this pictures suggests a map. Given a length vector $L$ the essential information is the collection of all so-called short subsets. I guess this data should help us identify which "popped out" $\mathbb{P}^1$s to contract and obtain a stable configuration (or the other way round?). I know what I have written here is pretty vague but I have a strong feeling that somebody must have thought about this earlier.
- The complement $M_L\setminus M_{0,n}$ is certainly not a normal crossing divisor. So, if one were to blow up along the subspaces of colliding points what would one get? Does this even make sense?

Motivation: For the past few months I have been working on some problems related to the moduli space of planar polygons. One day I realized that the planar polygons modulo $O(2)$ contain $M_{0, n}(\mathbb{R})$ as an open subset (which is just a disjoint union of $(n-1)!/2$ open discs). This observation lead me to above questions. I should also confess that I have no expertise in algebraic geometry. I am completely new to the world of moduli space of stable points and labeled points.