# Fulton-MacPherson spaces

Let $X$ be a smooth projective variety.Fulton and MacPherson define $X[n]$ to be the variety obtained by successively blowing up $X^n$ along nonsingular subvarieties $\Delta_I$ for all subsets $I\subset \{1,2,\dots n\}$ s.t. $|I|\geq 2$ where $\Delta_I$ are the proper transforms of the diagonals $\{(x_1,x_2,\dots x_n)\in X^n | x_i=x_j\}$ for all $i,j \in I$.

Now consider a sequence of blowups $f_n:X[n]^+\rightarrow Y_{n-2} \rightarrow\dots Y_0:=X[n]\times X$ ,where each $Y_{i+1}$ is the blowup of $Y_{i}$ along $\Delta _{I^{+}}$ where $I^{+}=I\cup \{n+1\}$ , $I\subset \{1,2,\dots n\}$ and $|I|=n-i$.

I would like to understand the following description on the fibers of the morphism $f_n$ (referred to in the original FM paper as the stable degenerations):

$\textbf Q1:$Any fiber should have a distinct component which is a blowup of $X$ at some points and then the rest of the components are blowups of $\mathbb{P}^n$ at a finite number of points,ending with components which are equal to $\mathbb{P}^n$ .How does one prove this by chasing through $f_n$ and what are the centers of the blowups of $\mathbb{P}^n$ above?

Also,the above gives a ''treelike'' description of the fibers . Namely we associate vertices with the components and edges with their intersections(components of the singular locus) . The distinct component above is the root of the tree. Cut the tree at an arbitrary vertex . That is , take an arbitrary vertex and assume you get rid of all of its descendants and the edges linking them.

$\textbf Q2:$ Is it true that you can get the new tree as the fiber of some map $B\rightarrow X[n]$, where $B$ is the blowup of $X[n]\times X$ along some of the $\Delta _{I^{+}}$?. Also , is there a universal family(possibly constructed as a blowup) whose fibers describe the pruned tree(the one we got rid off) ?

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