Let $X$ be an algebraic variety and $X[n]$ be the Fulton-MacPherson compactification of the configuration space $F(X,n)$ introduced in the paper "A compactification of configuration spaces". 


In this paper the authors give an explicit construction of the space $X[n]$ by a sequence of blow-ups, which is inductive. They assume that the space $X[n]$ is already constructed. They describe $X[n+1]$ as an explicit sequence of blow-ups of $X[n] \times X$.


As they mention in their paper their construction is not symmetric. For example, when $n=4$, if one starts by blowing up the small diagonal in $X^4$ and then blow-up proper transforms of the next larger diagonals, then the proper transform of succeeding diagonals will not be separated, so extra blow-ups are needed to get a smooth compactification. 


I am interested in the case where $X$ is a smooth curve. I was wondering if there is a construction of the space $X[n]$ as an explicit and symmetric sequence of blow-ups of $X^n$. 
I want to get a smooth space at each stage of the construction. 


Question: Is there any such sequence?



Symmetric sequence of blow-ups for the Fulton-MacPherson compactification
The interesting question of to what extent "wonderful compactifications" like the Fulton-MacPherson space depend on the order of blowups was studied -- and I think, mostly resolved -- by Li Li in his thesis. The paper <http://arxiv.org/abs/math/0611412> gives "a condition on the order of blow-ups in the construction....such that each blow-up is along a nonsingular center."


