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?