# Symmetric sequence of blow-ups for the Fulton-MacPherson compactification

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?

Another thing worth knowing about $C[n]$, the Fulton-MacPherson compactification for a curve $C$ with $g>1$, is that it is given by the fiber of the forgetful map $\pi : \overline{M}_{g,n}\to \overline{M}_g$ over the point in $\overline{M}_g$ corresponding to $C$.
Since the usual construction of $\overline{M}_{g,n}$ is symmetric in $n$, this gives a symmetric construction of $C[n]$ (although not by blowups).