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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?

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3 Answers

up vote 4 down vote accepted

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."

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Thanks very much for your answer. –  Passenger Nov 29 '10 at 17:21
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There is a slightly bigger compactification of the configuration space constructed by Ulyanov in http://arxiv.org/pdf/math/9904049v2. It dominates the Fulton-MacPherson compactification and it is again constructed inductively. It has the advantage of the blow-ups being symmetric on each stage. There is also a symmetric construction of the Fulton-MacPherson compactification that was pointed out by Dylan Thurston. You can find a brief description of a real version of this construction in section 3 of http://arxiv.org/pdf/math/9901110v2.

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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).

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