MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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 gives "a condition on the order of blow-ups in the construction....such that each blow-up is along a nonsingular center."

share|cite|improve this answer
Thanks very much for your answer. – Passenger Nov 29 '10 at 17:21

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).

share|cite|improve this answer

There is a slightly bigger compactification of the configuration space constructed by Ulyanov in 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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.