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Let $X$ be a (smooth complex projective) variety.

The Fulton-MacPherson compactification $X[n]$ is obtained from $X^n$ by blowing up the diagonals in a certain order. Is it possible to write down a (nice intelligible) graded ${\mathscr O}_{X^n}$-algebra ${\mathscr A}$ such that $$ X[n] = \mathrm{Proj}_{X^n}{\mathscr A}? $$ More generally, the same question can be asked for wonderful compactifications ( arXiv:math/0611412 ), since $X[n]$ is an example of such a thing.

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    $\begingroup$ Rather than the description via a sequence of blowups, it might be better to use the description as the closure of the configuration space in the product of blowups of all diagonals. Each blowup of a diagonal can be expressed as the Proj of an explicit graded ring. The fiber product of the blowups can be expressed as Proj of a certain tensor product of these rings. Then you have to find the right ideal to mod out by - I think it may just be the ideal of zero divisors with $\mathcal O_X$, which should be calculable. $\endgroup$
    – Will Sawin
    Commented May 8 at 17:23

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