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May 8 at 17:23 comment added Will Sawin 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.
May 8 at 16:51 history asked adrian CC BY-SA 4.0