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Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, or a similar result, hold over other fields? e.g. positive characteristic, non algebraically closed, etc.

ps I am afraid one could only dream of this, over non alg closed fields...

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what is the reference for Kapranov's result ? –  Alexander Chervov Oct 13 '12 at 12:50
    
The paper is called "chow quotients of grassmannians 1". –  Dan Petersen Oct 13 '12 at 17:37
    
@Alexander: there are also other realizations as a blow up, see Keel, Intersection theory of moduli spaces of stable n-pointed curves of genus 0. –  IMeasy Oct 14 '12 at 18:27
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up vote 4 down vote accepted

It seems to me Kapranov's methods are purely algebraic and that his description works verbatim over $\mathrm{Spec}(\mathbf Z)$.

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@Dan: thank you for answering, I have checked the paper and in fact it does seem that you are right. So probably he works over $\mathbb{C}$ just for sake of simplicity? –  IMeasy Oct 14 '12 at 18:28
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