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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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  • $\begingroup$ what is the reference for Kapranov's result ? $\endgroup$
    – Alexander Chervov
    Commented Oct 13, 2012 at 12:50
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    $\begingroup$ The paper is called "chow quotients of grassmannians 1". $\endgroup$
    – Dan Petersen
    Commented Oct 13, 2012 at 17:37
  • $\begingroup$ @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. $\endgroup$
    – IMeasy
    Commented Oct 14, 2012 at 18:27

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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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  • $\begingroup$ @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? $\endgroup$
    – IMeasy
    Commented Oct 14, 2012 at 18:28

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