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...