I'm reading on $\overline M_{0, n}$ and $\widetilde M_{0, n}$. I know that $\overline M_{0, n}$ is a Mori Dream space for $n \leq 6$ and not a Mori Dream space for $n \geq 13$. Is there a similar result about $\widetilde M_{0, n}$ ? I thought there should be some obvious statement of the type "GIT quotients of Mori Dream spaces are themselves Mori Dream spaces" but somehow I'm missing something. I would also like to understand a counterexample to the previous statement.

nota Mori dream space for $n\geq 13$. However, it remains possible that the quotient $\widetilde{M}_{0,n}$ is a Mori dream space. $\endgroup$3more comments