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The moduli spaceM  ¯    0,  n(    ℙ  1   ,  1   )${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$   of  n -pointed stable maps is a Mori dream space whenever the moduli spaceM  ¯    0  ,  n  +  3       of     (  n  +  3  )${{\bar{M}}_{0,n+3}}\; \text{of} \;(n+3)$   pointed rational curves is, andM  ¯    0  ,  n(    ℙ  1   ,  1   )${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$   is a log Fano variety for  n  ≤ 5.

When is M 0, n (ℙ 1 ,1) a Mori dream space?