Let $\overline{M}\_{0,n}$ be the usual Deligne-Mumford compactification of $M\_{0,n}$ the moduli space of smooth $n$-pointed rational curves. 


The canonical divisor $K\_{\overline{M}\_{0,n}}$ can be written as a combination of the irreducible components of the boundary of $\overline{M}\_{0,n}$. 


If $f:\overline{M}\_{0,n}\rightarrow\overline{M}\_{0,n}$ is an automorphisms then $f^{\*}K\_{\overline{M}\_{0,n}} = K\_{\overline{M}\_{0,n}}$. 


If $n\geq 5$, is this enough to conclude that $f$ has to map the boundary $\partial\overline{M}\_{0,n}$ to itself?



Canonical bundle of moduli space of rational curves and automorphisms
If $n\geq 5$ then $Aut(\overline{M}\_{0,n})\cong S\_n$ (<http://arxiv.org/abs/1006.0987>). For instance $\overline{M}\_{0,5}$ is a Del Pezzo surface of degree five. Its automorphism group is well known to be $S\_5$ (<http://arxiv.org/abs/math/0610595>).


In particular, any automorphism of $\overline{M}\_{0,5}$ preserves the boundary. 


