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Statement: the group of complex automorphisms of the moduli space $M_{0,n}$ of complex $n$-marked genus 0 curves is isomorphic to $\mathfrak S_n$: one has ${\rm Aut}(M_{0,n})=\mathfrak S_n$

I believe that this is true for $n\geq 5$ (what about the cases $n=3,4$?) and has been obtained in the 50's by peoples working in Teichmuller theory.

Questions:

  1. Is this statement correct ?
  2. If it is, who has to be credited for it?

Classical/canonical references as well as recent but very useful/relevant references are welcome !

Thanks in advance


Remark: if $\overline{M}_{0,n}$ stands for Deligne-Mumford-Knudsen moduli space of stable $n$-marked genus 0 curves, that ${\rm Aut}(\overline{M}_{0,n})=\mathfrak S_n$ has been proved only recently (in [A. Bruno, M. Mella]: The automorphism group of $\overline{M}_{0,n}$, J. Eur. Math. Soc., Volume 15 (2013), pp. 949-968).

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    $\begingroup$ $M_{0,3}$ is a point, so its automorphism group is trivial. $M_{0,4}$ is a projective line minus $3$ points, so its automorphism group is $S_3$. $\endgroup$
    – Will Sawin
    Feb 17, 2022 at 17:32

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