Let $M_{g,n}$ be the moduli space of $n$-pointed curves, and $M_g[m]$ the moduli space of (unpointed) curves with $m$-level structure.

Fix $m>0$. Is it true that for $n$ large enough, there is a dominant rational map $M_{g,n}\to M_g[m]$?

I am not aware of the existence of a "natural" (i.e. functorial) map $M_{g,n}\to M_g[m]$. For my purposes, any dominant rational map would work, but I understand that it is probably difficult to construct a "non-canonical" one.

  • 2
    That's an odd question, where does it come from? Here is an informal argument why there should be no "natural" map that one can write down just from the moduli functor: since there is no "natural" way to assign a curve $C′\neq C$ to a curve $C$ (the stack $M_g$ has no automorphisms), any "natural" map $M_{g,n}\to M_g[m]$ should be compatible with the projections down to $M_g$. But this is impossible, since $M_g[m]\to M_g$ is a covering space and $M_{g,n} \to M_g$ is surjective on fundamental groups. – Dan Petersen Oct 10 at 19:59

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