# Maps between moduli of curves

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.

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• 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