Fix a prime $p$, and let $\overline{M}$ be the Deligne-Mumford compactification of the moduli stack $M$ of elliptic curves (which, concretely, is the open substack of the stack of cubic curves which are smooth or have a nodal singularity) over $W(\overline{\mathbf{F}}_p)$. Is there a smooth projective scheme $X/W(\overline{\mathbf{F}}_p)$ with positive Kodaira dimension that admits a smooth map $f:X\to \overline{M}$? I had originally asked this question for the open substack of ordinary curves inside $M$, but, as follows from Jason Starr's comments below, the answer is relatively uninteresting (and, in particular, doesn't need the condition on the positivity of Kodaira dimension). I don't have much intuition for the smooth site of $M$ (hence of $\overline{M}$), so any insight into this would be much appreciated.

  • $\begingroup$ The image of a smooth morphism is open. For a $W$-morphism of finite type, separated $W$-schemes, if the domain is proper over $W$, then the image is also closed. The moduli stack of elliptic curves (with whatever level structure, marked points, etc. you like) is irreducible. Thus, the image of $f$ in $M$ (or $\overline{M}$) would be all of $M$ (or $\overline{M}$), contradicting that the image is contained in the proper subset $M^{\text{ord}}$. $\endgroup$ – Jason Starr Jun 30 '18 at 7:34
  • $\begingroup$ @JasonStarr That's a nice argument, thanks. In particular, this says that every smooth map from a smooth projective scheme to $M$ is surjective. If, instead of the target being $M^\mathrm{ord}$, I only asked for a smooth map $f:X\to M$, can one still exclude the existence of such an $X$? $\endgroup$ – skd Jun 30 '18 at 7:42
  • $\begingroup$ If you have a smooth morphism $f:X\to M$ with $X$ projective, then you can post-compose with the open immersion $M\hookrightarrow \overline{M}$. So you get the same conclusion for $X\to \overline{M}$, i.e., the image is all of $\overline{M}$. $\endgroup$ – Jason Starr Jun 30 '18 at 7:44
  • $\begingroup$ Sorry; I meant to ask the question in my comment above for a map $f:X\to \overline{M}$. $\endgroup$ – skd Jun 30 '18 at 7:45
  • $\begingroup$ There are no projective schemes mapping non constantly to the moduli stack of elliptic curves, because the latter has an affine coarse space $\endgroup$ – Ja ok Jun 30 '18 at 12:32

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