There are various frameworks around which enlarge the category of rings to include more exotic objects such as the 'field with one element,' $\mathbb{F}_1$.  While these frameworks differ in their details, there are certain things this should be true of any object that deserves to be called $\mathbb{F}_1$.  For example, The algebraic K-theory of $\mathbb{F}_1$ should be sphere spectrum, and the theory of toric varieties should be defined over $\mathbb{F}_1$.

**Question 1:** Is there a moral reason why the moduli space of curves should (or should not) be defined over Spec $\mathbb{F}_1$?

EDIT:  For anyone who would like to be more concrete, I'm happy to  take the Toen-Vaquie definition of schemes over $\mathbb{F}_1$.  (see [arXiv:math/0509684][1]).  In this setup (and most of the other frameworks I know) an affine scheme over $\mathbb{F}_1$ is just a commutative monoid $M$.  After base change to $\mathbb{Z}$ this becomes the monoid ring $\mathbb{Z}[M]$.  So here is a more precise question:


**Question 2:** Does the moduli space of curves $\mathcal{M}_{g,n}$ (over $\mathbb{Z}$, say) admit a covering by affine charts of the form spec $\mathbb{Z}[M_i]$ for commutative monoids $M_i$?  If so, can this covering be chosen so that (as in the case of toric varieties) the gluing is entirely determined by maps of monoids?

  [1]: http://arxiv.org/abs/math/0509684v4