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](http://arxiv.org/abs/math/0509684v4)). 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?



Is the moduli space of curves defined over the field with one element?
This "answer" will basically restate the comments of Marty and Jason Starr. 


Any variety covered by schemes of the form $\mathrm{Spec}(\mathbf Z[M\_i])$, or any torified variety, is rational. And indeed Severi conjectured at one point that $M\_g$ is rational for any $g$! But we know a lot about the Kodaira dimension of $M\_g$ by work of Harris--Mumford, Farkas, Eisenbud, Verra, ... in particular we know that Severi's conjecture is maximally false. We have $\kappa(M\_g) = -\infty$ for $g \leq 16$, we don't know anything for $17 \leq g \leq 21$, and $\kappa(M\_g) \geq 0$ for $g \geq 22$. In fact we know that $M\_{g}$ is of general type for $g = 22$ and $g \geq 24$. But if one only wants an example of a $g$ for which $M\_g$ is *not rational*, then I think one can find examples much earlier (maybe $g \approx 6,7$?).


See <http://arxiv.org/abs/0810.0702> for a survey by Farkas.


Moreover, the cohomology of a torified scheme can only contain mixed Tate motives. In particular, this implies properties like: the number of $\mathbf F\_q$-points is a polynomial function of $q$. As Marty says there are no problems in genus zero; these might be honest-to-God $\mathbf F\_1$ schemes. The cohomology of $M\_{1,n}$ contains only mixed Tate motives when $n \leq 10$, but for $n \geq 11$ one finds motives associated to cusp forms for $\mathrm{SL}(2,\mathbf Z)$. (The number 11 arises as one less than the smallest weight of a nonzero cusp form; the discriminant form $\Delta$.) This implies that the polynomiality behaviour changes drastically -- the number of $\mathbf F\_q$-points is now given by Fourier coefficients of modular forms, which are far more complicated and contain lots of arithmetic information. 


The connection with birational geometry is also very visible here. The cohomology classes on $M\_{1,n}$ associated to the cusp forms of weight $n+1$ are of type $(n,0)$ and $(0,n)$ in the Hodge realization, since given such a cusp form one can explicitly write down a corresponding differential form in coordinates. So by the very definition of Kodaira dimension we can not have $\kappa = -\infty$ anymore.


Bergström computed the Euler characteristic of $M\_{2,n}$ in the Grothendieck group of $\ell$-adic Galois representations by point counting techniques. That is, he found formulas for the number of $\mathbf F\_q$-points of $M\_{2,n}$ by working really explicitly with normal forms for hyperelliptic curves. These formulas turned out to be polynomial in $q$ for $n \leq 7$ (and conjecturally for $n \leq 9$), just as they would be if we knew that the cohomology of $M\_{2,n}$ contained only mixed Tate motives. By thinking a bit about the stratification of topological type, one concludes that this holds also for $\overline M\_{2,n}$ when $n \leq 7$, which is smooth and proper over the integers. Then a theorem of van den Bogaart and Edixhoven implies that the cohomology of $\overline M\_{2,n}$ is all of Tate type and with Betti numbers given by the coefficients of the polynomials. 


There are similar results by Bergström for $M\_{3,n}$ when $n \leq 5$ and Bergström--Tommasi for $M\_4$. But the general phenomenon is that increasing either $g$ or $n$ will rapidly take you out of the world of $\mathbf F\_1$-schemes, at least if this is taken to mean "commutative monoids". 


However, I don't know enough $\mathbf F\_1$--geometry to say what the answer is if one takes $\mathbf F\_1$-schemes in the sense of Borger. The first nontrivial case to answer would be: are the motives associated to cusp forms for $\mathrm{SL}(2,\mathbf Z)$ defined over $\mathbf F\_1$ in his set-up? To be clear, I don't believe that this is true, but I know almost nothing about $\lambda$-schemes.


Let me also make two small remarks: (i) Everything I have written above is *necessary* conditions for being defined over $\mathbf F\_1$. (ii) The fact that $M\_{g,n}$ is smooth over the integers says that the cohomology of $M\_{g,n}$ is at least quite "special", even though it is not defined over $\mathbf F\_1$. Smoothness is a strong restriction on the Galois representations that can occur in the cohomology.


