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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). 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?

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). 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?

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 $\mathbb{F}_1$.

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).

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). 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?

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 $\mathbb{F}_1$.

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

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 $\mathbb{F}_1$.

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).